|Course Archives Theoretical Statistics and Mathematics Unit|
Time: Currently not offered
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.
(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.
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[ Semester Schedule ][ Statmath Unit ] [Indian Statistical Institute]