GATE Physics Electromagnetic Theory Syllabus

State the relative orientation of $\vec{E}$, $\vec{B}$, and the propagation direction $\hat{k}$ for a plane EM wave in free space.

They are mutually perpendicular and form a right-handed set: $\hat{k} = \hat{E}\times\hat{B}$. The wave is transverse.

Define the Poynting vector and give its expression.

It is the energy flux (power per unit area) carried by the EM field: $$\vec{S} = \frac{1}{\mu_{0}}\vec{E}\times\vec{B} = \varepsilon_{0}c^{2}\,\vec{E}\times\vec{B}$$

What is the time-averaged intensity of a plane EM wave in free space with amplitude $E_{0}$?

$$\langle S\rangle = \frac{1}{2}\varepsilon_{0}c E_{0}^{2} = \frac{E_{0}^{2}}{2\mu_{0}c}$$

What is the intrinsic impedance of free space?

$$\eta_{0} = \sqrt{\frac{\mu_{0}}{\varepsilon_{0}}} \approx 377\ \Omega$$

Give the energy density of an EM wave in free space (electric plus magnetic).

$$u = \varepsilon_{0}E^{2} = \frac{1}{2}\varepsilon_{0}E^{2} + \frac{B^{2}}{2\mu_{0}}$$ where the electric and magnetic contributions are equal.

In a linear non-conducting (lossless dielectric) medium, what is the phase velocity of an EM wave?

$$v = \frac{1}{\sqrt{\mu\varepsilon}} = \frac{c}{n},\qquad n = \sqrt{\frac{\mu\varepsilon}{\mu_{0}\varepsilon_{0}}}\approx\sqrt{\varepsilon_{r}}$$

Define the refractive index $n$ of a non-magnetic dielectric in terms of relative permittivity.

$$n = \sqrt{\varepsilon_{r}\mu_{r}} \approx \sqrt{\varepsilon_{r}}\quad(\mu_{r}\approx 1)$$

What is the intrinsic impedance of a lossless non-conducting medium?

$$\eta = \sqrt{\frac{\mu}{\varepsilon}} = \frac{\eta_{0}}{n}\quad(\text{for }\mu=\mu_{0})$$

In a non-conducting medium, is the EM wave attenuated as it propagates? Why or why not?

No. With zero conductivity ($\sigma = 0$) there is no Ohmic loss, so the wave is undamped; the wave number is real and amplitude stays constant.

How does the wavelength of an EM wave in a dielectric compare to that in vacuum (same frequency)?

It is shorter by the factor $n$: $$\lambda_{\text{medium}} = \frac{\lambda_{0}}{n}$$ while the frequency is unchanged.

Write the general dispersion relation for a plane wave $e^{i(kz-\omega t)}$ in a conducting medium.

$$k^{2} = \mu\varepsilon\omega^{2} + i\mu\sigma\omega$$ giving a complex wave number $k = k_{r} + i\kappa$.

In a conducting medium, what is the physical meaning of the imaginary part $\kappa$ of the complex wave number?

It is the attenuation (absorption) coefficient: the wave amplitude decays as $e^{-\kappa z}$, representing Ohmic energy loss in the conductor.

Define the skin depth in a conductor and give its expression for a good conductor.

The skin depth $\delta$ is the distance over which the amplitude falls to $1/e$: $$\delta = \frac{1}{\kappa} \approx \sqrt{\frac{2}{\mu\sigma\omega}}$$

What dimensionless ratio distinguishes a good conductor from a good (poor-loss) dielectric?

The loss tangent $\frac{\sigma}{\varepsilon\omega}$. Good conductor: $\frac{\sigma}{\varepsilon\omega}\gg 1$; good dielectric: $\frac{\sigma}{\varepsilon\omega}\ll 1$.

In a good conductor, what is the phase relationship between $\vec{E}$ and $\vec{B}$?

$\vec{B}$ lags $\vec{E}$ by a phase of $45^{\circ}$ ($\frac{\pi}{4}$ radians); they are no longer in phase as in free space.

Write the complex (lossy) intrinsic impedance of a conducting medium.

$$\eta = \sqrt{\frac{i\mu\omega}{\sigma + i\varepsilon\omega}}$$ which is complex, giving the $45^{\circ}$ phase for a good conductor where $\eta \approx \sqrt{\frac{\mu\omega}{2\sigma}}(1+i)$.

For a good conductor, give the approximate real and imaginary parts of $k$.

$$k_{r} \approx \kappa \approx \sqrt{\frac{\mu\sigma\omega}{2}}$$ so the attenuation length and wavelength are comparable.

Why does an EM wave penetrate only a short distance into a good conductor at high frequency?

Skin depth $\delta = \sqrt{\frac{2}{\mu\sigma\omega}}$ decreases with increasing frequency $\omega$ and conductivity $\sigma$, so high-frequency fields are confined near the surface (skin effect).

At normal incidence on a boundary between two non-conducting media, write the reflection coefficient for the field amplitude.

$$r = \frac{E_{0R}}{E_{0I}} = \frac{n_{1}-n_{2}}{n_{1}+n_{2}} = \frac{\eta_{2}-\eta_{1}}{\eta_{2}+\eta_{1}}$$

At normal incidence between two dielectrics, write the transmission coefficient for the field amplitude.

$$t = \frac{E_{0T}}{E_{0I}} = \frac{2n_{1}}{n_{1}+n_{2}} = \frac{2\eta_{2}}{\eta_{2}+\eta_{1}}$$

Write the reflectance $R$ and transmittance $T$ at normal incidence in terms of refractive indices.

$$R = \left(\frac{n_{1}-n_{2}}{n_{1}+n_{2}}\right)^{2},\qquad T = \frac{4 n_{1} n_{2}}{(n_{1}+n_{2})^{2}},\qquad R+T=1$$

At normal incidence, what happens to the phase of the reflected wave when it goes from a rarer to a denser medium ($n_{2}>n_{1}$)?

The reflected electric field undergoes a phase change of $\pi$ (180°), since $r=\frac{n_{1}-n_{2}}{n_{1}+n_{2}}<0$.

What fraction of light is reflected at normal incidence from an air–glass interface ($n_{1}=1$, $n_{2}=1.5$)?

$$R = \left(\frac{1-1.5}{1+1.5}\right)^{2} = \left(\frac{-0.5}{2.5}\right)^{2} = 0.04\;(4\%)$$

State Snell's law for oblique incidence at a dielectric interface.

$$n_{1}\sin\theta_{i} = n_{2}\sin\theta_{t}$$ where $\theta_{i}$ is the incidence angle and $\theta_{t}$ the refraction (transmission) angle.

State the law of reflection arising from boundary matching at oblique incidence.

The angle of incidence equals the angle of reflection, $\theta_{i}=\theta_{r}$, and the incident, reflected, transmitted rays and the normal all lie in the same plane (plane of incidence).

Define Brewster's angle and give its formula.

The incidence angle at which the parallel-polarized (p) reflected wave vanishes: $$\tan\theta_{B} = \frac{n_{2}}{n_{1}}$$ At $\theta_{B}$ the reflected light is completely perpendicularly (s) polarized.

At Brewster's angle, what is the angular relationship between the reflected and transmitted (refracted) rays?

They are perpendicular: $\theta_{B} + \theta_{t} = 90^{\circ}$.

Write the Fresnel amplitude reflection coefficient for s-polarization (perpendicular, TE) at oblique incidence.

$$r_{s} = \frac{n_{1}\cos\theta_{i} - n_{2}\cos\theta_{t}}{n_{1}\cos\theta_{i} + n_{2}\cos\theta_{t}}$$

Write the Fresnel amplitude reflection coefficient for p-polarization (parallel, TM) at oblique incidence.

$$r_{p} = \frac{n_{2}\cos\theta_{i} - n_{1}\cos\theta_{t}}{n_{2}\cos\theta_{i} + n_{1}\cos\theta_{t}}$$

Define the critical angle for total internal reflection and give its formula.

For a wave going from a denser to a rarer medium ($n_{1}>n_{2}$), total internal reflection occurs for $\theta_{i}>\theta_{c}$ where $$\sin\theta_{c} = \frac{n_{2}}{n_{1}}$$

Which field components are continuous across a boundary in solving oblique-incidence EM problems?

The tangential components of $\vec{E}$ and $\vec{H}$ are continuous, and the normal components of $\vec{D}$ and $\vec{B}$ are continuous (in the absence of free surface charge/current).