Electromagnetics lecture guide, Lecture notes of Electromagnetic Engineering

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Electromagnetic

Theory

2

1 2

R

kQ Q

F 

It states that the force F between two point charges Q 1 and Q 2 is

Coulomb’s Law

In Vector form

Or

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.

Electric Flux Density

For an infinite sheet the electric flux density D is given by

For a volume charge distribution the electric flux density D is given by

In both the above equations D is a function of charge and position only (independent of medium)

Gauss Law

It states that the total electric flux ψ through any closed surface is equal to the total charge enclosed by that surface.

  Q enc

(i)

Electric Potential

Electric Field intensity, E due to a charge distribution can be obtained from Coulomb’s Law. or using Gauss Law when the charge distribution is symmetric. We can obtain E without involving vectors by using the electric scalar potential V. From Coulomb’s Law the force on point charge Q is

F  Q E

The work done in displacing the charge by length dl is

dW   F. dl   Q E. dl

The negative sign indicates that the work is being done by an external agent.

The total work done or the potential energy required in moving the point charge Q from A to B is

W Q E dl

B

A

  .

Dividing the above equation by Q gives the potential energy per unit charge.

E dl

Q

W B

A

 .  VAB

V AB is known as the potential difference between points A and B.

1. If is negative, there is loss in potential energy in moving Q from A to B (work is being done by the field), if is positive, there is a gain in potential energy in the movement (an external agent does the work).

V AB

V AB

2. It is independent of the path taken. It is measured in Joules per Coulomb referred as Volt.

For n point charges Q 1 , Q 2 , Q 3 …..Qn located at points with position

vectors the potential at is

If there is continuous charge distribution instead of point charges then

the potential at becomes

r (^) 1 , r 2 , r 3 ..... r n r



n

k (^) k

k

o r r

Q

V r

r

Relationship between E and V

The potential difference between points A and B is independent of the path taken

V E d l

B

A

AB  .

V AB   V BA

and V E dl

A

B

BA  .

V (^) AB  VBA   E. dl  0

 E. d^ l ^0

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

(i)

Also

It means Electric Field Intensity is the gradient of V.

E   V

The negative sign shows that the direction of is opposite to the direction in which V increases.

E

Consider an atom of the dielectric consisting of an electron cloud (-Q) and a positive nucleus (+Q).

Polarization in Dielectrics

When an electric field is applied, the positive charge is displaced

from its equilibrium position in the direction of by while

the negative charge is displaced by in the opposite

direction.

E

E F ^  QE

A dipole results from the displacement of charges and the dielectric is polarized. In polarized the electron cloud is distorted by the applied electric field.

F   Q E

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: H 2 , N 2 , O 2

Other types of molecules that have in-built permanent dipole moments are called polar. eg: H 2 O, 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

P dv^ '

P

The potential dV at an external point O due to

where R^2 = (x-x’)^2 +(y-y’)^2 +(z-z’)^2 and R is the distance between volume element dv’ and the point O.

But

Applying the vector identity

= -

(i)