Electricity and Magnetism

PH108: Electricity and Magnetism:Division 1

Instructor: Prof. Kumar Rao, Department of Physics, I.I.T.

Bombay

Address: Room 203, Physics Building

e-mail: kumar.rao@phy.iitb.ac.in

Phone: X-7587

Suggested Textbook: Introduction to Electrodynamics,

by D. J. Gri�ths

Syllabus

Review of vector calculus: Spherical polar and cylindrical

coordinates; gradient, divergence and curl; Divergence and

Stokes` theorems; Divergence and curl of electric �eld

Electric potential, properties of conductors; Poisson's and

Laplace's equations, uniqueness theorems, boundary value

problems, separation of variables, method of images, multipole

expansion

Polarization and bound charges, Gauss` law in the presence of

dielectrics, Electric displacement D and boundary conditions,

linear dielectrics

Syllabus Continued

Divergence and curl of magnetic �eld, Vector potential and its

applications; Magnetization, bound currents, Ampere`s law in

magnetic materials, Magnetic �eld H, boundary conditions,

classi�cation of magnetic materials; Faraday's law in integral

and di�erential forms, Motional emf, Energy in magnetic

�elds, Displacement current, Maxwell's equations.

Electromagnetic (EM) waves in vacuum and media, Energy

and momentum of EM waves, Poynting`s theorem; Re�ection

and transmission of EM waves across linear media.

Coordinate Systems in Two and Three Dimensions

Outline1 Coordinate systems in 2-dimensions: Cartesian and plane polar

coordinate systems and their relationship. Length and area

elements

2 Coordinate systems in 3-dimensions: Cylindrical and Spherical

Polar Coordinate systems, line, surface and volume elements

Objectives1 To learn to use symmetry adapted coordinate systems

2 To understand as to how to construct line, surface, and

volume elements for various coordinate systems

Using Symmetries in Physics

Using a coordinate system which is consistent with the

symmetry of the physical system simpli�es calculations

If a planar system has circular symmetry, use of plane-polar

coordinate system will simplify calculations

For systems with cylindrical symmetry, use of cylindrical polar

coordinates is advised

Likewise for spherical systems, use of spherical polar

coordinate system will be bene�cial

Coordinate Systems in Two Dimensions

Cartesian Coordinates:Location of a point in a �at plane is given by coordinates

(x ,y).

Di�erential line element dl is given by dl= dx i +dy j

2D Coordinates Continued

A general vector is given by A= Ax i +Ay j .

In�nitesimal area element dS12 in a plane described by

orthogonal coordinates 1 and 2 can be computed for any

coordinate system as

dS12 = dl1×dl2 (1)

For Cartesian coordinates it yields

dS= dx i ×dy j = dxdyk

or dS = dxdy

2D Coordinates...

Plane Polar Coordinates:Location of a point in a �at plane is given by coordinates

(ρ,θ).Di�erential line element dl is given by dl= dρρ +ρdθθ

In�nitesimal surface area is dS= dρρ ×ρdθθ = ρdρdθ k , ordS = ρdρdθ

Relationship between Cartesian and Plane Polar Coordinates

x = ρ cosθ , y = ρ sinθ

ρ =√

x2+ y2, θ = tan−1( yx

), where −∞ ≤ x ,y ≤ ∞;

0≤ ρ ≤ ∞, 0≤ θ ≤ 2π.

And unit vectors are related as ρ = cosθ i + sinθ j , andθ =−sinθ i + cosθ j

i = cosθρ − sinθθ , j = sinθρ + cosθθ

Using these relations, one can easily transform vectors

expressed in one coordinate system, into the other one.

Area of a circle of radius R , A=∫ R0

ρdρ∫2π

0dθ = πR2

Coordinate Systems in Three Dimensions

Cartesian Coordinates:Location of a point is is given by coordinates (x ,y ,z).

Di�erential line element dl is given by dl= dx i +dy j+dzk

In�nitesimal area element depends upon the plane. For xyplane it will be

dSxy = dxdyk

In�nitesimal volume element for any orthogonal 3D coordinate

system is given by

dV = dl1dl2dl3

for this case dV = dxdydz

3D Coordinate Systems...

Cylindrical Coordinates:Location of a point speci�ed by three coordinates (ρ,θ ,z)

3D Coordinate System....

Relationship with Cartesian coordinates x = ρ cosθ ,

y = ρ sinθ , z = z

Inverse relationship ρ =√

x2+ y2, θ = tan−1( yx

), z = z

Di�erential line element dl is given by dl= dρρ +ρdθθ +dzk

Area element in di�erent planes can be obtained using the

relation dS12 = dl1×dl2

Volume element dV = dl1dl2dl3 = ρdρdθdz

Volume of a cylinder of height L, and radius RV =

∫ Rρ=0

ρdρ∫ Lz=0

dz∫2π

θ=0dθ = πR2L

3D Coordinates...

Spherical Polar Coordinates:Location of a point is speci�ed by three coordinates (r ,θ ,φ),as shown below

What is the range of r , θ , and φ?

3D Coordinates...

Clearly 0≤ r ≤ ∞, 0≤ θ ≤ π, 0≤ φ ≤ 2π

Relationship with Cartesian coordinates x = r sinθ cosφ ,

y = r sinθ sinφ , z = r cosθ

Inverse relationship

r =√x2+ y2+ z2

θ = tan−1

(√x2+ y2

z

)φ = tan−1

y

x

Di�erential line element dl is given by

dl= dr r + rdθθ + r sinθdφφ

Cross products given by θ × φ = r , φ × r = θ , and r × θ = φ

Spherical Polar Coordinates...

Relationship between Cartesian and Spherical unit vectors

r = sinθ cosφ i + sinθ sinφ j+ cosθ k

θ = cosθ cosφ i + cosθ sinφ j− sinθ k

φ =−sinφ i + cosφ j

Area element on the surface of a sphere or radius R ,dSθφ = dlθ ×dlφ =Rdθθ ×R sinθdφφ = R2 sinθdθdφ r

Area of the surface of a sphere

A= R2∫

π

0sinθdθ

∫2π

0dφ = 4πR2

Spherical Polar Coordinates...

Elementary volume element

dV = dlrdlθdlφ = drrdθ r sinθdφ = r2 sinθdrdθdφ

Volume of a sphere of radius R

V =∫ R

0

r2dr∫

π

0

sinθdθ

∫2π

0

dφ =4

3πR3