Electricity and Magnetism
PH108: Electricity and Magnetism:
Division 1
Instructor:
Bombay
Prof. Kumar Rao, Department of Physics, I.I.T.
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
Address:
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 dierential 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
Outline
1
Coordinate systems in 2-dimensions: Cartesian and plane polar
coordinate systems and their relationship. Length and area
elements
Coordinate systems in 3-dimensions: Cylindrical and Spherical
Polar Coordinate systems, line, surface and volume elements
Objectives
1
2
To learn to use symmetry adapted coordinate systems
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 ).
Dierential 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
For Cartesian coordinates it yields
dS = dx i dy j = dxdy k
or dS = dxdy
(1)
2D Coordinates...
Plane Polar Coordinates:
Location of a point in a at plane is given by coordinates
(, ).
Dierential line element dl is given by dl = d + d
In nitesimal surface area is dS = d d = dd k, or
dS = dd
Relationship between Cartesian and Plane Polar Coordinates
x = cos , y = sin
p
= x 2 + y 2 , = tan1
0 , 0 2 .
y
x ,
where x, y ;
And unit vectors are related as = cos i+ sin j, and
= sin i+ cos j
i = cos sin , j = sin + cos
Using these relations, one can easily transform vectors
expressed in one coordinate system, into the other one.
R
R
Area of a circle of radius R , A = 0R d 02 d = R 2
Coordinate Systems in Three Dimensions
Cartesian Coordinates:
Location of a point is is given by coordinates (x, y , z).
Dierential line element dl is given by dl = dx i+ dy j+ dz k
In nitesimal area element depends upon the plane. For xy
plane it will be
dSxy = dxdy k
In nitesimal volume element for any orthogonal 3D coordinate
system is given by
dV = dl1 dl2 dl3
for this case dV = dxdydz
3D Coordinate Systems...
Cylindrical Coordinates:
Location of a point speci ed by three coordinates (, , z)
3D Coordinate System....
Relationship with Cartesian coordinates x = cos ,
y = sin , z = z
p
Inverse relationship = x 2 + y 2 , = tan1 yx , z = z
Dierential line element dl is given by dl = d + d + dz k
Area element in dierent planes can be obtained using the
relation dS12 = dl1 dl2
Volume element dV = dl1 dl2 dl3 = dd dz
Volume
of a cylinder
ofR height L, and radius R
R
R
V=
R
L
2
=0 d z=0 dz =0 d
= R 2 L
3D Coordinates...
Spherical Polar Coordinates:
Location of a point is speci ed by three coordinates (r , , ),
as shown below
What is the range of r , , and ?
3D Coordinates...
Clearly 0 r , 0 , 0 2
Relationship with Cartesian coordinates x = r sin cos ,
y = r sin sin , z = r cos
Inverse relationship
p
x2 + y2 + z2
!
p
x2 + y2
1
= tan
z
y
= tan1
x
r=
Dierential line element dl is given by
dl = dr r+ rd + r sin d
Cross products given by = r, r = , and r =
Spherical Polar Coordinates...
Relationship between Cartesian and Spherical unit vectors
r = sin cos i+ sin sin j+ cos k
= cos cos i+ cos sin j sin k
= sin i+ cos j
Area element on the surface of a sphere or radius R ,
dS = dl dl =Rd R sin d = R 2 sin d d r
Area of Rthe surfaceR of a sphere
A = R 2 0 sin d 02 d = 4R 2
Spherical Polar Coordinates...
Elementary volume element
dV = dlr dl dl = drrd r sin d = r 2 sin drd d
Volume of a sphere of radius R
Z R
Z
Z 2
4
2
V=
r dr
sin d
d = R 3
3
0
0
0
