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An inductor of \( 10 \, \text{mH} \), capacitor of \( 0.1 \, \mu \text{F} \) and a resistor of \( 100 \, \\Omega \) are connected in series across an a.c power supply \( 220 \, \text{V}, 70 \, \text{Hz} \). The power factor of the given circuit is \( 0.5 \). The difference in the inductive reactance and capacitance reactance is \( \\sqrt{3} \, \\alpha \, \\Omega \). The value of \( \\alpha \) is _______.

The equation of the electric field of an electromagnetic wave propagating through free space is given by:
\( E = \sqrt{377} \sin(6.27 \times 10^{3} t - 2.09 \times 10^{-5} x) \, \text{N}/\text{C} \)
The average power of the electromagnetic wave is \( \left(\frac{1}{\alpha}\right) \, \text{W}/\text{m}^{2} \). The value of \( \alpha \) is ______.
[Take \( \sqrt{\frac{\mu_{0}}{\varepsilon_{0}}} = 377 \) in SI units]

The figure given below shows an LCR series circuit with two switches \( S_1 \) and \( S_2 \). When switch \( S_1 \) is closed keeping \( S_2 \) open, the phase difference \( (\phi) \) between the current and source voltage is \( 30^\circ \) and phase difference is \( 60^\circ \) when \( S_2 \) is closed keeping \( S_1 \) open. The value of \( \left(3L_1 - L_2\right) \) is ________ \( \text{H} \\).

For the series \( LCR \) circuit connected with \( 220 \, \text{V} \), \( 50 \, \text{Hz} \) a.c. source as shown in the figure, the power factor is \( \frac{a}{10} \). The value of \( a \) is ______.

The electric field of an electromagnetic wave travelling through a medium is given by \( \vec{E}(x,t) = 25 \sin \left( 2.0 \times 10^{15} t - 10^{7} x \right) \, \hat{n} \) then the refractive index of the medium is ________. (All given measurement are in SI units)

Match the List-I with List-II.

Choose the correct answer from the options given below:

For an electromagnetic wave propagating through volume \( \vec{k} \), \( \vec{E} \) and \( \omega \) represent propagation vector, electric field and angular frequency, respectively. The magnetic field associated with this wave is represented by:

A series LCR circuit with \( R = 20 \, \Omega \), \( L = 1.6 \, \text{H} \) and \( C = 40 \, \mu \text{F} \) is connected to a variable frequency a.c. source. The inductive reactance at resonant frequency is ______________ \( \Omega \).

An electromagnetic wave travelling in x-direction is described by field equation \( E_y = 300 \sin \omega \left( t - \frac{x}{c} \right) \). If the electron is restricted to move in y-direction only with speed of \( 1.5 \times 10^6 \, \text{m/s} \) then ratio of maximum electric and magnetic forces acting on the electron is ______.

An electromagnetic wave of frequency \( 100 \, \text{MHz} \) propagates through a medium of conductivity, \( \sigma = 10 \, \text{mho/m} \). The ratio of maximum conduction current density to maximum displacement current density is ____________.
[Take \( \dfrac{1}{4\pi \varepsilon_{0}} = 9 \times 10^{9} \, \text{N m}^{2}/\text{C}^{2} \)]

Using a variable frequency a.c. voltage source the maximum current measured in the given LCR circuit is \( 50 \, \text{mA} \) for \( V = 5 \sin(100 t) \). The values of \( L \) and \( R \) are shown in the figure. The capacitance of the capacitor \( (C) \) used is ______ \( \mu\text{F} \).

A displacement current of \( 4.0 \, \text{A} \) can be set up in the space between two parallel plates of \( 6 \, \mu \text{F} \) capacitor. The rate of change of potential difference across the plates of the capacitor is nearly \( \alpha \times 10^{6} \) \( \text{V/s} \). The value of \( \alpha \) is ____.

The ratio of speeds of electromagnetic waves in vacuum and a medium, having dielectric constant \( k = 3 \) and permeability of \( \mu = 2\mu_0 \) is: \( (\mu_0 = \text{permeability of vacuum}) \)

A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first \( 2 \, \text{s} \) it rotates through an angle \( \theta_1 \) and in the next \( 2 \, \text{s} \) it rotates through an angle \( \theta_2 \). The ratio \( \frac{\theta_2}{\theta_1} \) is ______________ .

Inductance of a coil with $10^4$ turns is $10\text{ mH}$ and it is connected to a dc source of $10\text{ V}$ with internal resistance of $10\ \Omega$. The energy density in the inductor when the current reaches $\left( \frac{1}{\text{e}} \right)$ of its maximum value is $\alpha\pi \times \frac{1}{\text{e}^2}\text{ J/m}^3$. The value of $\alpha$ is _________.
$(\mu_0 = 4\pi \times 10^{-7}\text{ Tm / A})$

The electric current in the circuit is given as \( i = i_0 \left( \dfrac{t}{T} \right) \). The r.m.s current for the period \( t = 0 \) to \( t = T \) is _________.

An a.c. source of angular frequency \( \omega \) is connected across a resistor \( R \) and a capacitor \( C \) in series. The current is observed as \( I \). Now the frequency of the source is changed to \( \omega/4 \) (keeping the voltage unchanged) the current is found to be \( I/3 \). The ratio of resistance to reactance at frequency \( \omega \) is

A magnetic field vector in an electromagnetic wave is represented by \( \vec{B} = B_{0}\sin \left( 2\pi \nu t - \frac{2\pi x}{\lambda} \right) \hat{i} \). Its associated electric field vector is ________.

Match List-I with List-II.

Choose the correct answer from the options given below:

A laser beam has intensity of \( 4.0 \times 10^{14} \, \text{W}/\text{m}^{2} \). The amplitude of magnetic field associated with beam is ________ \( \text{T} \). (Take \( \varepsilon_{0} = 8.85 \times 10^{-12} \, \text{C}^{2}/\text{N m}^{2} \) and \( c = 3 \times 10^{8} \, \text{m}/\text{s} \))

A LCR series circuit driven with \( E_{\text{rms}}=90 \, \text{V} \) at frequency \( f_{d}=30 \, \text{Hz} \) has resistance \( R=80\Omega \), an inductance with inductive reactance \( X_{L}=20.0\Omega \) and capacitance with capacitive reactance \( X_{C}=80.0\Omega \). The power factor of the circuit is ____.

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JEE Main 2026 4th April Evening Shift

A magnetic field vector in an electromagnetic wave is represented by \( \vec{B} = B_{0}\sin \left( 2\pi \nu t - \frac{2\pi x}{\lambda} \right) \hat{i} \). Its associated electric field vector is ________.