Syllabus:
i) Brief review of vector calculus. Physical interpretation of gradient, divergence
and curl; Statement and physical interpretations of Greens theorem, Gauss divergence
theorem, Stokes curl theorem. Differential forms (in R3); gradient,
divergence, curl as co-boundaries (ds) of differential forms (in R3); Statement
of Stokes theorem for differential forms (in R3); Greens theorem, the divergence
theorem, the curl theorem as special cases (Generalized Fundamental
Theorem of Calculus). Spherical coordinates; Cylindrical coordinates. Dirac
delta function in in one/two/three dimensions; Delta function as divergence of a
radially outward vector field; Justification for treating Dirac delta as a function;
Remarks on Schwartzs distribution theory. Vector fields and potentials.
ii) Electrostatics. Coulombs Law for discrete and continuous charge distributions;
Divergence and curl of electrostatic fields. Electric potential; Poissons equation
and Laplaces equation; Electrostatic Boundary Conditions; General remarks
on Greens function (Impulse response). Work and Energy in Electrostatics.
Conductors; Surface Charge and the Force on a Conductor; Capacitors.
iii) Potential and field due to arrangement of charges. Solution to Laplaces equation;
Harmonic Functions; Mean-value property; Illustration in One Dimension,
Two Dimensions, Three Dimensions. Boundary Conditions and Uniqueness
Theorems for Laplaces equation; Application to conductors.
iv) The Method of Images. Separation of variables. Multipole Expansion; Monopole
and Dipole terms; The Electric Field of a Dipole. Dielectrics; Polarization;
Electric displacement.
v) Magnetostatics. Lorentz Force Law; Magnetic fields; Currents. Biot-Savart
Law; Steady Currents; Magnetic Field of a Steady Current. Divergence and
Curl of Magnetic field; Amperes Law; Maxwells Equations for Electrostatics
and Magnetostatics. Magnetic vector potential.
vi) Electromotive Force; Ohms Law. Electromagnetic Induction; Faradays Law;
Inductance; Energy in magnetic field. Maxwells correction to Amperes law
for magnetodynamics; Maxwells Equations - differential and integral form;
The Conundrum of Magnetic Charge/Monopole.
vii) Conservation Laws; The Continuity Equation; Poyntings work-energy theorem
of electrodynamics. Maxwells Stress Tensor; Conservation of Momentum.
Electromagnetic Waves. The Wave Equation; Sinusoidal Waves; General
remarks on the Fourier transform; Polarization. ElectromagneticWaves in Vacuum;
The Wave Equation for E and B; Monochromatic Plane Waves; Energy
and Momentum in Electromagnetic Waves.
viii) Special Theory of Relativity from Maxwells electrodynamics; Einsteins thoughtexperiment
and postulates. Relativity of simultaneity; Time dilation; Lorentz
length contraction. The Lorentz group of transformations; The Structure of
Spacetime; The Lorentz Metric; Space-time diagrams. Remarks on magnetism
as a relativistic phenomenon.
Reference Texts:
(a) Introduction to Electrodynamics - D. J. Griffiths.
(b) Foundations of Electromagnetic theory - J. R. Reitz, F. J. Milford andW. Charisty.
(c) (Chapter 5) A Visual Introduction to Differential Forms and Calculus on Manifolds
- J. P. Fortney.
(d) Theory and Problems of Electromagnetics (Schaums Outlines) - J. A. Edminister.
(e) A Guide to Physics Problems part 1: Mechanics, Relativity and Electrodynamics
- S. B. Cahn and B. E. Badgorny.
https://www.isibang.ac.in/~adean/infsys/database/Bmath/Ele.html
- Teacher: Prabuddha Chakraborty