List of top Electromagnetic Theory Questions asked in GATE Physics

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 _______.

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 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

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