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Q. No.A \(12 ~\mu \text{F}\) capacitor is charged by means of a \(6~\text{V}\) battery and the charged capacitor and the battery are connected in series so that their combined potential difference is twice as much. When a second unknown capacitor (initially uncharged) is connected across this combination, the first capacitor is observed to lose half of its initial charge.
The capacitance of the unknown capacitor is:
1. \(4 ~\mu \text{F}\)
2. \(6~ \mu \text{F}\)
3. \(24 ~\mu \text{F}\)
4. \(36 ~\mu \text{F}\)
Subtopic: Capacitance |
Level 3: 35%-60%
Hints
An arrangement consisting of two concentric spherical shells \(A\) and \(B\) has a capacitance \(C_0\) between them. If the upper 'hemispherical' space between them is filled by a dielectric of relative permittivity \(K_1\) and the lower by one of relative permittivity \(K_2,\) the new capacitance will be:
| 1. | \(\left(K_{1}+K_{2}\right) C_{0}\) | 2. | \( \dfrac{K_{1}+K_{2}}{2} C_{0}\) |
| 3. | \(\dfrac{1}{2}\left(\dfrac{1}{K_{1}}+\dfrac{1}{K_{2}}\right) C_{0}\) | 4. | \( \dfrac{2 K_{1} K_{2}}{K_{1}+K_{2}} C_{0}\) |
Subtopic: Dielectrics in Capacitors |
Level 3: 35%-60%
Hints
A uniform electric field exists in a certain region of space. The potential at the following points are given (all units are in SI):
• \(A \left ( 1, 0, 0 \right )\) \(V_{A}=2\) volt
• \(B \left ( 0, 2, 0 \right )\) \(V_{B}=4\) volt
• \(C \left ( 0, 0, 2 \right )\) \(V_{C}=6\) volt
• \(D \left ( 1, 1, 0 \right )\) \(V_{D}=-1\) volt
The component of the electric field along the \(x\text-\)axis is:
1. \(2\) V/m
2. \(8\) V/m
3. \(3\) V/m
4. \(-6\) V/m
• \(A \left ( 1, 0, 0 \right )\) \(V_{A}=2\) volt
• \(B \left ( 0, 2, 0 \right )\) \(V_{B}=4\) volt
• \(C \left ( 0, 0, 2 \right )\) \(V_{C}=6\) volt
• \(D \left ( 1, 1, 0 \right )\) \(V_{D}=-1\) volt
The component of the electric field along the \(x\text-\)axis is:
1. \(2\) V/m
2. \(8\) V/m
3. \(3\) V/m
4. \(-6\) V/m
Subtopic: Relation between Field & Potential |
Level 4: Below 35%
Hints
The arrangement shown in the figure is set up with capacitors initially uncharged, and the circuit is completed. A potential difference is imposed across \(AB\) so that the charge on the upper capacitor is doubled without changing its sign.
Then, \(V_{A}-V_{B}=\)
1. \(E_0\)
2. \(2E_0\)
3. \(-E_0\)
4. zero
Then, \(V_{A}-V_{B}=\)
1. \(E_0\)
2. \(2E_0\)
3. \(-E_0\)
4. zero
Subtopic: Capacitance |
Level 3: 35%-60%
Hints
Three metallic spheres of radii \(r_1,~ r_2,~ r_3\) are connected by very long conducting wires to form an equilateral triangle. The capacitance of the system is:
1. \(4 \pi \varepsilon_{0}\left(r_{1}+r_{2}+r_{3}\right)\)
2. \(4 \pi \varepsilon_{0} \dfrac{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}}{r_{1}+r_{2}+r_{3}}\)
3. \(4 \pi \varepsilon_{0}\left(\dfrac{1}{r_{1}}+\dfrac{1}{r_{2}}+\dfrac{1}{r_{3}}\right)^{-1}\)
4. \(4 \pi \varepsilon_{0} \sqrt{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}}\)
1. \(4 \pi \varepsilon_{0}\left(r_{1}+r_{2}+r_{3}\right)\)
2. \(4 \pi \varepsilon_{0} \dfrac{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}}{r_{1}+r_{2}+r_{3}}\)
3. \(4 \pi \varepsilon_{0}\left(\dfrac{1}{r_{1}}+\dfrac{1}{r_{2}}+\dfrac{1}{r_{3}}\right)^{-1}\)
4. \(4 \pi \varepsilon_{0} \sqrt{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}}\)
Subtopic: Capacitance |
51%
Level 3: 35%-60%
Hints
The left plate \(A\) of an air capacitor is connected to the positive terminal while the right plate \(B\) is connected to the negative terminal of a cell of voltage \(V_0.\) Assume that the plate area is \(A,\) and the plate separation is \(d.\) If a slab of dielectric constant \(K\) is inserted into the space between the plates, the electric field in the dielectric will be: (compared to the air capacitor)
| 1. | more. |
| 2. | less. |
| 3. | equal. |
| 4. | more or less or equal depending on the value of \(K\). |
Subtopic: Dielectrics in Capacitors |
Level 3: 35%-60%
Hints
A semi-circular wire carries a positive charge \(Q\) distributed uniformly over its circumference, and lies with its center at the origin and its ends on the x-axis, as shown in the figure. A single point charge is to be placed so that the net electric field due to the charge \(Q\) and the new charge \((q_1)\) is zero at the origin. Let the distance of \(q_1\) from O be \(r_1\). The potential at the origin is also zero. Which of the following is correct?
\(1.~ r_{1}< \frac{r}{2} \\ 2.~ \frac{r}{2}<r_{1}< r \\ 3.~ r<r_{1}<\frac{3 r}{2}\\ 4.~ \frac{3 r}{2}<r_{1}\)
\(1.~ r_{1}< \frac{r}{2} \\ 2.~ \frac{r}{2}<r_{1}< r \\ 3.~ r<r_{1}<\frac{3 r}{2}\\ 4.~ \frac{3 r}{2}<r_{1}\)
Subtopic: Electric Potential |
Level 3: 35%-60%
Please attempt this question first.
Hints
Please attempt this question first.
Two concentric metallic spheres, surface areas \(A_1,A_2\) and separation \(d\), have a capacitance \(C_0.\) If a parallel plate capacitor is built with the same separation \(d,\) and has the same capacitance \(C_0\) then its plate area will be:
| 1. | \(\dfrac{A_1+A_2}{2}\) | 2. | \(\sqrt{A_1A_2}\) |
| 3. | \(\dfrac{2A_1A_2}{A_1+A_2}\) | 4. | \(\dfrac{A_1^2A_2^2}{A_1+A_2}\) |
Subtopic: Capacitance |
Level 3: 35%-60%
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A charge is uniformly distributed on the circumference of a disc, and the potential at its centre is \(5\) volt. If the charge was uniformly distributed on the surface of this disc, the potential at a point \(P\) on its axis, at a distance equal to the disc's radius from its centre, equals:
1. \(10\) V
2. \(5 \sqrt 2\) V
3. \(10 \sqrt 2\) V
4. \(10 (\sqrt {2} -1)\) V
1. \(10\) V
2. \(5 \sqrt 2\) V
3. \(10 \sqrt 2\) V
4. \(10 (\sqrt {2} -1)\) V
Subtopic: Electric Potential |
Level 3: 35%-60%
Hints
A capacitance is formed by connecting two metallic balls of radius \(r\) by a conducting wire, and two oppositely charged identical metallic hemispheres \((A,B)\) slightly larger than the balls. The separation between the hemispheres and the respective balls is \(d.\) The capacitance between \(A,B\) is:
| 1. | \(\dfrac{4\pi\varepsilon_0r^2}{d}\) | 2. | \(\dfrac{2\pi\varepsilon_0r^2}{d}\) |
| 3. | \(\dfrac{\pi\varepsilon_0r^2}{d}\) | 4. | \(\dfrac{\pi\varepsilon_0r^2}{2d}\) |
Subtopic: Capacitance |
Level 4: Below 35%
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