Electromagnetic Theory

COORDINATE SYSTEMS

• RECTANGULAR or Cartesian

• CYLINDRICAL

• SPHERICAL

Choice is based on symmetry of problem

Examples:

Sheets - RECTANGULAR

Wires/Cables - CYLINDRICAL

Spheres - SPHERICAL

Vector Calculus

Cartesian Coordinates

P (x, y, z)

x

y

z

P(x,y,z)

Rectangular Coordinates Or

z

y

x

A vector A in Cartesian coordinates can be written as

),,( zyx AAA or zzyyxx aAaAaA

where ax,ay and az are unit vectors along x, y and z-directions.

Cylindrical Coordinates

P (ρ, Φ, z)

x= ρ cos Φ, y=ρ sin Φ, z=z

z

20

0

A vector A in Cylindrical coordinates can be written as

),,( zAAA or zzaAaAaA

where aρ,aΦ and az are unit vectors along ρ, Φ and z-directions.

zzx

yyx ,tan, 122

Spherical Coordinates

P (r, θ, Φ)

x=r sin θ cos Φ, y=r sin θ sin Φ, Z=r cos θ

20

0

0

r

A vector A in spherical coordinates can be written as

),,( AAAr or aAaAaA rr

where ar, aθ, and aΦ are unit vectors along r, θ, and Φ-directions.

x

y

z

yxzyxr 1

22

1222 tan,tan,

Differential Length, Area and

Volume

Cartesian Coordinate system

Cartesian Coordinate System

1- Differential displacement is given by

2- Differential normal surface area is given by

3- Differential volume is given by-

x y zdl dxa dya dza

x

y

z

dS dydza

dS dxdza

dS dxdya

dv dxdydz

Cylindrical Coordinate System

Cylindrical Coordinate System

1- Differential displacement is given by

2- Differential normal surface area is given by

3- Differential volume is given by-

zdl d a d a dza

z

dS d dza

dS d dza

dS d d a

dv d d dz

Spherical Coordinate System

Spherical Coordinate System

1- Differential displacement is given by

2- Differential normal surface area is given by

3- Differential volume is given by-

sinrdl dra rd a r d a

2 sin

sin

rdS r d d a

dS r drd a

dS rdrd a

2 sindv r drd d

Scalar and Vector Fields

• A scalar field is a function that gives us a single value

(of magnitude only) of some variable for every point in

space.

voltage, current, energy, temperature

• A vector is a quantity which has both a magnitude and

a direction in space.

velocity, momentum, acceleration and force

Del Operator “a vector differentiation operator”

Cartesian Coordinates

zyx az

ay

ax

Cylindrical Coordinates

Spherical Coordinates

zaz

aa

1

ar

ar

ar

r

sin

11

Gradient, Divergence and Curl

• Gradient of a scalar function is a

vector quantity.

• Divergence of a vector is a

scalar quantity.

• Curl of a vector is a vector

quantity.

• The Laplacian of a scalar A

is a scalar quantity.

f

A.

The Del Operator

A

A2

zyx az

ay

ax

The gradient of a scalar field V is a vector that represents

both the magnitude and the direction of the space rate of

change of V.

Gradient of a Scalar

zyx az

Va

y

Va

x

VV

The divergence of a vector A at a given point P is the volume density of the

outward flux of vector field of A from an infinitesimal volume around a

given point P.

Divergence of a Vector

.divA A

z

A

y

A

x

AA

.

Physical Significance:

The divergence measures how much a vector field ``spreads

out'' or diverges from a given point.

If the vector field lines are coming outwards from the given

point----- Divergence is positive.

If the vector field lines are coming in towards the point-----

Divergence is negative.

The curl of A is an axial vector whose magnitude is the maximum

circulation of A per unit area and whose direction is the normal

direction of the area.

Curl of a Vector

zyx

zyx

AAA

zyx

aaa

A

Physical Significance of Curl

The curl of a vector field measures the

tendency for the vector field to rotate around

the point.

If the value of curl is zero then the field is said

to be a non rotational field (in such condition,

divergence will have non-zero value)

The divergence theorem states that the total outward flux

of a vector field A through the closed surface S is the same

as the volume integral of the divergence of A.

Divergence Theorem or Gauss’ Theorem

. .S V

A dS Adv

L S

dSAdlA ).(.

Stokes’ Theorem

Stokes’s theorem states that the line integral of a vector field A around a

closed path L is equal to the surface integral of the curl of A over the open

surface S bounded by L, provided A and are continuous on S A

Q

FE

Electric Field Intensity is the force per unit charge when placed in the

electric field

Electric Field Intensity

In general, if we have more than two point

charges

If there is a continuous charge distribution say along a line, on a

surface, or in a volume

Electric Field due to Continuous Charge Distribution

The charge element dQ and the total charge Q due to these charge

distributions can be obtained by

The electric field intensity due to each charge distribution ρL, ρS and

ρV may be given by the summation of the field contributed by the

numerous point charges making up the charge distribution.

The value of R and aR vary as we evaluate above three integrals.

The electric field intensity E depends on the medium in which the

charges are placed.

Electric Flux Density

The electric flux ψ in terms of D can be defined as

ED o

The vector field D is called the electric displacement vector or

electric flux density and is measured in coulombs per square meter.

s

Gauss Law

It states that the total electric flux ψ through any closed surface is

equal to the total charge enclosed by that surface.

encQ

(i)

Using Divergence Theorem

(ii)

Comparing the two volume integrals in (i) and (ii)

It states that the volume charge density is the same as the

divergence of the electric flux density.

(This is also the first Maxwell’s equation.)

Relationship between E and V

The potential difference across a closed path is zero, i. e.

ldEV

B

A

AB .

BAAB VV

and ldEV

A

B

BA .

0. ldEVV BAAB

0. ldE

It means that the line integral of along a closed path must be zero. E

(i)

Physically it means that no net work is done in moving a charge along

a closed path in an electrostatic field.

0).(. SdEldE

Equation (i) and (ii) are known as Maxwell’s equation for static

electric fields.

(ii)

Applying Stokes’s theorem to equation (i)

0 E

Equation (i) is in integral form while equation (ii) is in differential

form, both depicting conservative nature of an electrostatic field.

An electric dipole is formed when two point charges of equal

magnitude but of opposite sign are separated by a small distance.

The potential at P (r, θ, Φ) is

Electric Potential due to an Electric Dipole

If r >> d, r2 - r1 = d cosθ

and r1r2 = r2 then

But where

The dipole moment is directed from –Q to +Q.

radd .cos

If we define as the dipole moment, then dQp

zadd

p

if the dipole center is not at the origin but at then 'r

• A dielectric substance is an electrical insulator that

can be polarized by an applied electric field.

• When a dielectric is placed in an electric field, electric

charges do not flow through the material (as they do in

a conductor), but they only slightly shift from their

average equilibrium positions causing dielectric

polarization.

• Because of dielectric polarization, positive charges are

displaced toward the field and negative charges shift in

the opposite direction.

• This creates an internal electric field which reduces the

overall field within the dielectric itself.

Polarization in Dielectrics

P

For the measurement of intensity of polarization, we define

polarization as dipole moment per unit volume (Unit: coulomb per

square meter)

The major effect of the electric field on the dielectric is the creation of

dipole moments that align themselves in the direction of electric field.

This type of dielectrics are said to be non-polar. eg: H2, N2, O2

Other types of molecules that have in-built permanent dipole

moments are called polar. eg: H2O, HCl

When electric field is applied to a polar material then its permanent

dipole experiences a torque that tends to align its dipole moment in

the direction of the electric field.

Consider a dielectric material consisting of dipoles with Dipole

moment per unit volume.

Field due to a Polarized Dielectric

'dvP

P

The potential dV at an external point O due to

where R is the distance between volume

element dv’ and the point O, i.e.

R2 = (x-x’)2+(y-y’)2+(z-z’)2

But

(Vector Identity)

= -

(i)

Here primed quantities are having primed del operator

Now, since

or

Put this in (i) and integrate over the entire volume v’ of the dielectric

Applying Divergence Theorem to the first term

where an’ is the outward unit normal to the surface dS’ of the dielectric

The two terms in (ii) denote the potentials due to surface and

volume charge distributions with densities given as-

(ii)

Where

ρps = Bound surface charge density

and

ρpv = Volume charge density.

Bound charges are those which are not free to move in the

dielectric material.

Free charges are those that are capable of moving over

macroscopic distance. (The charge due to polarization is known as bound charge, while

charge on an object produced by electrons gained or lost from outside

the object is called free charge. )

Whenever, polarization occurs, an equivalent volume charge

density, ρpv is formed throughout the dielectric while an

equivalent surface charge density, ρps is formed over the surface

of dielectric.

Now, let us take the case when dielectric contains free charge

If ρv is the free volume charge density then the total volume charge

density ρt

Hence

Where

If ρv is the free volume charge density then the total volume charge

density ρt

Hence

Where

Dielectric Constant and Strength

Here = Charge density arising due to polarization (bound charges)

No dielectric is ideal. When the electric field in a dielectric is

sufficiently high then it begins to pull electrons completely out of the

molecules, and the dielectric starts conducting.

The effect of the dielectric on electric field , is to increase

inside it by an amount . P

DE

Further, the polarization would vary directly as the applied electric

field. So one can write-

Where is known as the electric susceptibility of the material e

It is a measure of how susceptible a given dielectric is to electric fields.

We know that

or

where є is the permittivity of the dielectric, єo is the permittivity of the

free space and єr is the dielectric constant or relative permittivity.

and

Thus

where ro

and

When a dielectric becomes conducting then it is called dielectric

breakdown. It depends on the type of material, humidity, temperature

and the amount of time for which the field is applied.

The minimum value of the electric field at which the dielectric

breakdown occurs is called the dielectric strength of the dielectric

material.

or

The dielectric strength is the maximum value of the electric field that a

dielectric can tolerate or withstand without breakdown.

Continuity Equation and Relaxation Time

According to principle of charge conservation, the time rate of

decrease of charge within a given volume must be equal to the net

outward current flow through the closed surface of the volume.

The current Iout coming out of the closed surface

where Qin is the total charge enclosed by the closed surface.

Using divergence theorem

But

(i)

Equation (i) now becomes

This is called the continuity of current equation.

Effect of introducing charge at some interior point of a

conductor/dielectric

or

According to Ohm’s law

According to Gauss’s law

(ii)

Equation (ii) now becomes

Integrating both sides

or

This is homogeneous linear ordinary differential equation. By

separating variables we get

where

(iii)

Equation (iii) shows that as a result of introducing charge at some

interior point of the material there is a decay of the volume charge

density ρv.

The time constant Tr is known as the relaxation time .

Relaxation time is the time in which a charge placed in the interior of a

material to drop to e-1 = 36.8 % of its initial value.

For Copper Tr = 1.53 x 10-19 sec (Tr -short for good conductors)

For fused Quartz Tr = 51.2 days (Tr -large for good dielectrics)

Electric Field: Boundary Conditions

If the electric field exists in a region consisting of two different media,

the conditions that the field must satisfy at the interface separating the

media are called boundary conditions

These conditions are helpful in determining the field on one side of the

boundary when the field on other side is known.

We will consider the boundary conditions at an interface separating

1. Dielectric (єr1) and Dielectric (єr2)

2. Conductor and Dielectric

3. Conductor and free space

For determining boundary conditions we will use Maxwell’s equations

and

• Also we need to decompose E to the

interface of the interest-

Where and are the normal and

tangential components of E to the

interface of the medium

t nE E E

nEtE

Boundary Conditions

Dielectric- Dielectric boundary conditions

• Let us consider the E field exists in a region that consists

of two different dielectrics characterized by and

as shown below- 1 0 1r

2 0 2r

In media 1 and 2, Electric field can be decomposed as

Using Maxwell Equation along the closed path abcda-

As ∆h→0, above equation becomes

Thus the tangential components are the same on the two sides of the boundary,

i.e is continuous across the boundary. tE

----------------- (i)

Since n tD E D D

Thus, undergoes some change, hence it is

said to be discontinuous across the interface. tD

We can also apply first

Maxwell equation, to the

Cylindrical Gaussian

Surface of the adjacent

figure.

The contribution due to

curved sides vanishes.

Allowing ∆h→0 gives

or

here is the free charge

density at boundary

----------------- (ii)

• Generally = 0 (until and unless, we are not

putting free charge at the interface deliberately), the

above equation become-

• Thus the normal component of D is continuous

across the interface. Further since , we can

write-

• Showing that the normal component of the E is

discontinuous at the boundary.

• Equations (i), (ii), (iii) and (iv) are known as -

BOUNDARY CONDITIONS. (They must be satisfied at

the dielectric- dielectric boundary)

D E

----------------- (iii)

----------------- (iv)