Alternating Current

Consider a circuit consisting of a capacitor of capacitance $C$ and a coil with $N$ turns per unit length, cross sectional area $S$ and length $d$, where $d^2\gg S$. There is another coil of length $d/2$, cross sectional area $S/2$ and $2N$ turns per unit length completely inside the larger coil, as shown in the figure. The ends of this smaller coil are connected with each other by an insulated conducting wire. The self-inductance of the larger coil is $L$. Neglecting edge effects and all the Ohmic resistances, the resonant frequency of the circuit is:

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° and phase difference is 60° when $S_2$ is closed keeping $S_1$ open. The value of $(3L_1-L_2)$ is ________ H.

An inductor of 10 mH, capacitor of 0.1 μF and a resistor of 100 Ω are connected in series across an $a.c$ power supply 220 V, 70 Hz. The power factor of the given circuit is 0.5. The difference in the inductive reactance and capacitance reactance is $\sqrt3\,\alpha$ Ω. The value of $\alpha$ 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 series LCR circuit with $R=20$ Ω, $L=1.6$ H and $C=40$ μF is connected to a variable frequency a.c. source. The inductive reactance at resonant frequency is ________ Ω.

A LCR series circuit driven with $E_{rms}=90$ V at frequency $f_d=30$ Hz has resistance $R=80$ Ω, an inductance with inductive reactance $X_L=20.0$ Ω and capacitance with capacitive reactance $X_C=80.0$ Ω. The power factor of the circuit is ________.

A circuit with an electrical load having impedance $Z$ is connected with an AC source as shown in the diagram. The source voltage varies in time as $V(t) = 300\sin(400t)$ V, where $t$ is time in s. List-I shows various options for the load. The possible currents $i(t)$ in the circuit as a function of time are given in List-II.

Choose the option that describes the correct match between the entries in List-I to those in List-II.

The circuit shown in the figure contains an inductor $L$, a capacitor $C_0$, a resistor $R_0$ and an ideal battery. The circuit also contains two keys K₁ and K₂. Initially, both the keys are open and there is no charge on the capacitor. At an instant, key K₁ is closed and immediately after this the current in $R_0$ is found to be $I_1$. After a long time, the current attains a steady state value $I_2$. Thereafter, K₂ is closed and simultaneously K₁ is opened and the voltage across $C_0$ oscillates with amplitude $V_0$ and angular frequency $\omega_0$.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

A series LCR circuit is connected to a $45\sin(\omega t)$ Volt source. The resonant angular frequency of the circuit is $10^5\ \mathrm{rad\,s^{-1}}$ and current amplitude at resonance is $I_0$. When the angular frequency of the source is $\omega = 8\times10^4\ \mathrm{rad\,s^{-1}}$, the current amplitude in the circuit is $0.05\,I_0$. If $L = 50$ mH, match each entry in List-I with an appropriate value from List-II and choose the correct option.

Consider an LC circuit, with inductance $L = 0.1$ H and capacitance $C = 10^{-3}$ F, kept on a plane. The area of the circuit is $1\ \mathrm m^2$. It is placed in a constant magnetic field of strength $B_0$ which is perpendicular to the plane of the circuit. At time $t = 0$, the magnetic field strength starts increasing linearly as $B = B_0 + \beta t$ with $\beta = 0.04\ \mathrm{T\,s^{-1}}$. The maximum magnitude of the current in the circuit is _____ mA.

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

As shown in the figure an inductor of inductance 200 mH is connected to an AC source of emf 220 V and frequency 50 Hz. The instantaneous voltage of the source is 0 V when the peak value of current is $\dfrac{\sqrt a}\pi$ A. The value of $a$ 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 circular coil of 1000 turns each with area $1\ \mathrm m^2$ is rotated about its vertical diameter at the rate of one revolution per second in a uniform horizontal magnetic field of 0.07 T. The maximum voltage generation will be ______ V.

Match 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 sinusoidal voltage $V(t)=210\sin3000t$ volt is applied to a series LCR circuit in which $L=10$ mH, $C=25\ \mu$F and $R=100\ \Omega$. The phase difference ($\Phi$) between the applied voltage and resultant current will be :

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

A 110 V, 50 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 ______ A.

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