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/ TEAL /

Flash MIT TEAL 3D

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: Vector Fields

A Particle Sink

The animation shows a sink of particles. The particles appear on the rim of the cone, and then move down the cone under the influence of gravity. They disappear out the bottom of the cone. The vector velocities of the particles as seen from above are all directed toward the center of the cone. A Particle Source

The animation shows a source of particles. The particles appear at the center of the cone, and then move down the cone under the influence of gravity. The vector velocities of the particles as seen from above are all directed outwards from the center of the cone.

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A Circulating Flow of Particles

The animation shows a set of particles which have neither a sink nor a source (except at the beginning of the animation). Once created, the particles circulate about the center of the cone at different radii, and are neither created nor destroyed. The vector velocities of the particles as seen from above are directed counterclockwise about the center of the cone. A Fluid Flow with a Source

The animation shows a flow field using animated textures. The direction of the velocity field is indicated by the correlation in the textures. When animated, the texture patterns move in the direction of the velocity field. This flow field has a source at the origin, and the texture patterns move away from that point.

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A Fluid Flow with a Source and a Sink

The animation shows a flow field using animated textures. The direction of the velocity field is indicated by the correlation in the textures. When animated, the texture patterns move in the direction of the velocity field. This flow field has a both a source and a sink. Near the source or sink, the texture patterns move away from the source or towards the sink. A Fluid Flow with Two Sources

The animation shows a flow field using animated textures. The direction of the velocity field is indicated by the correlation in the textures. When animated, the texture patterns move in the direction of the velocity field. This flow field has two sources located at different points and with different strengths. Near the either source, the texture patterns move away from that source.

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A Fluid Flow with a Circulation

The animation shows a flow field using animated textures. The direction of the velocity field is indicated by the correlation in the textures. When animated, the texture patterns move in the direction of the velocity field. This flow field is a pure circulation. There is no source here, the fluid simply moves in circles with no destruction or creation of fluid particles.

A Fluid Flow with Two Circulations in the Same Sense

The animation shows a flow field using animated textures. The direction of the velocity field is indicated by the correlation in the textures. When animated, the texture patterns move in the direction of the velocity field. This flow field has two different circulations located at different points and with different strengths. The sense of the circulation is the same for both, however.

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A Fluid Flow with Two Circulations in Opposite Senses

The animation shows a flow field using animated textures. The direction of the velocity field is indicated by the correlation in the textures. When animated, the texture patterns move in the direction of the velocity field. This flow field has two different circulations located at different points and with different strengths. The sense of the circulation is counterclockwise for one and clockwise for the other.

A Fluid Flow with a Circulations and a Source

The animation shows a flow field using animated textures. The direction of the velocity field is indicated by the correlation in the textures. When animated, the texture patterns move in the direction of the velocity field. This flow field has both a circulation and a source.

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A Source and a Constant Flow

The animation shows a flow field using animated textures. The direction of the velocity field is indicated by the correlation in the textures. When animated, the texture patterns move in the direction of the velocity field. This flow field has a source sitting in a constant downward flow. A Circulation and a Constant Flow

The animation shows a flow field using animated textures. The direction of the velocity field is indicated by the correlation in the textures. When animated, the texture patterns move in the direction of the velocity field. This flow field has a circulation sitting in a constant downward flow.

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The Dot Product of Two Vectors

This interactive animation illustrates the concept of the dot product of two vectors. The red vector is "dotted" in to the green vector, and the scalar result is represented by the length of the red highlight along the green vector. The arc traced out by the red vector represents the angle between the two vectors, and the transparent green vector represents the negative of the solid green vector. The length and direction of the red vector can be changed by using the arrow keys. The Cross Product of Two Vectors

0 360

This interactive animation illustrates the concept of the cross product of two vectors. By definition, the cross product of two vectors is a mutually

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perpendicular vector whose direction is given by the "Right Hand Rule": when you point the fingers of your open hand in the direction of the first vector (green), and then curl them in the direction of the second vector (red) by way of the smallest angle between them, your thumb points in the direction of the cross product of those two vectors (orange). As seen in the animation, the hand points itself in the proper direction according to this rule as you rotate the red vector through an angle theta. Note that though the angle theta goes from zero to 360 degrees, the angle used in the Right Hand Rule is always the smallest angle between the two vectors. Coordinate Systems

C

xyz rz r

This interactive visualization illustrates the different types of coordinate systems often used in studying electromagnetism: Cartesian, cylindrical (polar), and spherical. Pressing "C" cycles between them. Each system has a distinct set of principle axes, represented by the three surfaces. For Cartesian, x, y, and z. For cylindrical, r, theta, and z. For spherical, r, theta, and phi. By using the arrow keys (and control + up/down arrow keys), you can move the observation point in the three different principle directions for the current coordinate system displayed. The small arrows on the observation point display the unit vectors for each system.

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Line Integrals

. This applet illustrates the concept of line integrals. By selecting a line type and manipulating the source charge, you can see the effect on the line integral of dot product of the field and line element for the various configurations. The blue arrows represent the direction of the line elements and the red arrows represent the electric field at that line element. The line types are circular, rectangular, and random. Surface Integrals Shockwave

This Shockwave visualization illustrates the concept of surface integrals. By selecting a surface type and manipulating the source charge, you can see the effect on the surface integral of the dot product of the field and the surface element for the various configurations. The red arrows represent the electric field at that surface element. The surface types are icosahedrons, geodome, and rectangular.

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TEAL Version 1.00

Mapping Fields Applet

X Y "*" , "/" , "^"grass seedsequipotentials This applet allows you to see how various analytic functions look in a "grass seeds" representation, letting you visualize the functions more clearly. You can also explore the set of curves which are perpendicular to your analytic function (these are equipotentials if the your function corresponds to an electrostatic field). Enter the x-component of your analytic function in the left text box, using standard convections (i.e. "*" for multiplication, "/" for division, "^" for exponents). Enter the y-component of your analytic function in the right text box. Then choose either "grass seeds" or "equipotentials" depending on what you want displayed.

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: Electrostatics

van de Graaff A van de Graaff Generator Attracting a Charge Van de Graaff

An animation of the motion of a negatively charged particle attracted by the positively charged sphere of a van de Graaff generator. When the charge is moving away from the sphere and slowing down, its kinetic energy is decreasing and the energy is being stored in the local electric field as that field is stretched. When the charge is moving toward from the sphere and speeding up, its kinetic energy is increasing as the energy previously stored in the stretched field is released. van de Graaff A van de Graaff Generator Repelling a Charge

Van de Graaff

An animation of the motion of a positively charged particle repelled by the positively charged sphere of a van de Graaff generator. When the charge is

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moving toward the sphere and slowing down, its kinetic energy is decreasing and the energy is being stored in the local electric field as that field is compressed. When the charge is moving away from the sphere and speeding up, its kinetic energy is increasing as the energy previously stored in the compressed field is released. The Electric

Field of a Positive Charge

simulation of the electric field generated by a point charge. The observation

Electric Field of a Moving Positive Charge

he electric field of a moving charge is small compared to the speed

()

Apoint (black sphere) can be moved using the arrow keys to sample the field at different positions relative to the charge.

Tpositive charge when the speed of theof light.

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Electric Field of a Moving Negative Charge

he electric field of a moving charge is small compared to the speed

The Electric Field of a Dipole

n interactive simulation of the electric field of two equal and opposite charges.

Tnegative charge when the speed of theof light.

AThe "grass seeds" respresentation shows the total electric field. We can move the observation point (black sphere) around in space to see how the total field at various points arises from the individual fields of each charge.

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Integrating Along a Line of Charge

his simulation illustrates the electric field generated by a line of charge, and

Tshows how, by the principle of superposition, a continuous charge distribution can be thought of as the sum of many discrete charge elements. Each element generates its own field, described by Coulomb's Law (and represented here by the small vectors attached to the observation point), which, when added to the contribution from all the other elements, results in the total field of the line (given by the large resultant vector). In this animation, each element is being added up one by one (indicated by the highlighted portion of the line), and the total field changes accordingly. As the entire line is integrated, the components of the field contributions in the direction of the line are cancelled out, leaving a total field that is perpendicular to the ring on its perpendicular bisector. If this were truly an infinite line of charge, the total field of the line would be perpendicular to the line at all points.

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The Line of Charge

This simulation illustrates the electric field generated by a line of charge, and

if this were truly an "infinite line" of charge, the components of the

shows how, by the principle of superposition, a continuous charge distribution can be thought of as the sum of many discrete charge elements. Each element generates its own field, described by Coulomb's Law (and represented here by the small vectors attached to the observation point), which, when added to the contribution from all the other elements, results in the total field of the ring (given by the large resultant vector, and by the large two dimensional field map). By moving the observation point around with the arrow keys, changes in field magnitude and direction can be observed at different positions relative to the line. Note that field in the direction of the line would cancel each other out. The resultant field would thus be described only by the contributions perpendicular to the axis of the line.

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Integrating Around a Ring of Charge

his simulation illustrates the electric field generated by a charged ring, and

Tshows how, by the principle of superposition, a continuous charge distribution can be thought of as the sum of many discrete charge elements (in this case, thirty). Each element generates its own field, described by Coulomb's Law (and represented here by the small vectors attached to the observation point), which, when added to the contribution from all the other elements, results in the total field of the ring (given by the large resultant vector). In this animation, each element is being added up one by one (indicated by the highlighted portion of the ring), and the total field changes accordingly. As the entire ring is integrated, the components of the field contributions in the plane of the ring are cancelled out; leaving a total field that is perpendicular to the ring on its central axis.

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The Charged Ring

This simulation illustrates the electric field generated by a charged ring, and

the perpendicular component of each

shows how, by the principle of superposition, a continuous charge distribution can be thought of as the sum of many discrete charge elements (in this case, thirty). Each element generates its own field, described by Coulomb's Law (and represented here by the small vectors attached to the observation point), which, when added to the contribution from all the other elements, results in the total field of the ring (given by the large resultant vector, and by the large two dimensional field map). By moving the observation point around with the arrow keys, changes in field magnitude and direction can be observed at different positions relative to the ring. Note that along the axis of the ring, element's field is cancelled out by the corresponding element directly opposite it on the other side of the ring. The resultant field is thus described only by the contributions along that axis. At the center of the ring, where those components are zero, the resultant field is also exactly zero.

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-Changing Field

positive point charge sits at rest at t

Repulsion of Charges with Opposite Sign

wo charges hang from pendulums whose supports can be moved closer or

The Force on a Charge in a Time

A he origin in a time-changing external field. This external field is uniform in space but varies in time as indicated by the orange vector in the animation. The animation shows the dramatic inflow of energy into the neighborhood of the charge as the magnitude of the external electric field grows, with a resulting build-up of stress that transmits a downward force to the positive charge. The downward force is indicated by the white vector.

Tfurther apart by an external agent. The charges have the same sign, and

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therefore repel. An external agent tries to move the supports (from which the two positive charges are suspended) together. The force of gravity is pulling the charges down, and the force of electrostatic repulsion is pushing them apart on the radial line joining them. The behavior of the electric fields in this situation is an example of an electrostatic pressure transmitted perpendicular to the field. That pressure tries to keep the two charges apart in this situation, as the external agent controlling the pendulum supports tries to move them together Attraction of Charges with Opposite Sign

wo charges hang from pendulums whose supports can be moved closer or

Tfurther apart by an external agent. The charges have opposite signs, and therefore attract. An external agent moves the supports (from which the two charges are suspended) together. The force of gravity is pulling the charges down, and the force of electrostatic attraction is pulling them together on the radial line joining them. The behavior of the electric fields in this situation is an example of an electrostatic tension transmitted parallel to the field. That tension tries to pull the two charges together in this situation.

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Creation of an Electric Dipole

uppose we have a positive point charge sitting right on top of a negative

Selectric charge, so that the total charge exactly cancels out. Then there is no electric field anywhere in space. Now suppose we pull these two charges apart. In this animation, we ignore the effects of radiation. As the charges are pulled apart, we initially see the classic electric dipole field pattern. As the charges are moved further apart, we see the usual electric field of a point charge. We artificially terminate the field lines at a fixed distance from the charges to avoid visual confusion.

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Creating an Electric Field

An animation of the creation of an electric field by an external agent who

magnetic

L

separates charges. We start out with five negative electric charges and five positive charges, all at the same point in space. Since there is no net charge, there is no electric field. Now the agent moves one of the positive charges at constant velocity from its initial position to a distance L away along the horizontal axis. After doing that, the agent moves the second positive charge in the same manner to the position where the first positive charge sits. The agent continues on with the rest of the positive charges in the same manner, until all the positive charges are sitting a distance L from their initial position along the horizontal axis. We have color coded the "grass seeds" representation to represent the strength of the electric field. Very strong fields are white, very weak fields are black, and fields of intermediate strength are yellow. The field lines move in the direction of the energy flow of the electrofield. Over the course of the animation, the strength of the electric field grows as each positive charge is moved into place. That energy flows out from the path along which the charges move, and is being provided by the agent moving the charge against the electric field of the other charges. The work that this agent does to separate the charges against their electric attraction appears as energy in the electric field.

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Destroying an Electric Field

An animation of the destruction of an electric field. We start out with five

on of the energy flow of the electromagnetic

L

negative electric charges and five positive charges, separated by a distance L in space. An external agent moves one of the positive charges at constant velocity from its initial position to the position of the negative charges. After doing that, the agent moves the second positive charge in the same manner. The agent continues on with the rest of the positive charges, until all the positive charges are sitting on top of the negative charges. We have color coded the "grass seeds" representation to represent the strength of the electric field. Very strong fields are white, very weak fields are black, and fields of intermediate strength are yellow. The field lines move in the directifield. Over the course of the animation, the strength of the electric field decreases as each positive charge is moved into place. That energy flows into the path along which the charges move, and is being provided to the agent moving the charges at constant speed with the electric field of the other charges. The work done by this agent at the end of this process is equal in magnitude and opposite in sign to the energy originally stored in the electric field.

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Scattering of Charges with Same Sign

n animation of the motion of a positi

Scattering of Charges of Opposite Sign

n animation of the motion of a negative charge moving past a massive

A ve charge moving past a massive charge which is also positive. The smaller charge is deflected away from the larger charge because of their mutual repulsion. This repulsion is primarily due to a pressure transmitted by the electric fields surrounding the charges.

Apositive charge. The negative charge is deflected toward the positive charge because of the attraction between them. This attraction is primarily due to a tension transmitted by the electric fields surrounding the charges.

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Molecules

We show the interaction of four charges of equal mass. Two of the charges are

e first thing

Pauli

Pauli

i.e.

NaCl

positively charged and two of the charges are negatively charged, and all have the same magnitude of charge. The particles interact via the Coulomb force. We also introduce a quantum-mechanical "Pauli" force, which is always repulsive and becomes very important at small distances, but is negligible at large distances. This critical distance is about the radius of the spheres shown in the animation. This "Pauli" force is quantum mechanical in origin, and keeps the charges from collapsing into a point (i.e., it keeps a negative particle and a positive particle from sitting exactly on top of one another). Additionally, the motion of the particles is damped by a term proportional to their velocity, allowing them to "settle down" into stable (or meta-stable) states. When these charges are allowed to evolve from the initial state, ththat happens (very quickly) is that the charges pair off into dipoles. This is a

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rapid process because the Coulomb attraction between unbalanced charges is very large. This process is called "ionic binding", and is responsible for the intermolecular forces in ordinary table salt, NaCl. After the dipoles form, there is still an interaction between neighboring dipoles, but this is a much weaker interaction because the electric field of the dipoles falls off much faster than that of a single charge. This is because the net charge of the dipole is zero. Although in principle the dipole-dipole interaction can be either repulsive or

2D Interactive Molecules 2D

s 2D

Molecules 2D simulates the interaction of charged particles in a two

attractive, in practice there is a torque that rotates the dipoles so that the dipole-dipole force is attractive. This dipole-dipole attraction eventually brings the two dipoles together in a bound state. The force of attraction between two dipoles is termed a "van der Waals" force, and is responsible for the intermolecular forces that bind the molecules of some substances into a solid.

Molecule

Pauli

m

dimensional plane. The particles interact via the classical Coulomb force, as well as the repulsive quantum-mechanical Pauli force, which acts at close distances (accounting for the "collisions" between them). Additionally, the motion of the particles is damped by a term proportional to their velocity, allowing them to "settle down" into stable (or meta-stable) states.

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Moving in response to these forces, the particles will eventually end up in a

3D Interactive Molecules 3D

s 3D

olecules 3D simulates the interaction of charged particles in three

ill eventually end up in a

configuration of minimum potential, where the net force on any given particle is essentially zero. Generally, individual particles will first pair off into dipoles and then slowly combine into larger structures. Rings and straight lines are the most common configurations, but by clicking and dragging particles around, they can be coaxed into more complex meta-stable formations. Pressing "m" while the simulation is paused will generate a two-dimensional map of the potential field generated by the particles, where red represents a strong positive potential and black represents a strong negative potential. This process may take a few seconds when a large number of particles are involved.

Molecule

Pauli

w 3d

Mdimensional space. The particles interact via the classical Coulomb force, as well as the repulsive quantum-mechanical Pauli force, which acts at close distances (accounting for the "collisions" between them). The motion of the particles is also damped by a term proportional to their velocity, allowing them to "settle down" into stable (or meta-stable) states. Moving in response to these forces, the particles wconfiguration of minimum potential, where the net force on any given particle is

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essentially zero. Generally, individual particles will first pair off into dipoles and then slowly combine into larger structures. Rings and straight lines are the most common configurations, but by manipulating particles manually, they can be coaxed into more complex meta-stable formations. Additionally, pressing "w" will toggle the presence of potential well directed radically inward toward the center of the 3d space. This can be used to bring the particles closer to one another if they start to drift apart. Interactive Dipoles

eractive simulation in two dimensions of a group of electric dipoles. The ipoles are created with random positions and orientations, with all the electric

An int

ddipole vectors in the plane of the display. Although in principle the dipole-dipole interaction can be either repulsive or attractive, in practice there is a torque that rotates the dipoles so that the dipole-dipole force is attractive. In the simulation we see this behavior-that is, the dipoles orient themselves so as to attract, and then the attraction gathers them together into bound structures. The dipoles can be repositioned by clicking and dragging them with the mouse.

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The Ion Trap

al well

particles in a potential well. The particles interact via the classical Coulomb force, as well as the repulsive

ven particle is

potenti

Pauli

potential well

s

The Ion Trap simulates the interaction of charged

quantum-mechanical Pauli force, which acts at close distances (accounting for the "collisions" between them). The potential well is given by a force directed radially inward that is proportional to the distance from the origin. In addition, the motion of the particles is damped by a term proportional to velocity, which allows them to "settle down" into stable and meta-stable states. Moving in response to these forces, the particles will eventually end up in a configuration of minimum potential, where the net force on any giessentially zero. Interestingly, these configurations are often highly symmetrical, and for any given combination of particles there are often several different stable (or meta-stable) configurations. Press "s" to illustrate these symmetries by generating a surface based on the positions of the particles! Rotate the camera by clicking and dragging the mouse. Zoom in and out by control-clicking and dragging the mouse.

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Simulation of a Nucleus and Four Electrons

simulation of how an atom might rm. A nucleus with a positive net char

a stable configuration. The quantum

2D The Suspension Bridge

is created by attaching a series of positive and negatively charged particles to two fixed

Pauli Afo ge attracts four electrons, which are eventually "captured" and held in mechanical "Pauli" force prevents the particles from collapsing in on one another, while a damping term allows them to "settle down".

Pauli

In this simulation, an "electromagnetic suspension bridge"

o

endpoints, and adding a downward gravitational force. The tension in the "bridge" is supplied simply by the Coulomb interaction of its constituent parts and the Pauli force keeping them from collapsing in on each other. Initially, the bridge only sags slightly under the weight of gravity, but what would happen to it under a rain of massive neutral particles? Press "o" to find out!

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Charge Moving in a Constant Field

tric fields, and of the exchange of energy between fields and particles. An electric charge with charge q > 0

in the animation that the electric field lines are generally compressed above the charge and stretched below the

q>0 z

z

:

An example of the stresses transmitted by elec

moves in a constant electric field. The charge is initially moving upward along the negative z-axis in a constant background field. The charge feels a constant downward force. The charge eventually comes to rest at the origin, and then moves back down the negative z-axis. This motion, and the fields that accompany it, are shown in the animation.

As the charge moves upward, it is apparent

charge. This field configuration enables the transmission of a downward force to the moving charge we can see as well as an upward force to the charges that produce the constant field, which we cannot see. The overall appearance of the upward motion of the charge through the electric field is that of a point being forced into a resisting medium, with stresses arising in that medium as a result of that encroachment.

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Faraday would have described the downward force on the charge as follows: Surround the charge by an imaginary sphere centered on it. The field lines piercing the lower half of the sphere transmit a tension that is parallel to the

3D The Suspension Bridge 3D

netic suspension bridge" is created by ing a lattice of positive and negatively charged particles between four

xed corners, and adding a downward gravitational force. The tension in the

field. This is a stress pulling downward on the charge from below. The field lines draped over the top of the imaginary sphere transmit a pressure perpendicular to themselves. This is a stress pushing down on the charge from above. The total effect of these stresses is a net downward force on the charge.

Pauli

o In this simulation, an "electromagattachfi"bridge" is supplied simply by the Coulomb interaction of its constituent parts with the Pauli force keeping them from collapsing in on each other. Initially, the bridge only sags slightly under the weight of gravity, but what would happen to it under a rain of massive neutral particles? Press "o" to find out!

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Van de Graff Charging a Van de Graff Generator (applet)

of a Van de Graff Generator to a positive otential when there is a stationary positive charge sitting above it (resting on a

Van de GraffVan de Graff

q a Q

-J dot E Start

Q2

Q This applet shows the chargingpplastic square). The presence of that charge means that we have to do more work than normal to charge up the generator. If the positive charge above the generator has charge q and the generator has radius a and is charged to a total charge Q, the energy we must put in to charge up the generator is:

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is stored in the electrostatic field surrounding the

or releases the stationary positive

d in the electrostatic field (not including the

This total amount of energygenerator after it is charged. As we charge it, there is a Poynting flux outward from the region where we are creating electromagnetic energy. This region is where the charge is being moved against the electric field on the conveyer belt (inside the cylindrical shaft of the generator). This region is where the creation rate per unit volume for electromagnetic energy (-J dot E), is positive. Energy is created there and flows out to fill the space around the generator. This energy flow can be seen in the motion of the electric field lines, which always move in the direction of the Poynting flux. Pressing "Start" after charging the generatcharge sitting above the generator. If there is enough electrostatic repulsion to overcome gravity, that charge will move upward. We can see that the source of its increasing kinetic energy and gravitational potential energy is the electrostatic field, because again we see the flow of energy out of that field, as indicated by the motion of the electric field lines. Since we have no energy dissipation in the system, the charged particle will eventually come to rest at some distance above the generator, and then start to fall back, transferring gravitational potential energy and kinetic energy back into the electrostatic field, as shown by the field line motion. The graph shows the energy storeself energy term proportional to Q^2) and the sum of the gravitational potential energy and the kinetic energy of the charge. The sum of these is the total energy of the charge: gravitational potential, plus electrostatic potential, plus kinetic energy, and is constant as long as we keep Q constant.

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Van de Graff

tion)

Van

Start

his animation shows the process of charging a Van de Graff generator in the

Charging a Van de Graff Generator (anima

de Graff Van de Graff

q a Q

-J dot E

Tpresence of a stationary positive point charge. The presence of that charge means that we have to do more work than we would normally have to do to charge up the generator. If the point charge above has charge q and the generator has radius a and is charged to a total charge Q, the energy we must put in to charge up the generator is:

y is stored in the electrostatic field surrounding the This total amount of energ

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generator after it is charged. As we charge it, there is a Poynting flux outward from the region where we are creating electromagnetic energy. The region is where the charge is being moved against the electric field on the conveyer belt (inside the cylindrical shaft of the generator). This region is where the creation rate per unit volume for electromagnetic energy (-J dot E), is positive. Energy is created there and flows out to fill the space around the generator. This flow can be seen in the motion of the electric field lines, which is always in the direction of the Poynting flux. After the generator is fully charged, we release the stationary positive charge

Van de Graff ff Generators

an

sitting above it. In the case shown here, there is enough electrostatic repulsion to overcome gravity, and the charge moves upward. We can see that the source of its increasing kinetic energy and gravitational potential energy is the electrostatic field, because again we see the flow of energy out of that field, as indicated by the motion of the electric field lines. Since we have no energy dissipation in the system, the charged particle will eventually come to rest at some distance above the generator, and then start to fall back, transferring gravitational potential energy and kinetic energy back into the electrostatic field as shown by the field line motion.

Charging and Discharging Two Van de Gra

Q Vde Graff

-J dot E

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In this animation we show the charging of two Van de Graff generators sitting

enerators, we turn off the conveyer belts. We then ionize

2D Lattices 2D

Pauli

attice 2D simulates the interaction of charged particles in a two dimensional

side by side to equal and opposite charges Q. As we charge them, there is a Poynting flux outward from the region where we are creating electromagnetic energy. This region is where the charge is being moved against the electric field on the conveyer belts of the two generators (inside the cylindrical shafts of the generators). This is where the creation rate per unit volume for electromagnetic energy (-J dot E) is positive. Energy is created in those regions and flows out to fill the space around the generators. That flow can be seen in the motion of the electric field lines, which is always in the direction of the Poynting flux. After charging the gthe air between the two spheres so that they discharge along the line joining them. We see the spark discharge and the inflow of energy from the electrostatic field powers that spark. Again we see the energy drain out of the electrostatic field as indicated by field line motion, which is in the direction of the Poynting flux.

2D

Pauli

"f"

/

Lplane. The particles interact via the classical Coulomb force, as well as the repulsive quantum-mechanical Pauli force, which acts at close distances

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(accounting for the "collisions" between them). Additionally, the motion of the particles is damped by a term proportional to their velocity, allowing them to "settle down" into stable (or meta-stable) states. In this simulation, the proportionality of the Coulomb and Pauli forces has been

lines

3D Lattices 3D

Pauli

attice 3D simulates the interaction of charged particles in three dimensions.

of the Coulomb and Pauli forces has been

adjusted to allow for lattice formation, as one might see in a crystal. The "preferred" stable state is a rectangular lattice, although other formations are possible depending on the number of particles and their initial positions. New feature: Selecting a particle and pressing "f" will toggle fieldillustrating the local field around that particle. Performance varies depending on the number of particles / fieldlines in the simulation.

3D

Pauli

"f"

/

LThe particles interact via the classical Coulomb force, as well as the repulsive quantum-mechanical Pauli force, which acts at close distances (accounting for the "collisions" between them). Additionally, the motion of the particles is damped by a term proportional to their velocity, allowing them to "settle down" into stable (or meta-stable) states. In this simulation, the proportionalityadjusted to allow for lattice formation, as one might see in a crystal. The "preferred" stable state is a rectangular (cubic) lattice, although other formations are possible depending on the number of particles and their initial

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positions. New feature: Selecting a particle and pressing "f" will toggle fieldlines illustrating the local field around that particle. Performance varies depending on the number of particles / fieldlines in the simulation. Colliding Lattices

lliding Lattices simulates the interaction of charged particles in two

of a collision between two

Pauli

""

"f"/

Codimensions. The particles interact via the classical Coulomb force, as well as the repulsive quantum-mechanical Pauli force, which acts at close distances (accounting for the "collisions" between them). Additionally, the motion of the particles is damped by a term proportional to their velocity, allowing them to "settle down" into stable (or meta-stable) states. In this program, you can observe the results rectangular lattices of variable dimensions and initial velocities. After selecting a dimension, position, and velocity for each, clicks the "Start" button to run the simulation. Depending on your initial conditions, the two lattices may combine in to one larger lattice, or perhaps some more exotic formation. Experiment with various values and see what happens, although note that collisions above a certain speed will likely cause the particles to annihilate! The simulation is

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fully interactive, so feel free to manipulate the particles as they evolve. New feature: Selecting a particle and pressing "f" will toggle fieldlines

a Pentagon-shaped Box

this simulation, ten positively charged particles interact in two dimensions

The Capacitor

illustrating the local field around that particle. Performance varies depending on the number of particles / fieldlines in the simulation.

Charges Interacting Inside 10

Pauli

Ininside a pentagon-shaped box via the classical Coulomb force and the quantum mechanical Pauli force. The mutual repulsion felt by the charges compels them to maximize the distance between each other. In this case, maximal separation puts them at the corners of the pentagon and at the halfway points along the sides that connect them.

12

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This simulation illustrates the interaction of charged particles inside the two plates of a capacitor. Each plate contains twelve charges interacting via the Coulomb and Pauli forces, where one plate contains positive charges and the other contains negative charges. Because of their mutual repulsion, the particles in each plate are compelled to maximize the distance between one another, and thus spread themselves evenly around the outer edge of their enclosure. However, the particles in one plate are attracted to the particles in the other, so they attempt to minimize the distance between themselves and their oppositely charged correspondents. Thus, they distribute themselves along the surface of their bounding box closest to the other plate. The Electrostatic Force Experiment

,

This applet is a simulation of an experiment in which an aluminum sphere sitting on the bottom plate of a capacitor is lifted to the top plate by the

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electrostatic attraction generated as the capacitor is charged. While the sphere is in contact with the lower plate, their surface charge densities are approximately equal. Thus, as the capacitor is charged, the charge density on the sphere increases proportional to the potential difference between the plates. In addition, energy flows into the region between the plates as the electric field builds up. This can be seen in the motion of the electric field lines as they move from the edge to the center of the capacitor. As the potential difference between the plates increases, the sphere feels an increasing attraction towards the top plate, indicated by the increasing tension in the field as more field lines "attach" to it. Eventually this tension is enough to overcome the downward force of gravity, and the sphere is "levitated". Once separated from the lower plate, the sphere's charge density no longer increases and it feels both an attractive force towards the upper plate and a repulsive force from the lower one. The result is a net force upwards. In this simulation we have placed a non-conducting barrier just below the upper plate to prevent the sphere from touching it and discharging. The Electrostatic Zoo

Pauli

""(

"Grass Seeds") ("")

"

""" ""

"" """

"""

CTRL The Electrostatic Zoo simulates the interaction between charged particles, and illustrates the electric fields generated as a result. Charges in the Zoo interact

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via the Coulomb force, with a Pauli repulsive force at close distances and a damping force proportional to velocity. The fields can be visualized in two ways: a "grass-seeds" representation (click on "Grass Seeds "), or a traditional vector field representation (adjust the "Field Visualization" slider), in addition to the real time renderings of the local field around each particle. Use Start button to start simulation. Use Pause button to pause simulation. Click and drag on any particle to change its position. Use "Grass Seeds" to get a "grass-seeds" representation of the electric field. Use "Equipotential Lines" to get a representation of the electric potential field. Select and click "remove" to remove charge. Use "Add Random Positive" and "Add Random Negative" to add a positive or negative charge, respectively. Use "Toggle Field Lines on Selected" to toggle fieldlines on the selected charge(s). Control-click to select multiple charges. The Electrostatic Videogame

Pauli

()

"" . The Electrostatic Video Game is a simulation consisting of one charge that is free to move and two charges that are fixed at specific locations. Using the principles of Coulomb's Law, Pauli repulsion at close distances, and damping proportional to velocity, the objective of the game is to steer the moving charge around the maze to the exit in the lower right wall. This is accomplished by dynamically (in real time) changing the value of the charge on the moving particle in response to the forces acting on it due to the local electric field. By thinking about the the charges involved, and resulting fields, you should be

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able devise a strategy for guiding the particle to the exit. Use the Start button to start simulation. Use Pause button to pause simulation. Use "Player Charge" slider in panel labeled "Point Charge" to change the charge of moving particle. Steer the moving particle around the maze (exit at lower right) by changing its charge. Two Point Charges java

3

""

This java simulation illustrates the field pattern created by two point charges with opposite signs of charge. In this simulation, the position and charge of each particle can be modified in real time, and the field configuration will update itself accordingly. All three field visualization techniques can be applied to show the overall electric field of the two-charge configuration: vector field, field lines, and "grass seeds". Charging by Induction java

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This java simulation illustrates how it is possible to charge a conductor without

gh an Electric Field

hese animations show the field and force on a point charge as it moves from

having direct contact. We charge by induction by first bringing the conductor to the proximity of a charged object. If the object's charge is large enough, the charges within the conductor will be separated. Next, we ground the conductor, which allows charges of the same sign as the object to flow away. Finally, we unground the conductor, and discard the object. The conductor is now charged, and has the opposite sign of the original object.

The Force on a Charge Moving Throu

(

)

Ta region with no external electric field to a region with a constant electric field (the field points upwards, or out of the screen in the top view). As indicated by the arrow, the particle feels no force while in the empty region, but feels a constant upwards force for the entire time it spends in the region of constant external field. Note that unlike the case of a particle moving through a magnetic field, the force felt by the particle in an electric field is not proportional to its velocity.

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The Charged Metal Slab

his applet simulates the movement

le in a Constant Field

his applet simulates the field of an

Tof electrons in a charged metal slab.

Torque on an Electric Dipo Telectric dipole rotating in a constant magnetic field.

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: Magnetostatics

Moving Positive Charge

he magnetic field of a moving

Moving Negative Charge

he magnetic field of a moving

The Magnetic Field of a Current Element

The Magnetic Field of a Tpositive charge when the speed of the charge is small compared to the speed of light.

The Magnetic Field of a Tnegative charge when the speed of the charge is small compared to the speed of light.

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A simulation of the magnetic field generated by a current element. The

ge Moving in a Circle

he animation shows the magnetic

s Moving in a Circle

he animation shows the magnetic

observation point (black sphere) can be moved using the arrow keys to sample the field at different positions relative to the current element. The red arrow at the current element and the transparent red arrow at the observation point indicate the direction of current flow. The transparent orange arrow at the observation point indicates the direction from the current element to the observation point. The blue arrow represents the magnetic field at the observation point.

Magnetic Field of One Char

Tfield of a single charge moving in a circle. We show the magnetic field vector directions in only one plane.

Magnetic Field of Two Charge

Tfield of two charges moving in a circle. We show the magnetic field vector directions in only one plane.

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es Moving in a Circle

he animation shows the magnetic

es Moving in a Circle

he animation shows the magnetic

Magnetic Field of Four Charg

Tfield of four charges moving in a circle. We show the magnetic field vector directions in only one plane.

Magnetic Field of Eight Charg

Tfield of eight charges moving in a circle. We show the magnetic field vector directions in only one plane.

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Integrating Around a Ring of Current

This simulation illustrates the magnetic field generated by a ring of current, and

Biot-Savart

shows how, by the principle of superposition, a continuous current distribution can be thought of as the sum of many discrete current elements (in this case, thirty). Each element generates its own field, described by the Biot-Savart Law (and represented here by the small vectors attached to the observation point), which, when added to the contribution from all the other elements, results in the total field of the ring (given by the large resultant vector). In this animation, each element is being added up one by one (indicated by the highlighted portion of the ring), and the total field changes accordingly. As the entire ring is integrated, the components of the field contributions in the plane of the ring are cancelled out; leaving a total field that is perpendicular to the ring on its central axis.

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The Ring of Current

This simulation illustrates the magnetic field generated by a ring of current, and

Biot-Savart

shows how, by the principle of superposition, a continuous current distribution can be thought of as the sum of many discrete current elements (in this case, thirty). Each element generates its own field, described by the Biot-Savart Law (and represented here by the small vectors attached to the observation point), which, when added to the contribution from all the other elements, results in the total field of the ring (given by the large resultant vector, and by the large two dimensional field map). By moving the observation point around with the arrow keys, changes in field magnitude and direction can be observed at different positions relative to the ring.Note that along the axis of the ring, the perpendicular component of each element's field is cancelled out by the corresponding element directly opposite it on the other side of the ring. The resultant field is thus described only by the contributions along that axis.

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e-Changing Field

positive point charge moves out of the page origin in a time-changing

ero, we see only the magnetic field of the

e neighborhood

The Force on a Moving Charge in a Tim

T

Aexternal field. This external field is uniform in space but varies in time with a period T as indicated by the blue vector in the animation. We assume that the variation of the external field is so rapid that the charge moves only a negligible distance in one period of this field. When the external magnetic field is zmoving charge. Since the charge is moving out of the page, its field circulates clockwise. When the vertically downward external magnetic field is at its maximum, we see to the left of the charge an enhancement of the magnetic field. This enhancement arises because the field of the charge is in the same direction as the external magnetic field (downward) on that side. To the right of the charge, where the field of the charge is opposite that of the external magnetic field, we see a reduction in the total magnetic field. The animation shows dramatically the inflow of energy into thof the charge as the external magnetic field grows, with a resulting build-up of stress that transmits a sideways force to the moving positive charge. The white vector indicates direction and magnitude of this force.

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front and back

The animation shows a charge moving toward a region where the magnetic

Charge Moving in a Magnetic Field

90

field is vertically upward. When the charge enters the region where the external magnetic field is non-zero, it is deflected in a direction perpendicular to that field and to its velocity as it enters the field. This causes the charge to move in an arc that is a segment of a circle, until the charge exits the region where the external magnetic field in non-zero. We show in the animation the total magnetic field-that is the magnetic field of the moving charge in addition to that of the external magnetic field. The bulging of the total field on the side opposite the direction in which the particle is pushed is due to the build up in magnetic pressure on that side. It is this pressure that causes the charge to move in the arc of a circle.

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The moving charge in the animation changes its direction of motion by ninety

sely at the field stresses where the external field lines enter

Two Wires in Parallel

he animation shows the magnetic field configuration around two wires

degrees over the course of the animation. How do we conserve momentum in this process? Momentum is conserved because momentum is transmitted from the moving charge to the currents that are generating the constant external field. This is plausible given the field configuration shown in the animation. The magnetic field stress, which pushes the moving charge sideways, is accompanied by a tension pulling the current source in the opposite direction. To see this, look clothe region where the currents that produce them are hidden, and remember that the magnetic field acts as if it were exerting a tension parallel to itself. The momentum loss by the moving charge is transmitted to the hidden currents producing the constant field in this manner.

Maxwell Tcarrying current in the same direction (i.e. wired in parallel). The Maxwell stresses associated with the magnetic fields cause the wires to feel a mutual attraction, and pull towards each other as a result.

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Two Wires in Series

he animation shows the magnetic fiel

Two Rings of Current Attracting

he animation shows two co-axial wire

Maxwell T d configuration around two wirescarrying current in opposite directions. The Maxwell stresses associated with the magnetic fields cause the wires to feel a mutual repulsion, and they spread apart as a result.

T loops carrying current in the same sense. The loops attract one another. We show the field configuration here using the "iron filings" representation. The bottom wire loop carries three times the current of the top wire loop.

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Two Rings of Current Repelling

he animation shows two co-axial wire loop

TeachSpin(tm) The TeachSpin(tm) Apparatus

Tea

The animation the magnetic field of a spring in the TeachSpinTM apparatus,

The magnet is partially levitated by the magnetic field of the coil.

T s carrying current in opposite senses. The loops repel one another. We show the field configuration here using the "iron filings" representation. The bottom wire loop carries three times the current of the top wire loop.

chSpin(tm)

permanent magnet suspended by a plus the magnetic field due to a current in the top coil. The magnet is fixed so that its North Pole points downward and the current in the coil is counter-clockwise when seen from above. The resulting force on the magnet is upwards, and the magnet moves upward as the current in the coil is increased.

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The Dip Needle

"

, ic field of the

arth is predominantly downward and northward at these Northern latitudes.

rth's field again (where the torque goes to zero), and

"

[ ]

A magnetic dipole in a "dip needle" oscillating in the magnetic field of the earthat latitude approximately the same as that of Boston. The magneteTo explain what is going on in this visualization, suppose that the magnetic dipole vector is initially along the direction of the earth's field and rotating clockwise. As the dipole rotates, the magnetic field lines are compressed and stretched. The tensions and pressures associated with this field line stretching and compression results in an electromagnetic torque on the dipole that slows its clockwise rotation. Eventually the dipole comes to rest. But the counterclockwise torque still exists, and the dipole then starts to rotate counterclockwise, passing back through being parallel to the eaovershooting. As the dipole continues to rotate counterclockwise, the magnetic field lines are now compressed and stretched in the opposite sense. The

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electromagnetic torque has reversed sign, now slowing the dipole in its counterclockwise rotation. Eventually the dipole will come to rest, start rotating clockwise once more, and pass back through being parallel to the field, as in the beginning. If there is no damping in the system, this motion continues indefinitely. Faraday understood the oscillations of a compass needle in exactly the way we describe here. In his words, "To understand this point, we have to

The Earth and a Giant Dip Needle (Far and Near)

his animation illustrates a hypotheticeedle (a compass whose needle can oscillate in a vertical plane) is placed on

's

consider that a [compass] needle vibrates by gathering upon itself, because of it magnetic condition and polarity, a certain amount of the lines of force, which would otherwise traverse the space about it"

Maxwell T al scenario in which an enormous dip nthe surface of the earth and oscillates undamped in the earth's field. The oscillation is of course a result of the interaction between the magnetic field of the earth and the magnetic field generated by the dip needle. If the needle

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oscillations were damped, it would align itself along the local magnetic field. Note how the total field is stretched and compressed as the needle oscillatesin keeping with the Maxwell stresses that cause the oscillation of the dip needle.

,

Magnet Oscillating Between Two Coils

TeachSpinTM

shows the magnetic field of a permanent magnet suspended by spring in the TeachSpinTM apparatus (see TeachSpin visualization), plus the

f the resulting field. When the dipole moment

( TeachSpin )

(

)

This animationamagnetic field due to current in the two coils (here we see a "cutaway" cross-section of the apparatus). The magnet is fixed so that its north pole points upward, and the current in the two coils is sinusoidal and 180 degrees out of phase. When the effective dipole moment of the top coil points upwards, the dipole moment of the bottom coil points downwards. Thus, the magnet is attracted to the upper coil and repelled by the lower coil, causing it to move upwards. When the conditions are reversed during the second half of the cycle, the magnet moves downwards. This process can also be described in terms of tension along, and pressure perpendicular to, the fieldlines oof one of the coils is aligned with that of the magnet, there is a tension along

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the fieldlines as they attempt to "connect" the coil and magnet. Conversely, when their moments are anti-aligned, there is a pressure perpendicular to the fieldlines as they try to keep the coil and magnet apart. Magnet Suspended Between Two Coils

TeachSpinTM

magnetic field of a permanent magnet suspended by spring in the TeachSpinTM apparatus (see TeachSpin visualization), plus the

(TeachSpin)

(

)

sinusoidal

(

)

This animation shows the amagnetic field due to current in the two coils (here we see a "cutaway" cross-section of the apparatus). The magnet is fixed so that its North Pole points upward, and the current in the two coils is sinusoidal and in phase. When the effective dipole moment of the top coil points upwards, the dipole moment of the bottom coil points upwards as well. Thus, the magnet the magnet is attracted to both coils, and as a result feels no net force (although it does feel a torque, not shown here since the direction of the magnet is fixed to point upwards). When the dipole moments are reversed during the second half of the cycle, the magnet is repelled by both coils, again resulting in no net force.

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This process can also be described in terms of tension along, and pressure perpen

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dicular to, the fieldlines of the resulting field. When the dipole moment

Helmholtz () The Magnetic Field of a Helmholtz Coil (aligned)

tic field of a Helmholtz coil when the current in e top and bottom coils flows in the same direction (i.e. their dipole moments

Helmholtz ()

The Magnetic Field of a Helmholtz Coil (anti-aligned)

of the coils is aligned with that of the magnet, there is a tension along the fieldlines as they are "pulled" from both sides. Conversely, when their moments are anti-aligned, there is a pressure perpendicular to the fieldlines as they are "squeezed" from both sides.

Helmholtz

(

)

This animation shows the magnethare aligned). The fields from the two coils add up to create a net field that is constant in the center of the coils. Though the coils are fixed, they are attracted to one another, resulting in a tension in the field between them. This is illustrated by fieldlines as they stretch to enclose both coils.

Helmholtz

(

)

86

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This animation shows the magnetic field of a Helmholtz coil when the current in

The Magnetosphere of the Earth

""

he earth sits in the magnetic field of the solar wind, which drags out the

the top and bottom coils flows in opposite directions (i.e. their dipole moments are anti-aligned). The fields from the two point in opposite directions, creating a net field that is zero at the center of the coils. Though the coils are fixed, they are repelled from one another, resulting in a pressure in the field between them. This is illustrated by fieldlines as they are compressed along the central horizontal axis between the coils.

()

"

"

""

Tmagnetic field of the sun to the neighborhood of the earth. The magnetic field lines in the polar regions of the earth connect to the interplanetary magnetic field lines, which are being carried past the earth away from the sun at high speeds by the solar wind. When the interplanetary magnetic field is southward (as is the case here) it can connect easily to the earth's magnetic field, which emerges from the south geographic pole of the earth. Energy from the solar wind flow then goes into stretching the reconnected field lines into the "magnetic tail" of the earth. Eventually those field lines "break" and snap back toward the earth on the nightside. It is this process transferring solar wind

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energy into magnetic energy and thence into energy flowing into the earth's atmosphere as the magnetic field lines "snap" that drives the aurora. The aurora occur at the feet of the "last closed field line", which defines the auroral oval. Magnetic Merging

nergy in solar flares comes from the abrupt release of magnetic energy that

th's Magnetic Field

Ehas built up over a long time into energetic particles and plasma heating. This release takes place via a process called magnetic merging, or magnetic annihilation. In this process, magnetic energy flows in horizontally from the sides, the field lines "reconnect", and then energy in the form of energetic particles or accelerated plasma flows vertically upwards and downwards.

A Bar Magnet in the Ear

TEAL

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60

CTRL+

his model shows a bar magnet and compass sitting on a table in the TEAL

ee that a compass needle will always align

ouse to rotate the scene. Control-click and drag to zoom

Constant Magnetic Field

)

current carrying wire in a constant magnetic field. We show here a moving

Tclassroom. The interaction between the magnetic field of the bar magnet and the magnetic field of the earth is illustrated by the fieldlines that extend out from the bar magnet. Fieldlines that emerge towards the edges of the magnet generally reconnect to the magnet near the opposite pole. However, fieldlines that emerge near the poles tend to wander off and reconnect to the magnetic field of the earth, which, in this case, is approximately a constant field coming at 60 degrees from the horizontal. Looking at the compass, one can sitself in the direction of the local field. In this case, the local field is dominated by the bar magnet. Click and drag the min and out.

A Current Carrying Wire in a

()

(

Awire which is coming out of the screen, and carries current out of the screen, as it moves into a constant vertical magnetic field. The wire feels a force to the left, which is a combination of a magnetic pressure to the right in front of the wire which pushes it back to the left, and a magnetic tension to the left behind

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the wire, which pulls it back to the left. Because of the stresses associated with the magnetic field (both its own field and the constant field), the wire slows down and comes to rest, and then accelerates back in the direction from whence it came. The currents which produce the constant field (not shown) absorb the momentum of the wire as it reverses, and this produces a force on those sources which is to the right. Two Current Carrying Rings

his is a simulation of the magnetic field generated by two rings of current. By

The Floating Coil

""

Tmanipulating the properties of the rings (position, orientation, radius, and current) and observing the fieldlines, you can see how the net field changes. Click the "Grass Seeds" button for a high resolution image of the field.

(

)

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This animation shows the magnetic field of a current-carrying coil suspended

The Floating Coil Applet

his applet illustrates the forces on a current carrying coil sitting on the axis of

by a spring above a permanent magnet. Depending on the direction of current flow in the wire (i.e. the direction in which the batteries are connected), the coil will either be repelled upwards away from the magnet, or attracted downwards towards it. In the first case, the effective dipole moment of the coil opposes the dipole moment of the magnet, resulting in a pressure in the magnetic field between them, which pushes against the coil. When the batteries are reversed, the effective dipole moment of the coil points upwards, in the same direction as the dipole moment of the magnet. The result is a tension in the field that pulls the coil downwards.

Ta permanent magnet. For current flowing in one direction in the coil the force on the coil will be upward, and if the current is strong enough the coil will levitate, floating on the magnetic fields of the coil plus the magnet. This is the way one form of Maglev works. For the other direction of current the coil is attracted to the magnet the magnet has its north pole at the top, and the direction of positive current is counterclockwise when viewed from above. The coil initially rests at one coil radius above the magnet on a platform. There is a marker at two coil radii above the magnet. A quantitative problem at this link gives you an idea of the currents you need to get levitation in the simple experiments which this applet illustrates.

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http://jlearn.mit.edu/simulations/floatingcoil.jnlphttp://web.mit.edu/8.02T/www/802TEAL3D/visualizations/magnetostatics/floatingcoilapp/floatingCoil.pdf

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hrough a Magnetic Field

hese animations show the field and

etic field to a region with a constant

The Force on a Charge Moving T

(

)

Tforce on a point charge as it moves from a region with no external magnmagnetic field (the field points upwards, or out of the screen in the top view). As indicated by the arrow, the particle feels no force while in the empty region, but feels a varying force in the region of constant external field. A moving charge generates a magnetic field whose magnitude and direction are proportional to the velocity of the charge. Thus, the particle feels maximal force when it is moving fastest through the external field, in the direction given by the cross product of its velocity and the direction of the external field. When it slows to a stop, its field collapses and the force felt reduces to zero. When it starts moving again, in the opposite direction, the force returns. However, since the direction of the velocity has reversed, so has the direction of the force?

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http://web.mit.edu/8.02T/www/802TEAL3D/visualizations/magnetostatics/force_in_bfield/force_in_bfield_p.mpg

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in a Constant Field

his applet simulates the field of a magnetic dipole rotating in a constant

TeachSpin(tm) t

TeachSpin

his applet is a simulation of the TeachSpin experiment, in which a magnet is

Torque on a Magnetic Dipole

Tmagnetic field.

The TeachSpin(tm) Apple

(Helmholtz)

() Tsuspended by a spring between two current-carrying coils (in Helmholtz configuration). As a current is run through the coils, the magnet, whose orientation has been fixed such that its north pole points upwards, feels a force dependent on the direction in which the current is flowing in each coil. The two

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http://jlearn.mit.edu/simulations/torqueondipb.jnlphttp://jlearn.mit.edu/simulations/teachspin.jnlp

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basic configurations are one in which the current in one coil flows opposite the direction of the current in the other coil, and one in which the currents in both coils flow in the same direction. In the first case, the magnet will feel a repulsive force from one coil and an attractive force from the other, causing it to be displaced vertically. In the second case, the magnet feels repulsion or attraction from both coils simultaneously, which cancel each other out, leaving the magnet undisplaced between the coils. In the applet, you can vary several parameters, including the relative direction

The Magnetic Field of a Wire and a Compass

his applet simulates the magnetic field of a long current-carrying wire and a

of the currents and their magnitudes. In addition to varying the current manually, you can turn on a signal generator that will produce a sinusoidal current that causes the magnet to oscillate (or not) at the given frequency and amplitude.

Tcompass needle, along with the dynamics of their interaction.

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: Faraday's Law

s the magnetic fielon-magnet ring (e.g. copper) as it falls under gravity in the magnetic field of a

The Suspended Ring

he animation shows the magnetic fielon-magnetic ring (e.g. copper) as it falls under gravity in the magnetic field of

The Levitating Ring

Poynting The animation show d configuration around a conducting nfixed permanent magnet. The current in the ring is indicated by the small moving spheres. In this case, the ring is light and has zero resistance, and levitates above the magnet. The motions of the field lines are in the direction of the local Poynting flux vector.

Poynting T d configuration around a conducting na fixed permanent magnet. The current in the ring is indicated by the small moving spheres. In this case, the ring is light and has zero resistance, and is

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suspended below the magnet. The motions of the field lines are in the direction of the local Poynting flux vector. The Falling Ring with Finite Resistance

he animation shows the magnetic fielon-magnetic ring (e.g. copper) as it falls under gravity in the magnetic field of

The Falling Ring with Zero Resistance

he magnetic eld configuration around a conduct

permanent magnet. The current in the

Poynting T d configuration around a conducting na fixed permanent magnet. The current in the ring is indicated by the small moving spheres. In this case, the ring has finite resistance and falls past the magnet. The motions of the field lines are in the direction of the local Poynting flux vector.

Poynting The animation shows tfi ing non-magnetic ring as it falls undergravity in the magnetic field of a fixed

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ring is indicated by the small moving spheres. In this case, the ring is heavy and has zero resistance, and falls past the magnet. The motions of the field lines are in the direction of the local Poynting flux vector. The Levitating Magnet

etic eld configuration around a

rrent in the ring is indicated by the small

The Suspended Magnet

he animation shows the magnetic fills under gravity through a non-magnetic conducting ring. The current in the

Poynting The animation shows the magnfi permanent magnet as it falls under gravity through a conducting non-magnet ring. The cumoving spheres. In this case, the magnet is light, the ring has zero resistance, and the magnet levitates above the ring. The motions of the field lines are in the direction of the local Poynting flux vector.

Poynting T eld configuration around a magnet as it fa

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ring is indicated by the small moving spheres. In this case, the magnet is light and the ring has zero resistance, and the magnet is suspended below the ring. The motions of the field lines are in the direction of the local Poynting flux vector.

The Falling Magnet with Finite Resistance Ring

he animation shows the magnetic fills under gravity through a conducting non-magnetic ring. The current in the

The Falling Magnet with a Zero Resistance Ring

he animation shows the magnetic fi

Poynting T eld configuration around a magnet as it faring is indicated by the small moving spheres. In this case, the ring has finite resistance and the magnet falls through it. The motions of the field lines are in the direction of the local Poynting flux vector.

Poynting T eld configuration around a magnet as it

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falls under gravity through a conducting non-magnetic ring. The current in the

Creating a Magnetic Field

umber of free positive charges that re not moving. Since there is no current, there is no magnetic field. Now

ring is indicated by the small moving spheres. In this case, the magnet is heavy and the ring has zero resistance, so the magnet falls through the ring. The motions of the field lines are in the direction of the local Poynting flux vector.

Suppose we have five rings that carry a nasuppose a set of external agents come along (one for each charge) and simultaneously spin up the charges counterclockwise as seen from above, at the same time and at the same rate, in a manner that has been pre-arranged. Once the charges on the rings start to accelerate, there is a magnetic field in the space between the rings, mostly parallel to their common axis, which is stronger inside the rings than outside. This is the solenoid configuration. As the magnetic flux through the rings grows, Faraday's Law tells us that there

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is an electric field induced by the time-changing magnetic field that is

peed against the induced electric field

Destroying a Magnetic Field

at re moving counter-clockwise. This current results in a magnetic field that is

circulating clockwise as seen from above. The force on the charges due to this electric field is thus opposite the direction the external agents are trying to spin the rings up in (counterclockwise), and thus the agents have to do additional work to spin up the charges because of their charge. This is the source of the energy that is appearing in the magnetic field between the rings-the work done by the agents against the "back emf". Over the time whe