List of top Electromagnetic Theory Questions asked in GATE PH

When an unpolarized plane electromagnetic wave in a dielectric medium with refractive index \( n_1 \) is incident on a plane interface separating it from another dielectric medium with refractive index \( n_2 \) (where \( n_2 > n_1 \)), and the angle of incidence is \(\tan^{-1} \left( \frac{n_2}{n_1} \right) \), the following statement is true:

Consider the Lagrangian \( L = m\dot{x}\dot{y} - mw_0^2 zxy \). If \( p_x \) and \( p_y \) denote the generalized momenta conjugate to \( x \) and \( y \), respectively, then the canonical equations of motion are _______.

The X-ray diffraction pattern of a monatomic cubic crystal with rigid spherical atoms of radius 1.56 Å shows several Bragg reflections of which the reflection appearing at the lowest 2𝜃 value is from (111) plane. If the wavelength of X-ray used is 0.78 Å, the Bragg angle (in 2𝜃, rounded off to one decimal place) corresponding to this reflection and the crystal structure, respectively, are

An extrinsic semiconductor shown in the figure carries a current of 2 mA along its length parallel to the \( +x \) axis.

When the majority charge carrier concentration is \( 12.5 \times 10^{13} \, \text{cm}^{-3} \) and the sample is exposed to a constant magnetic field applied along the \( +z \) direction, a Hall voltage of 20 mV is measured with the negative polarity at the \( y = 0 \) plane. Take the electric charge as \( 1.6 \times 10^{-19} \) C. The concentration of minority charge carriers is negligible. Which of the following statements is/are true?

The equation of motion for the forced simple harmonic oscillator is \( \ddot{x}(t) + \omega^2 x(t) = F \cos(\omega t) \) where \( x(t = 0) = 0 \) and \( \dot{x}(t = 0) = 0 \). Which one of the following options is correct?

The electric field in a region depends only on \( x \) and \( y \) coordinates as \( \vec{E} = k \frac{x \hat{x} + y \hat{y}}{x^2 + y^2} \) where \( k \) is a constant. The flux of \( \vec{E} \) through the surface of a sphere of radius \( R \) with its center at the origin is \( n\pi k R \), where the value of \( n \) is ___ (in integer).

Following trial wave functions \( \phi_1 = e^{-Z' (r_1 + r_2)} \) and \( \phi_2 = e^{-Z' (r_1 + r_2) (1 + g |\vec{r}_1 - \bar{\vec{r}}_2|)} \) are used to obtain variational estimates \( E_1 \) and \( E_2 \) for the ground state energy \( E_0 \) of the helium atom. Here, \( Z' \) and \( g \) are variational parameters, \( \bar{\vec{r}}_1 \) and \( \bar{\vec{r}}_2 \) are the position vectors of the electrons. Let \( E_0 \) be the exact ground state energy of the helium atom.\( E_1 \) and \( E_2 \) are the variational estimates of the ground state energy of the helium atom corresponding to \( \phi_1 \) and \( \phi_2 \), respectively, Which one of the following options is true?

The wavefunction for a particle is given by the form \( e^{-i(ax+\beta)} \), where \( a \) and \( \beta \) are real constants. In which one of the following potentials \( V(x) \) does the particle move?

An infinitely long cylinder of radius \( R \) carries a frozen-in magnetization \(\vec{M} = ke^{-s}\hat{z}\), where \( k \) is a constant and \( s \) is the distance from the axis of the cylinder. The magnetic permeability of free space is \(\mu_0\). There is no free current present anywhere. The magnetic flux density (\(\vec{B}\)) inside the cylinder is

A particle is subjected to a potential described as follows:
\[ V(x) = \begin{cases} \infty, & \text{if } x \leq 0 \\ V_0, & \text{if } a \leq x \leq b \\ 0, & \text{elsewhere} \end{cases} \]
Here, \(a > 0\) and \(b > a\). If the energy of the particle \(E < V_0\), which one of the following schematics is a valid quantum mechanical wavefunction (\(\Psi\)) for the system?

An inertial observer sees two spacecrafts S and T flying away from each other along x-axis with individual speed 0.5c, where c is the speed of light. The speed of T with respect to S is

The wavefunction of a particle in an infinite one-dimensional potential well at time \( t \) is
\[ \Psi(x, t) = \sqrt{\frac{2}{3}} e^{-iE_1 t/\hbar}\psi_1(x) + \frac{1}{\sqrt{6}} e^{i\pi/6} e^{-iE_2 t/\hbar} \psi_2(x) + \frac{1}{\sqrt{6}} e^{i\pi/4} e^{-iE_3 t/\hbar} \psi_3(x) \]where \(\psi_1\), \(\psi_2\), and \(\psi_3\) are the normalized ground state, the normalized first excited state, and the normalized second excited state, respectively. \(E_1\), \(E_2\), and \(E_3\) are the eigen-energies corresponding to \(\psi_1\), \(\psi_2\), and \(\psi_3\), respectively. The expectation value of energy of the particle in state \(\Psi(x,t)\) is