Energy Stored in an Inductor

Q. The current in a coil of self-inductance $2.0\text{H}$ is increasing according to equation $I=2\sin t^2\text{A}$. Find the amount of energy spent during the period when the current changes from $0$ to $2\text{A}$.

1. $3\text{J}$
2. $4\text{J}$
3. $2\text{J}$
4. $\text{None of these}$

Q. A short-circuited coil is placed in a time-varying magnetic field. Electrical power is dissipated due to the current induced in the coil. If the number of turns were to be quadrupled and the wire radius halved, the electrical power dissipated would be

1. Halved
2. The same
3. Doubled
4. Quadrupled

Q. A coil of resistance $20\Omega$ and inductance $5H$ has been connected to a $100V$ battery The energy stored in the coil after a long time is

1. $31.25J$
2. $125J$
3. $62.5J$
4. $250J$

Q. Two different coils have self-inductance $L_1=8mH,L_2=2mH$. The current in one coil is increased at a constant rate. The current in the second coil is also increased at the same rate. At a certain instant of time, the power given to the two coils is the same. At that time the current, the induced voltage and the energy stored in the first coil are $i_1,V_{1}and{W}_1$ respectively. Corresponding values for the second coil at the same instant are $i_2,V_{2}and{W}_2$ respectively. Then

1. $\frac{i_1}{i_2}=\frac{1}{4}$
   
2. $\frac{V_2}{V_1}=\frac{1}{2}$
3. $\frac{i_1}{i_2}=4$
   
4. $\frac{W_2}{W_1}=2$
   

Q. Shown in the figure is a circular loop of radius r and resistance R. A variable magnetic field of induction B = $B_0{e}^{-t}$ is established inside the coil. If the key (K) is closed, the electrical power developed right after closing the switch is equal to

1. 
2. 
3. 
4. 

Q.

A coil of inductance L is carrying a steady current i. What is the nature of its stored energy?

1. Magnetic

2. Electrical

3. Heat

4. Both magnetic and electrical

Q. An electron is shot into one end of a solenoid at the speed of 800 m/s making an angle $30°$ with the axis of solenoid. Current in solenoid is 4 A and there are total 8000 turns over solenoid. If the total number of revolutions made by electron within solenoid before emerging from another end of solenoid is $n\times {10}^6$, then the value of n is
$\left(\frac{e}{m}=\sqrt{3}\times {10}^{11}\text{C/kg (write two digits after the decimal point.)}\right]$

Q. A circular loop of radius $r$ is made of a wire of circular cross section of diameter $a$. When current $I$ flows through the loop the magnetic flux linked with the loop due to self induced magnetic field is given by, $\varphi ={\mu }_{o}r[ln\left(\frac{16r}{a}\right)-\frac{7}{4}]I.$ The resistivity of the material of the wire is $\rho$ and $r>>a$. Switch $S$ is closed at time $t=0$ so as to connect the loop to a cell of emf $V$.
Choose the correct statements

1. Self inductance of the loop is, $L={\mu }_{o}r[ln\left(\frac{16r}{a}\right)-\frac{7}{4}]$
2. Resistance of the loop is, $R=\frac{8\rho r}{a^2}$
3. Current in the loop is, $i=\frac{V{a}^2}{8\rho r}[1-e^{\frac{-t}{\tau }}]$ where $\tau =\frac{L}{R}=\frac{{\mu }_0{a}^2}{8\rho }[ln\left(\frac{16r}{a}\right)-\frac{7}{4}]$
4. Current in the loop is, $i=\frac{V{a}^2}{8\rho r}[1-e^{\frac{-2t}{\tau }}]$
   where $\tau =\frac{L}{R}=\frac{{\mu }_o{a}^2}{\rho }[ln\left(\frac{16r}{a}\right)-\frac{7}{4}]$
   

Q. The current $i$ in an induction coil varies with time $t$ according to the graph shown in the figure. Which of the following graphs shows the induced emf in the coil with time ?

1. 
2. 
3. 
4. 

Q. An inductor of inductance $\left(L\right)$ and a resistor of resistance $\left(R\right)$ are initially connected with the battery in series to steady state and then the battery is removed. The total heat produced in the resistor $R$, when the current in the circuit decreases from maximum value $i_0$ to $0$ is -

1. $L{i}_0^2$
2. $\frac{1}{2}L{i}_0^2$
3. $\frac{3}{2}L{i}_0^2$
4. $\frac{1}{3}L{i}_0^2$

Q.

An average emf of 20 V is induced in an inductor when the current in it is changed from 2.5 A in one direction to the same value in the opposite direction in 0.1 s. Find the self-inductance of the inductor.

Q. A coil of inductance $L=2.0\mu H$ and resistance $R=1.0\Omega$ is connected to a source of constant emf $ϵ=3.0V$.
Resistance $R_0=2.0\Omega$ is connected in parallel with the coil.The amount of heat generated in the coil after the switch $S_w$ is disconected is (in microjoule)? (The internal resistance of the source is negligible.)

Q. A $40$ $mH$ coil carries a current of $8A$, the energy stored in the coil is

1. $240$ $mJ$
2. $280$ $mJ$
3. $320$ $mJ$
4. $360$ $mJ$

Q. What is the basic use of capacitor ?

Q. A square loop of side 20 cm is placed in a magnetic field $B=(5+2t^2)$; such that, the magnetic field is perpendicular to the plane of the loop. If resistance of loop is $2\Omega$, heat generated in 5 sec in joules is (write until two digits after the decimal point)

Q. In the circuit shown in the figure. The coil has inductance $L$ and resistance $R$. When point $X$ is joined to $Y$, the time constant is $\tau$ during the growth of current. When the steady state is reached, heat is produced in the coil at a rate of $P$. Now, point $X$ is joined to $Z$, then, the total heat produced when current in the circuit reduced to zero is

1. $P\tau$
2. $\frac{1}{2}P\tau$
3. $2P\tau$
4. $\frac{3}{2}P\tau$

Q.

An inductor of inductance 2.00 H is joined in series with a resistor of resistance $200\Omega$ and a battery of emf 2.00 V. At t = 10 ms, find (a) The curent in the circuit, (b) the power delivered by the battery, (c) the power dissipated in heating the resistor and (d) the rate at which energy is being stored in magnetic field.

Q.

Explain why the efficiency of a machine is never 100%.

Q. Energy stored in a coil of self – inductance 40mH carrying a steady current of 2 A is

1. 0.8 J
2. 8 J
3. 0.08 J
4. 80 J

Q. Two coils $A$ and $B$ having the radii in the ratio $1:2$ carrying currents in the ratio $5:1$ and have the number of turns in the ratio $1:5.$ The ratio of magnetic inductions at their centers is

1. $5:1$
2. $1:5$
3. $2:1$
4. $1:2$

Q. The current in the outer coil is varying with time as $I=2t^2$. The heat developed in the inner coil between $t=0$ to $t$ seconds is $\frac{k{\mu }_0^2{\pi }^2{a}^4{t}^3}{3b^{2}R}$, where the resistance of the inner coil is $R$ and $b>>a$. Find $k$.

Q. An inductor coil stores U energy when I current is passed through it and dissipates energy at the rate of P.Find the time cons†an t of the circuit , when this coil is connected across a battery of zero internal resis†an ce

Q. Consider a coaxial cable which consists of an inner wire of radius $a$ surrounded by an outer shell of inner and outer radii $b$ and $c$ respectively. The inner wire carries an electric current $i_0$ and the outer shell carries an equal current in the same direction. Find the magnetic field at a distance $x$ from the axis where $b<x<c$.
(Assume current density to be uniform in the inner wire and outer shell.)

1. $\frac{{\mu }_0{i}_{0}x}{2\pi a^2}$
2. $\frac{{\mu }_0{i}_0(2c^2+x^2)}{\pi x(c^2-x^2)}$
3. $\frac{{\mu }_0{i}_0}{2\pi x}$
4. $\frac{{\mu }_0{i}_0(c^2+x^2-2b^2)}{2\pi x(c^2-b^2)}$

Q. In the steady state of circuit, ratio of energy stored in capacitor to the energy stored in inductor is Here L = 0.2 mH and C = 500 $\mu F$

Q. In an LC circuit, $C=4.00\mu \text{F}$, the maximum potential difference across the capacitor during the oscillations is $1.50\text{V}$ and the maximum current in the circuit is $50.0\text{mA}$. What is the value of inductance $L$?

1. $2.4\text{mH}$
2. $3.6\text{mH}$
3. $4.8\text{mH}$
4. $6.0\text{mH}$

Q.

Suppose the loop in Exercise 6.4 is stationary but the current feeding the electromagnet that produces the magnetic field is gradually reduced so that the field decreases from its initial value of 0.3 T at the rate of 0.02 T s⁻¹. If the cut is joined and the loop has a resistance of 1.6 Ω how much power is dissipated by the loop as heat? What is the source of this power?

Q. An average emf of $20\text{V}$ is induced in an inductor when the current in it is changed from $2.5\text{A}$ in one direction to $5.5\text{A}$ in the opposite direction, in $0.1\text{s}$. Find the self-inductance of the inductor.

1. $0.75\text{H}$
2. $0.85\text{H}$
3. $0.25\text{H}$
4. $0.45\text{H}$

Q. The magnetic field at a point inside a $2\text{mH}$ inductor coil becomes $0.8$ of its maximum value in $20\mu \text{s}$ when the inductor is joined to a battery. Find the resistance of the circuit.
[Take $\ln \left(0.2\right)=-1.609$]

1. $144\Omega$
2. $127\Omega$
3. $133\Omega$
4. $161\Omega$

Q. A coil of inductance $L=2\mu \text{H}$ and resistance $R_1=1\Omega$ is connected to a source of constant emf $E=3\text{V}$. A resistance of $R_2=2\Omega$ is connected in parallel with the coil. Find the amount of heat generated in the coil after the switch is disconnected. The internal resistance of the source is negligible,

1. $3\mu \text{J}$
2. $9\mu \text{J}$
3. $4.5\mu \text{J}$
4. $30\mu \text{J}$

Q. In the circuit shown in figure, $X$ is joined to $Y$ for a long time and then $X$ is joined to $Z$. The total heat produced in $R_2$ is

1. $\frac{L{E}^2}{2R_2^2}$
2. $\frac{L{E}^2}{2R_1{R}_2}$
3. $\frac{L{E}^2{R}_2}{2R_1^3}$
4. $\frac{L{E}^2}{2R_1^2}$