CSIR NET Physical Sciences Electromagnetic Theory Syllabus

Electromagnetic Theory syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Electromagnetic Theory in CSIR NET Physical Sciences, not a summary of it.

1. 1

   Electrostatics

   2 topics

   • Gauss’s Law and its Applications
   • Laplace and Poisson Equations
     • Boundary Value Problems
2. 2

   Magnetostatics

   2 topics

   • Biot-Savart Law
   • Ampere's Theorem
3. 3

   Electromagnetic Induction

   overview

   Examined as a single unit within Electromagnetic Theory — no further topic split in the official outline.

4. 4

   Maxwell's Equations in Free Space and Linear Isotropic Media

   1 topic

   • Boundary Conditions on the Fields at Interfaces
5. 5

   Scalar and Vector Potentials

   1 topic

   • Gauge Invariance
6. 6

   Electromagnetic Waves in Free Space

   overview

   Examined as a single unit within Electromagnetic Theory — no further topic split in the official outline.

7. 7

   Dielectrics and Conductors

   overview

   Examined as a single unit within Electromagnetic Theory — no further topic split in the official outline.

8. 8

   Reflection and Refraction

   2 topics

   • Polarization
   • Fresnel’s Law
9. 9

   Interference, Coherence, and Diffraction

   overview

   Examined as a single unit within Electromagnetic Theory — no further topic split in the official outline.

10. 10

    Dynamics of Charged Particles in Static and Uniform Electromagnetic Fields

    overview

    Examined as a single unit within Electromagnetic Theory — no further topic split in the official outline.

Electromagnetic Theory flashcards for CSIR NET Physical Sciences

23 of 50 cards from the Electromagnetic Theory deck — real questions with worked answers.

1. What is a dispersion relation in the context of electromagnetic wave propagation?

   A dispersion relation is the functional relationship between the angular frequency $\omega$ and the wave vector $k$, i.e. $\omega = \omega(k)$. It determines how the phase and group velocities depend on frequency, and hence how a medium spreads (disperses) a wave packet.

2. Define phase velocity and group velocity in terms of the dispersion relation.

   Phase velocity: $v_{p} = \dfrac{\omega}{k}$ (speed of a constant-phase point). Group velocity: $v_{g} = \dfrac{d\omega}{dk}$ (speed of energy/information and of the wave-packet envelope).

3. What distinguishes a non-dispersive medium from a dispersive medium?

   In a non-dispersive medium $\omega = vk$ with $v$ constant, so $v_{p} = v_{g}$ and a pulse keeps its shape. In a dispersive medium $\omega(k)$ is nonlinear, so $v_{p} \neq v_{g}$ and a wave packet broadens as it propagates.

4. State the dispersion relation for a collisionless cold plasma and define the plasma frequency.

   $$\omega^{2} = \omega_{p}^{2} + c^{2}k^{2}, \qquad \omega_{p} = \sqrt{\frac{n e^{2}}{m \varepsilon_{0}}}$$ where $n$ is electron density. Waves propagate only for $\omega > \omega_{p}$; below $\omega_{p}$, $k$ is imaginary and the wave is evanescent (reflected).

5. For the plasma dispersion relation $\omega^{2} = \omega_{p}^{2} + c^{2}k^{2}$, show the relation between phase and group velocity.

   $v_{p} = \dfrac{\omega}{k} = \dfrac{c}{\sqrt{1 - \omega_{p}^{2}/\omega^{2}}} > c$, and $v_{g} = \dfrac{d\omega}{dk} = \dfrac{c^{2}k}{\omega} = c\sqrt{1 - \omega_{p}^{2}/\omega^{2}} < c$. Their product is $v_{p}v_{g} = c^{2}$.

6. What is anomalous dispersion?

   Anomalous dispersion is the frequency region (near a resonance/absorption line) where the refractive index decreases with increasing frequency, $\dfrac{dn}{d\omega} < 0$. Outside this region (normal dispersion) $n$ increases with $\omega$.

7. State the Kramers–Kronig relations and what they connect.

   They connect the real and imaginary parts of a causal response function such as $\varepsilon(\omega) = \varepsilon_{1} + i\varepsilon_{2}$: $$\varepsilon_{1}(\omega) - 1 = \frac{2}{\pi}\,\mathcal{P}\!\int_{0}^{\infty}\frac{\omega' \varepsilon_{2}(\omega')}{\omega'^{2}-\omega^{2}}\,d\omega'.$$ They follow from causality (dispersion implies absorption and vice versa).

8. In the Lorentz (Drude–Lorentz) oscillator model, what is the frequency dependence of the dielectric function?

   $$\varepsilon(\omega) = 1 + \frac{N e^{2}}{m \varepsilon_{0}} \sum_{j} \frac{f_{j}}{\omega_{j}^{2} - \omega^{2} - i\gamma_{j}\omega}$$ where $\omega_{j}$ are resonance frequencies, $\gamma_{j}$ damping rates, and $f_{j}$ oscillator strengths.

9. Why can the phase velocity exceed $c$ without violating special relativity?

   Because no energy or information travels at the phase velocity. Signals and energy propagate at the group velocity (or signal/front velocity), which remains $\leq c$. Phase velocity is merely the speed of a pure, infinitely long sinusoid's wavefronts.

10. Write the four Maxwell equations in vacuum in differential form.

    $$\nabla\cdot\vec{E} = \frac{\rho}{\varepsilon_{0}}, \quad \nabla\cdot\vec{B} = 0,$$ $$\nabla\times\vec{E} = -\frac{\partial \vec{B}}{\partial t}, \quad \nabla\times\vec{B} = \mu_{0}\vec{J} + \mu_{0}\varepsilon_{0}\frac{\partial \vec{E}}{\partial t}.$$

11. What is the electromagnetic field strength tensor $F^{\mu\nu}$ and why is it introduced?

    $F^{\mu\nu} = \partial^{\mu}A^{\nu} - \partial^{\nu}A^{\mu}$ is an antisymmetric rank-2 tensor whose components are the $\vec{E}$ and $\vec{B}$ fields, e.g. $F^{0i} = -E^{i}/c$ and $F^{ij} = -\varepsilon^{ijk}B_{k}$. It makes the Lorentz covariance of electromagnetism manifest.

12. Write Maxwell's equations in manifestly covariant (tensor) form.

    Inhomogeneous: $\partial_{\mu}F^{\mu\nu} = \mu_{0} J^{\nu}$. Homogeneous (Bianchi): $\partial_{\mu}F_{\nu\lambda} + \partial_{\nu}F_{\lambda\mu} + \partial_{\lambda}F_{\mu\nu} = 0$, equivalently $\partial_{\mu}\tilde{F}^{\mu\nu} = 0$ with $\tilde{F}$ the dual tensor.

13. What is the four-current $J^{\mu}$ and what conservation law does $\partial_{\mu}J^{\mu}=0$ express?

    $J^{\mu} = (c\rho, \vec{J})$. The condition $\partial_{\mu}J^{\mu} = 0$ is the covariant continuity equation $\dfrac{\partial \rho}{\partial t} + \nabla\cdot\vec{J} = 0$, expressing local conservation of electric charge.

14. Name the two Lorentz invariants formed from the electromagnetic field tensor and give their field expressions.

    $$F_{\mu\nu}F^{\mu\nu} = 2\left(B^{2} - \frac{E^{2}}{c^{2}}\right), \qquad F_{\mu\nu}\tilde{F}^{\mu\nu} \propto \vec{E}\cdot\vec{B}.$$ Thus $B^{2} - E^{2}/c^{2}$ and $\vec{E}\cdot\vec{B}$ are the same in all inertial frames.

15. How do the parallel and perpendicular components of $\vec{E}$ transform under a Lorentz boost with velocity $\vec{v}$?

    $E_{\parallel}' = E_{\parallel}$, and $\vec{E}_{\perp}' = \gamma\left(\vec{E} + \vec{v}\times\vec{B}\right)_{\perp}$, where $\gamma = (1 - v^{2}/c^{2})^{-1/2}$. The component along the boost is unchanged; the transverse component mixes with $\vec{B}$.

16. How do the parallel and perpendicular components of $\vec{B}$ transform under a Lorentz boost with velocity $\vec{v}$?

    $B_{\parallel}' = B_{\parallel}$, and $\vec{B}_{\perp}' = \gamma\left(\vec{B} - \dfrac{\vec{v}\times\vec{E}}{c^{2}}\right)_{\perp}$. A pure electric field in one frame can appear as a combination of $\vec{E}$ and $\vec{B}$ in another.

17. What is the four-potential $A^{\mu}$, and write the Lorenz gauge condition covariantly.

    $A^{\mu} = (\phi/c, \vec{A})$. The Lorenz gauge condition is $\partial_{\mu}A^{\mu} = 0$, i.e. $\dfrac{1}{c^{2}}\dfrac{\partial \phi}{\partial t} + \nabla\cdot\vec{A} = 0$. In this gauge $\Box A^{\mu} = \mu_{0}J^{\mu}$.

18. Write the covariant form of the Lorentz force law (equation of motion of a charge).

    $$\frac{dp^{\mu}}{d\tau} = q\,F^{\mu\nu} u_{\nu}$$ where $p^{\mu}$ is the four-momentum, $u_{\nu}$ the four-velocity, $\tau$ proper time. Its spatial part reduces to $\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})$.

19. What is the relativistic Doppler shift formula for light?

    For a source receding with speed $v$ (longitudinal): $$\omega_{\text{obs}} = \omega_{0}\sqrt{\frac{1-\beta}{1+\beta}}, \quad \beta = v/c.$$ Transverse Doppler effect: $\omega_{\text{obs}} = \omega_{0}/\gamma$, a purely relativistic time-dilation redshift.

20. What is the characteristic impedance $Z_{0}$ of a lossless transmission line in terms of its distributed parameters?

    $$Z_{0} = \sqrt{\frac{L}{C}}$$ where $L$ and $C$ are the inductance and capacitance per unit length. For a general (lossy) line, $Z_{0} = \sqrt{\dfrac{R + i\omega L}{G + i\omega C}}$.

21. Write the telegrapher's equations for a transmission line.

    $$\frac{\partial V}{\partial z} = -\left(R + L\frac{\partial}{\partial t}\right)I, \qquad \frac{\partial I}{\partial z} = -\left(G + C\frac{\partial}{\partial t}\right)V.$$ They describe voltage and current as functions of position $z$ and time on the line.

22. Give the propagation constant $\gamma$ of a transmission line and identify its parts.

    $$\gamma = \sqrt{(R + i\omega L)(G + i\omega C)} = \alpha + i\beta$$ where $\alpha$ is the attenuation constant (Np/m) and $\beta$ is the phase constant (rad/m). For a lossless line $\alpha = 0$, $\beta = \omega\sqrt{LC}$.

23. Define the reflection coefficient $\Gamma$ at a transmission-line load.

    $$\Gamma = \frac{Z_{L} - Z_{0}}{Z_{L} + Z_{0}}$$ where $Z_{L}$ is the load impedance and $Z_{0}$ the line's characteristic impedance. $\Gamma = 0$ for a matched load, $\Gamma = -1$ for a short, $\Gamma = +1$ for an open.

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