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JEE Mains | PYQ

In the given electromagnetic wave \( E_{y} = 600 \, \sin(\omega t - kx) \, \text{V m}^{-1} \), intensity of the associated light beam is 
(in \( \frac{W}{m^{2}} \) : Given \( \epsilon_{0} = 9 \times 10^{-12} \, \text{C}^{2} \, \text{N}^{-1} \, \text{m}^{-2} \))

A circuit element \( X \) when connected to an a.c. supply of peak voltage \( 100\,V \) gives a peak current of \( 5\,A \) which is in phase with the voltage. A second element \( Y \) when connected to the same a.c. supply also gives the same value of peak current which lags behind the voltage by \( \dfrac{\pi}{2} \). If \( X \) and \( Y \) are connected in series to the same supply, what will be the rms value of the current in ampere?

If frequency of electromagnetic wave is \( 60 \, \text{MHz} \) and it travels in air along \( z \) direction then the corresponding electric and magnetic field vectors will be mutually perpendicular to each other and the wavelength of the wave (in \( \text{m} \)) is:

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Choke coil is simply a coil having a large inductance but a small resistance. Choke coils are used with fluorescent mercury-tube fittings. If household electric power is directly connected to a mercury tube, the tube will be damaged.
Reason (R): By using the choke coil, the voltage across the tube is reduced by a factor \( \left(\frac{R}{\sqrt{R^{2}+\omega^{2} L^{2}}}\right) \), where \( \omega \) is frequency of the supply across resistor \( R \) and inductor \( L \). If the choke coil were not used, the voltage across the resistor would be the same as the applied voltage.
In the light of the above statements, choose the most appropriate answer from the options given below:

A telegraph line of length \( 100 \, \mathrm{km} \) has a capacity of \( 0.01 \, \mu \mathrm{F}/\mathrm{km} \) and it carries an alternating current at \( 0.5 \) kilo cycle per second. If minimum impedance is required, then the value of the inductance that needs to be introduced in series is ________ \( \mathrm{mH} \). (if \( \pi = \sqrt{10} \) ).

An alternating current is given by \( I = I_A \sin \omega t + I_B \cos \omega t \). Then r.m.s current will be:

Light wave traveling in air along \( x \)-direction is given by \( E_y = 540\sin \pi \times 10^4(x-ct)\,\text{V m}^{-1} \). Then, the peak value of magnetic field of wave will be: (Given \( c=3\times 10^8 \,\text{ms}^{-1} \))

In an a.c. circuit, voltage and current are given by: \( V = 100 \sin(100 t) \, \text{V} \) and \( I = 100 \sin(100 t + \dfrac{\pi}{3}) \, \text{mA} \) respectively. The average power dissipated in one cycle is:

A series LCR circuit with \(\displaystyle  L = \frac{100}{\pi} \, \text{mH} \), \(\displaystyle C = \frac{10^{-3}}{\pi} \, \text{F} \) and \( R = 10 \, \Omega \), is connected across an ac source of \( 220 \, \text{V} \), \( 50 \, \text{Hz} \) supply. The power factor of the circuit would be __________ .

An ac source is connected in given series LCR circuit. The rms potential difference across the capacitor of \( 20 \, \mu \text{F} \) is __________ \( \text{V} \).

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]

Match List-I with List-II:

Choose the most appropriate answer from the options given below:

An AC voltage \( V = 20 \sin 200\pi t \) is applied to a series LCR circuit which drives a current \( I = 10 \sin\left(200\pi t + \dfrac{\pi}{3}\right) \). The average power dissipated is:

Given below are two statements:
Statement-I: The reactance of an ac circuit is zero. It is possible that the circuit contains a capacitor and an inductor.
Statement-II: In ac circuit, the average power delivered by the source never becomes zero.
In the light of the above statements, choose the correct answer from the options given below:

A series \( LCR \) circuit is connected to an alternating source of emf \( E \). The current amplitude at resonant frequency is \( I_0 \). If the value of resistance \( R \) becomes twice of its initial value then amplitude of current at resonance will be:

For the given figures, choose the correct options:

In a series LCR circuit, a resistor of \( 300 \, \Omega \), a capacitor of \( 25 \, \text{nF} \) and an inductor of \( 100 \, \text{mH} \) are used. For maximum current in the circuit, the angular frequency of the ac source is __________ \( \times 10^{4} \) radians \( \text{s}^{-1} \).

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} \\).

An LCR series circuit of capacitance \( 62.5 \, \text{nF} \) and resistance of \( 50 \, \Omega \), is connected to an A.C. source of frequency \( 2.0 \, \text{kHz} \). For maximum value of amplitude of current in circuit, the value of inductance is ______________ \( \text{mH} \). (Take \( \pi^{2} = 10 \))

An AC source is connected to an inductance of \( 100 \, \text{mH} \), a capacitance of \( 100 \, \mu \text{F} \) and a resistance of \( 120 \, \Omega \) as shown in figure. The time in which the resistance having a thermal capacity \( 2 \, \text{J}/^\circ\text{C} \) will get heated by \( 16^\circ \text{C} \) is ______ \( \text{s} \).

The electric field of an electromagnetic wave in free space is \( \vec{E} = 57 \cos \!\left[ 7.5 \times 10^{6} t - 5 \times 10^{-3}(3x + 4y) \right] (4\hat{i} - 3\hat{j}) \, \text{N} / \text{C} \). The associated magnetic field in Tesla is:

A resistance of \( 40 \, \Omega \) is connected to a source of alternating current rated \( 220 \, \text{V} \), \( 50 \, \text{Hz} \). Find the time taken by the current to change from its maximum value to the rms value:

Primary coil of a transformer is connected to \( 220 \, \text{V} \) a.c. Primary and secondary turns of the transforms are \( 100 \) and \( 10 \) respectively. Secondary coil of transformer is connected to two series resistances shown in figure. The output voltage \( (V_0) \) is:

An alternating current is represented by the equation, \( i=100\sqrt{2}\sin(100\pi t) \) ampere. The RMS value of current and the frequency of the given alternating current are:

In the given circuit, the magnitude of \( V_L \) and \( V_C \) are twice that of \( V_R \). Given that \( f = 50 \, \text{Hz} \), the inductance of the coil is \( \dfrac{1}{K \pi} \, \text{mH} \). The value of \( K \) is __________.

Which of the following Maxwell’s equation is valid for time varying conditions but not valid for static conditions:

A series L.R circuit connected with an ac source \( E = (25 \sin 1000 t) \, \text{V} \) has a power factor of \( \frac{1}{\sqrt{2}} \). If the source of emf is changed to \( E = (20 \sin 2000 t) \, \text{V} \), the new power factor of the circuit will be:

A capacitor reactance \( 4\sqrt{3} \, \Omega \) and a resistor of resistance \( 4 \, \Omega \) are connected in series with an ac source of peak value \( 8\sqrt{2} \, V \). The power dissipation in the circuit is ......\( W \).

An inductor of \( 0.5 \, \text{mH} \), a capacitor of \( 20 \, \mu \text{F} \) and resistance of \( 20 \, \Omega \) are connected in series with a \( 220 \, \text{V} \) \( \text{ac} \) source. If the current is in phase with the emf, the amplitude of current of the circuit is \( \sqrt{x} \) \( \text{A} \). The value of \( x \) is:

Given below are two statements:
Statement I: In an LCR series circuit, current is maximum at resonance.
Statement II: Current in a purely resistive circuit can never be less than that in a series LCR circuit when connected to same voltage source.
In the light of the above statements, choose the correct from the options given below:

A plane electromagnetic wave propagates along the \( +x \) direction in free space. The components of the electric field, \( \vec{E} \) and magnetic field, \( \vec{B} \) vectors associated with the wave in Cartesian frame are:

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 ______.

What happens to the inductive reactance and the current in a purely inductive circuit if the frequency is halved?

In the circuit shown in the figure, the ratio of the quality factor and the band width is ______ \( \text{s} \).

Match List I with List II.

Choose the correct answer from the options given below:

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)

In a series LCR resonance circuit, if we change the resistance only, from a lower to higher value:

As shown in the figure an inductor of inductance \( 200 \, \text{mH} \) is connected to an AC source of emf \( 220 \, \text{V} \) and frequency \( 50 \, \text{Hz} \). The instantaneous voltage of the source is \( 0 \, \text{V} \) when the peak value of current is \( \frac{\sqrt{a}}{\pi} \, \text{A} \). The value of \( a \) is ________.

Match the List-I with List-II.

Choose the correct answer from the options given below:

If wattless current flows in the AC circuit, then the circuit is:

A capacitor of capacitance \( 150.0 \, \mu \text{F} \) is connected to an alternating source of emf given by \( E = 36 \sin(120 \pi t) \, \text{V} \). The maximum value of current in the circuit is approximately equal to:

An ac current is represented as \( i = 5\sqrt{2} + 10\cos\!\left(650\pi t + \frac{\pi}{6}\right)\ \text{Amp} \). The r.m.s value of the current is:

The waves emitted when a metal target is bombarded with high energy electrons are:

An ac circuit has an inductor and a resistor of resistance \( R \) in series, such that \( X_L = 3R \). Now, a capacitor is added in series such that \( X_C = 2R \). The ratio of new power factor with the old power factor of the circuit is \( \sqrt{5} : x \). The value of \( x \) is ________.

An alternating emf \( E = 440\sin 100\pi t \) is applied to a circuit containing an inductance of \( \dfrac{\sqrt{2}}{\pi} \, \text{H} \). If an a.c. ammeter is connected in the circuit, its reading will be:

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:

The series LCR circuit is connected to an ac source of \( 220 \, \text{V} \), \( 50 \, \text{Hz} \). The circuit contain a resistance \( R = 100\Omega \) and an inductor of inductive reactance \( X_L = 79.6\Omega \). The capacitance of the capacitor needed to maximize the average rate at which energy is supplied will be \( \_\_\_\_\_ \) \( \mu \text{F} \).

In a series LR circuit \( X_L = R \) and power factor of the circuit is \( P_1 \). When capacitor with capacitance \( C \) such that \( X_L = X_C \) is put in series, the power factor becomes \( P_2 \). Then ratio \( \dfrac{P_1}{P_2} \) is:

The current flowing through an ac circuit is given by \( I = 5 \sin (120\pi t) \, \text{A} \)
How long will the current take to reach the peak value starting from zero?

A series combination of resistor of resistance \( 100 \, \Omega \), inductor of inductance \( 1 \, \text{H} \) and capacitor of capacitance \( 6.25 \, \mu \text{F} \) if connected to an ac source. The quality factor of the circuit will be ________.

Given below are two statements:
Statement I : Maximum power is dissipated in a circuit containing an inductor, a capacitor and a resistor connected in series with an AC source, when resonance occurs
Statement II : Maximum power is dissipated in a circuit containing pure resistor due to zero phase difference between current and voltage.
In the light of the above statement, choose the correct answer from the potions given below:

To light, a \( 50 \, \text{W} \), \( 100 \, \text{V} \) lamp is connected, in series with a capacitor of capacitance \( \dfrac{50}{\pi \sqrt{x}} \, \mu \text{F} \), with \( 200 \, \text{V} \), \( 50 \, \text{HZ} \) AC source. The value of \( x \) will be ______.

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 ______________ \( \Omega \).

To increase the resonant frequency in series LCR circuit.

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 alternating voltage source \( V = 260 \sin(628 t) \) is connected across a pure inductor of \( 5 \, \text{mH} \). Inductive reactance in the circuit is:

Two circuits are shown in the figure (a) & (h). At a frequency of ______ \( \text{rad /s} \) the average power dissipated in one cycle will be same in both the circuits.

An AC source rated \( 220 \, \text{V} \), \( 50 \, \text{Hz} \) is connected to a resistor. The time taken by the current to change from its maximum to the rms value is:

A series LCR circuit is subjected to an ac signal of \( 200\,\text{V} \), \( 50\,\text{Hz} \). If the voltage across the inductor (\( L = 10\,\text{mH} \)) is \( 31.4\,\text{V} \), then the current in this circuit is ________.

In a series LCR circuit, the inductive reactance \( (X_L) \) is \( 10 \, \Omega \) and the capacitive reactance \( (X_C) \) is \( 4 \, \Omega \). The resistance \( (R) \) in the circuit is \( 6 \, \Omega \). The power factor of the circuit 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 \( \dfrac{1}{4\pi \varepsilon_{0}} = 9 \times 10^{9} \, \text{N m}^{2}/\text{C}^{2} \)]

The magnetic field of an E.M. wave is given by \( \vec{B} = \left(\displaystyle \frac{\sqrt{3}}{2}\,\hat{i} + \frac{1}{2}\,\hat{j} \right) \, 30 \, \sin \left[ \omega \left(\displaystyle t - \frac{z}{c} \right) \right] \) (S.I. Units). The corresponding electric field in S.I. units is:

A \( 72 \, \Omega \) galvanometer is shunted by a resistance of \( 8 \, \Omega \). The percentage of the total current which passes through the galvanometer is:

The magnetic field in a plane electromagnetic wave is \( B_{y} = (3.5 \times 10^{-7}) \sin (1.5 \times 10^{3} x + 0.5 \times 10^{11} t) \, \text{T} \). The corresponding electric field will be

Intensity of sunlight is observed as \( 0.092 \, \mathrm{W \, m^{-2}} \) at a point in free space. What will be the peak value of magnetic field at the point? \(( \varepsilon_{0} = 8.85 \times 10^{-12} \, \mathrm{C^{2} \, N^{-1} \, m^{-2}}) \)

The energy density associated with electric field \( \vec{E} \) and magnetic field \( \vec{B} \) of an electromagnetic wave in free space is given by ( \( \varepsilon_{0} \) – permittivity of free space, \( \mu_{0} \) – permeability of free space)

In a circuit consisting of a capacitance and a generator with alternating emf \( E_g = E_g \sin \omega t \), \( V_c \) and \( I_c \) are the voltage and current. Correct phasor diagram for such circuit is:

A \( 220 \, \text{V}, 50 \, \text{Hz} \) AC source is connected to a \( 25 \, \text{V}, 5 \, \text{W} \) lamp and an additional resistance \( R \) in series (as shown in figure) to run the lamp at its peak brightness, then the value of \( R \) (in ohm) will be ______.

As per the given graph, choose the correct representation for curve A and curve B.
{Where \( X_C \) = reactance of pure capacitive circuit connected with A.C. source
\( X_L \) = reactance of pure inductive circuit connected with A.C. source
\( R \) = impedance of pure resistive circuit connected with A.C. source.
\( Z \) = Impedance of the LCR series circuit}

A \( 0.07 \, \text{H} \) inductor and a \( 12 \, \Omega \) resistor are correct in series to a \( 220 \, \text{V} \), \( 50 \, \text{Hz} \) ac source. The approximate current in the circuit and the phase angle between current and source voltage are respectively. \([\text{Take } \pi \text{ as } \tfrac{22}{7}]\)

A \( \text{LCR} \) circuit is at resonance for a capacitor \( C \), inductance \( L \) and resistance \( R \). Now the value of resistance is halved keeping all other parameters same. The current amplitude at resonance will be now:

An alternating emf \( E = 110\sqrt{2} \, \sin100 t \) volt is applied to a capacitor of \( 2\,\mu \text{F} \), the rms value of current in the circuit is ______ \( \text{mA} \).

A series LCR circuit is designed to resonate at an angular frequency \( \omega_{0}=10^{5} \, \text{rad/s} \). The circuit draws \( 16 \, \text{W} \) power from \( 120 \, \text{V} \) source at resonance. The value of resistance ‘\( R \)’ in the circuit is ________ \( \Omega \).

In a series LCR circuit, the inductance, capacitance and resistance are \( L = 100 \, \text{mH} \), \( C = 100 \, \mu \text{F} \) and \( R = 10 \, \Omega \) respectively. They are connected to an AC source of voltage \( 220 \, \text{V} \) and frequency of \( 50 \, \text{Hz} \). The approximate value of current in the circuit will be __________ \( \text{A} \).

Given below are two statements:
Statement I: When the frequency of an A.C source in a series LCR circuit increases, the current in the circuit first increases, attains a maximum value and then decreases.
Statement II: In a series LCR circuit, the value of power factor at resonance is one.
In the light of given statements, choose the most appropriate answer from the options given below.

For a given series LCR circuit it is found that maximum current is drawn when value of variable capacitance is \( 2.5\,\text{nF} \). If resistance of \( 200\,\Omega \) and \( 100\,\text{mH} \) inductor is being used in the given circuit. The frequency of ac source is ______ \( \times 10^{3} \) \( \text{Hz} \). (given \( \pi^{2}=10 \) )

Match List I with List II:

Choose the correct answer from the options given below:

As per the given figure, if \( \frac{dI}{dt} = -1 \, \text{A/s} \) then the value of \( V_{AB} \) at this instant will be _______ \( \text{V} \).

A series \( LCR \) circuit driven by \( 300 \, \text{V} \) at a frequency of \( 50 \, \text{Hz} \) contains a resistance \( R = 3 \, \text{k}\Omega \), an inductor of inductive reactance \( X_L = 250\pi \, \Omega \) and an unknown capacitor. The value of capacitance to maximize the average power should be: (Take \( \pi^2 = 10 \) )

If an optical medium possesses a relative permeability of \( \frac{10}{\pi} \) and relative permittivity of \( \frac{1}{0.0885} \), then the velocity of light is greater in vacuum than that in this medium by _______ times.
\( (\mu_0 = 4\pi \times 10^{-7} \, \text{H/m} \;, \; \varepsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \;, \; c = 3 \times 10^{8} \, \text{m/s} )\)

A coil of negligible resistance is connected in series with \( 90 \, \Omega \) resistor across \( 120 \, \text{V} \), \( 60 \, \text{Hz} \) supply. A voltmeter reads \( 36 \, \text{V} \) across resistance. Inductance of the coil is:

When a dc voltage of \( 100\,\text{V} \) is applied to an inductor a dc current of \( 5\,\text{A} \) flows through it. When an ac voltage of \( 200\,\text{V} \) peak value is connected to inductor, its inductive reactance is found to be \( 20\sqrt{3}\,\Omega \). The power dissipated in the circuit is ________ \( \text{W} \).

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 ____.

An inductor of reactance \( 100\,\Omega \), a capacitor of reactance \( 50\,\Omega \) and a resistor of resistance \( 50\,\Omega \) are connected in series with an AC source of \( 10 \, \text{V} \), \( 50 \, \text{Hz} \). Average power dissipated by the circuit is _____ \( \text{W} \).

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: \( (\mu_0 = \text{permeability of vacuum}) \)

An ideal transformer with purely resistive load operates at \( 12 \, \text{kV} \) on the primary side. It supplies electrical energy to a number of nearby houses at \( 120 \, \text{V} \). The average rate of energy consumption in the houses served by the transformer is \( 60 \, \text{kW} \). The value of resistive load \( (R_s) \) required in the secondary circuit will be \( \text{m}\Omega \).

A power transmission line feeds input power at \( 2.3\,\text{kV} \) to a step down transformer with its primary winding having \( 3000 \) turns. The output power is delivered at \( 230\,\text{V} \) by the transformer. The current in the primary of the transformer is \( 5\,\text{A} \) and it efficiency is \( 90\% \). The winding of transformer is made of copper. The output current of transformer is ______ \( \text{A} \).

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 ______________ .

Match List-I with List-II:

Choose the correct answer from the options given below:

At very high frequencies, the effective impedance of the given circuit will be ______ \( \Omega \) .

The radiation exerted by a \( 450\, \text{W} \) light source on a perfectly surface placed at \( 2\, \text{m} \) away from it, is:

A \( 10\,\Omega \) resistance is connected across \( 220 \, \text{V} \) - \( 50 \, \text{Hz} \) AC supply. The time taken by the current to change from its maximum value to the rms value is:

The alternating current is given by
\( i = \left( \sqrt{42} \sin\!\left( \dfrac{2\pi}{T} t \right) + 10 \right) \, \text{A} \).
The r.m.s. value of this current is__________ \( \text{A} \).

The frequencies at which the current amplitude in an \( LCR \) series circuit becomes \( \dfrac{1}{\sqrt{2}} \) times its maximum value, are \( 212 \, \text{rad s}^{-1} \) and \( 232 \, \text{rad s}^{-1} \). The value of resistance in the circuit is \( R = 5 \, \Omega \). The self inductance in the circuit is __________ \( \text{mH} \).

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 _________.
$(\mu_0 = 4\pi \times 10^{-7}\text{ Tm / A})$

For ac circuit shown in figure, \( R = 100 \, \text{k}\Omega \) and \( C = 100 \, \text{pF} \) and the phase difference between \( V_m \) and \( (V_B - V_A) \) is \( 90^\circ \). The input signal frequency is \( 10^{x} \, \text{rad/sec} \), where ‘x’ is __________.

Given below are two statements:
Statement I: An AC circuit undergoes electrical resonance if it contains either a capacitor or an inductor.
Statement II: An AC circuit containing a pure capacitor or a pure inductor consumes high power due to its non-zero power factor.
In the light of above statements, choose the correct answer from the options given below:

In series LCR circuit, the capacitance is changed from \( C \) to \( 4C \). To keep the resonance frequency unchanged, the new inductance should be:

In a series LR circuit with \( X_{L} = R \), power factor is \( P_{1} \). If a capacitance \( C \) with \( X_{C} = X_{L} \) is added to the circuit the power factor becomes \( P_{2} \). The ratio of \( P_{1} \) to \( P_{2} \) will be:

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 ________.

In an ac circuit, the instantaneous current is zero, when the instantaneous voltage is maximum. In this case, the source may be connected to:
A. Pure inductor.
B. Pure capacitor.
C. Pure resistor.
D. Combination of an inductor and capacitor.
Choose the correct answer from the options given below:

In the given circuit the AC source has \( \omega = 100 \, \text{rad} \, \text{s}^{-1} \). Considering the inductor and capacitor to be ideal, what will be the current \( I \) flowing through the circuit?

A series LCR circuit of \( R = 5 \Omega \), \( L = 20 \, \text{mH} \) and \( C = 0.5 \, \mu \text{F} \) is connected across an AC supply of \( 250 \, \text{V} \), having variable frequency. The power dissipated at resonance condition is ______________ \( \times 10^{2} \, \text{W} \) .

A series LCR circuit is connected to an AC source of \( 220 \, \text{V} \), \( 50 \, \text{Hz} \). The circuit contains a resistance \( R = 80 \, \Omega \), an inductor of inductive reactance \( X_L = 70 \, \Omega \) and a capacitor of capacitive reactance \( X_C = 130 \, \Omega \). The power factor of circuit is \( \dfrac{x}{10} \). The value of \( x \) is:

The equation of current in a purely inductive circuit is \( 5 \sin(49\pi t - 30^\circ) \). If the inductance is \( 30 \, \text{mH} \) then the equation for the voltage across the inductor, will be:
{Let \( \pi = \dfrac{22}{7} \)}

Arrange the following in the ascending order of wavelength \( \lambda \):
(A) Microwaves \( \left(\lambda_{1}\right) \)    (B) Ultraviolet rays \( \left(\lambda_{2}\right) \)    (C) Infrared rays \( \left(\lambda_{3}\right) \)    (D) X-rays \( \left(\lambda_{4}\right) \)
Choose the most appropriate answer from the options given below:

AC voltage \( V(t) = 20 \sin \omega t \) of frequency \( 50 \, \text{Hz} \) is applied to a parallel plate capacitor. The separation between the plates is \( 2 \, \text{mm} \) and the area is \( 1 \, \text{m}^2 \). The amplitude of the oscillating displacement current for the applied AC voltage is ________.

For a series LCR circuit, \( I \) vs \( \omega \) curve is shown:
(a) To the left of \( \omega_r \), the circuit is mainly capacitive
(b) To the left of \( \omega_r \), the circuit is mainly inductive
(c) At \( \omega_r \), impedance of the circuit is equal to the resistance of the circuit
(d) At \( \omega_r \), impedance of the circuit is \( 0 \).

Choose the most appropriate answer from the options given below:

A plane electromagnetic wave of frequency \( 20 \, \text{MHz} \) travels in free space along the \( +x \) direction. At a particular point in space and time, the electric field vector of the wave is \( E_y = 9.3 \, \text{V m}^{-1} \). Then, the magnetic field vector of the wave at that point is:

An inductor of \( 0.5 \, \text{mH} \), a capacitor of \( 200 \, \mu\text{F} \) and a resistor of \( 2 \, \Omega \) are connected in series with a \( 220 \, \text{V} \) ac source. If the current is in phase with the emf, the frequency of ac source will be ______ \( \times 10^{2} \, \text{Hz} \).

Match List-I with List-II.

Choose the correct answer from the options given below:

A sinusoidal voltage \( V(t) = 210\sin(3000t) \) volt is applied to a series LCR circuit in which \( L = 10\,\text{mH} \), \( C = 25\,\mu\text{F} \) and \( R = 100\,\Omega \). The phase difference \( (\phi) \) between the applied voltage and resultant current will be:

A series LCR circuit has \( L = 0.01 \, \text{H} , \ R = 10 \, \Omega \) and \( C = 1 \, \mu \text{F} \) and it is connected to ac voltage of amplitude \( (V_m) \) \( 50 \, \text{V} \). At frequency \( 60 \% \) lower than resonant frequency, the amplitude of current will be approximately:

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 \( c = 3 \times 10^{8} \, \text{m}/\text{s} \))

Electromagnetic waves travel in a medium with speed of \( 1.5 \times 10^{8} \, \text{m s}^{-1} \). The relative permeability of the medium is \( 2.0 \). The relative permittivity will be:

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 ____.

A \( 100 \,\mathrm{V} \), \( 50 \,\mathrm{Hz} \), AC source is connected in the circuit (as shown in figure). The current through the resistance \( 55 \,\Omega \), at resonance in the circuit, will be __________ \( \mathrm{A} \).

In the given circuit, rms value of current \( (I_{\mathrm{rms}}) \) through the resistor \( R \) is:

An AC current is given by \( I = I_1 \sin \omega t + I_2 \cos \omega t \). A hot wire ammeter will give a reading:

A parallel plate capacitor of area \( A = 16 \, \text{cm}^2 \) and separation between the plates \( 10 \, \text{cm} \), is charged by a DC current Consider a hypothetical plane surface of area \( A_0 = 3.2 \, \text{cm}^2 \) inside the capacitor and parallel to the plates. At an instant, the current through the circuit is \( 6 \, \text{A} \). At the same instant the displacement current through \( A_0 \) is __________ \( \text{mA} \).

Match List-I with List-II:

Choose the correct answer from the options given below:

A alternating current at any instant is given by \( i = \left[ 6 + \sqrt{56}\,\sin\!\left(100\pi t + \frac{\pi}{3}\right) \right] \,\text{A} \). The rms value of the current is _______ \( \text{A} \).