-Magnetic Flux-Gauss’s Law for Magnetism

-“Ampere-Maxwell” Law

AP Physics C

Mrs. Coyle

Magnetic Flux

θ

Magnetic Flux, The number of magnetic (flux) field

lines which pass through a given cross-sectional area A

Units:

webers

B Tesla

A area m2 angle formed between B and the normal to

the loop (area vector A)The area vector A is perpendicular to the surface A and

has a magnitude equal to the area A.

For constant B and A:

cos

B dA

BA

When B is perpendicular to the loop?

= BA

Why?

Which has the largest magnetic flux?

Answer: A

When B is along the plane of the loop?

• Hint: is the angle formed between B and the normal to the loop.

Gauss’ Law in Magnetism

• Magnetic fields do not begin or end at any point– The number of lines entering a surface equals

the number of lines leaving the surface

• Gauss’ law in magnetism says:

0d B A

Ampere’s Law – General Form

• Also known as the Ampere-Maxwell law

• Where is the electric flux.

• The second term Id is called displacement current and is caused by electric fields that vary with time as in a capacitor.

I I I

Eo d o o o

dd μ μ μ ε

dtB s

E

Example: Capacitor

• Consider surfaces S 1 and S2 .

• The current through S1 is I.

• There is no conducting current through S2

• The electric flux through

S2 is EA – A is the area of the capacitor

plates– E is the electric field between the

plates

• If q is the charge on the plate at any time, E = EA = q/o

Example: Capacitor cont’d

• The displacement current is

• The displacement current is the same as the conduction current through S1

• The displacement current on S2 is the source of the magnetic field on the surface boundary

Ed o

d dqI ε

dt dt

• Magnetic fields are produced both by conduction currents and by time-varying electric fields

Classification of Magnetic Substances

• Paramagnetic and ferromagnetic materials are made of atoms that have permanent magnetic moments

• Diamagnetic materials are those made of atoms that do not have permanent magnetic moments

Classifying Materials by Permeability

• Materials can be classified by how their permeability compares with the permeability of free space (o)

• Paramagnetic: m > o

• Diamagnetic: m < o

Examples of Ferromagnetic Materials

– iron– cobalt– nickel– gadolinium– dysprosium

Domains

The domain is an area in a material within which all magnetic moments are aligned

Domains, Unmagnetized Material

• The magnetic moments in the domains are randomly aligned

• The net magnetic moment is zero

Domains, External Field Applied

Meissner Effect

• Certain types of superconductors also exhibit perfect diamagnetism – This is called the

Meissner effect

• If a permanent magnet is brought near a superconductor, the two objects repel each other

Earth’s Magnetic Field

• Like a bar magnet• The Earth’s south magnetic

pole is located near the north geographic pole

• The Earth’s north magnetic pole is located near the south geographic pole

• Magnetic Declination

Dip Angle of Earth’s Magnetic Field

• If a compass is free to rotate vertically as well as horizontally, it points to the Earth’s surface

• The angle between the horizontal and the direction of the magnetic field is called the dip angle– The farther north the device is moved, the farther

from horizontal the compass needle would be• The compass needle would be horizontal at the equator and

the dip angle would be 0°• The compass needle would point straight down at the south

magnetic pole and the dip angle would be 90°

Reversals of the Earth’s Magnetic Field

• The direction of the Earth’s magnetic field reverses every few million years– Evidence of these reversals are found in

basalts resulting from volcanic activity– The origin of the reversals is not understood