Energy Stored in an Inductor

Q. A conducting rod $PQ$ rotates with a constant angular velocity $\omega$ about the axis which passes through the point $O$ and is perpendicular to its length. A uniform magnetic field $B$ exists in this region, as shown in the figure. The potential difference between the two ends of the rod is :

1. Zero
2. $5B\omega l^2$
3. $4B\omega l^2$
4. $6B\omega l^2$

Q. Initially both the capacitors are having charges +Q, +2Q respectively as shown. Find heat loss in the resistance R after the switch is closed.

प्रारम्भ में दोनों संधारित्रों पर आवेश दर्शाए अनुसार क्रमशः +Q, +2Q हैं। स्विच को बंद करने के बाद, प्रतिरोध R में ऊष्मा हानि ज्ञात कीजिए।

1. $\frac{Q^2}{2C}$
2. $\frac{2Q^2}{C}$
3. $\frac{Q^2}{C}$
4. $\frac{3Q^2}{2C}$

Q. In a long cylindrical wire of radius $R$, magnetic induction varies with the distance $r$ from axis as $B=C{r}^{\alpha }$, where $C\&\alpha$ are constant. Find the function of current density in wire with the distance from the axis of wire.

1. $\left[\frac{(\alpha +1)C}{{\mu }_0}\right]r^{\alpha -1}$
2. $\left[\frac{(\alpha -1)}{{\mu }_0}\right]r^{1-\alpha }$
3. $\left[\frac{(\alpha +1)C}{{\mu }_0}\right]r^{\alpha +1}$
4. $\left[\frac{(\alpha -1)}{{\mu }_0}\right]r^{\alpha +1}$

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

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

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

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

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 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