Q. No.In a certain region of space, equipotential surfaces of the electric field are drawn - corresponding to \(V=10\) volt and \(V=9.9\) volts. There is no field along the \(z\text-\)direction. At a certain point \(P,\) on the \(10-\)volt surface, the distance \(PQ_1,\) to the \(9.9\) volt surface is \(2\) mm when \(\overrightarrow{P Q}_{1}\) is along the \(x\text-\)axis. On the other hand, if \(\overrightarrow{P Q}_{2}\) is taken parallel to the \(y\text-\)axis, the corresponding distance \(PQ_2=1\) mm. The electric field at \(P\) is along:
1.
\(2 \hat{i}+\hat{j}\)
2.
\(2 \hat{j}+\hat{i}\)
3.
\(\dfrac{1}{4} \hat{i}+\hat{j}\)
4.
\( \dfrac{1}{4} \hat{j}+\hat{i}\)
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The capacitance is now \(C_2.\)
\(\dfrac{C_1} { C_2}\) equals:
| 1. | \(3\) | 2. | \(2\) |
| 3. | \(\dfrac{3}{2}\) | 4. | \(\dfrac{4}{3}\) |
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The capacitance of the unknown capacitor is:
1. \(4 ~\mu \text{F}\)
2. \(6~ \mu \text{F}\)
3. \(24 ~\mu \text{F}\)
4. \(36 ~\mu \text{F}\)
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| 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. | \( \dfrac{2 K_{1} K_{2}}{K_{1}+K_{2}} C_{0}\) |
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• \(A \left ( 1, 0, 0 \right )\) \(V_{A}=2\) 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
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| 1. | \(\dfrac Ud\) | 2. | \(\dfrac {2U}d\) |
| 3. | \(\dfrac U{2d}\) | 4. | \(\dfrac {\sqrt2U}d\) |
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It is observed that \(V_A=V_B\) after a sufficiently long time.
Then:
| 1. | \(\dfrac{C_1}{C_2}=\dfrac{C_3}{C_4}\) | 2. | \(\dfrac{C_1}{C_2}=\dfrac{R_3~C_3}{R_4~C_4}\) |
| 3. | \(\dfrac{C_1}{C_2}=\dfrac{R_4}{R_3}\) | 4. | \(\dfrac{C_1}{C_2}=\dfrac{R_4~C_3}{R_3~C_4}\) |
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Then, \(V_{A}-V_{B}=\)
1. \(E_0\)
2. \(2E_0\)
3. \(-E_0\)
4. zero
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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. \(4 \pi \varepsilon_{0} \sqrt{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}}\)
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\(\Big(k=\dfrac{1}{4\pi\varepsilon_0}\Big)\)
| 1. | \(\dfrac{8~kq}{d}\) | 2. | \(\dfrac{4~kq}{d}\) |
| 3. | \(\dfrac{4\sqrt2~kq}{d}\) | 4. | \(\dfrac{\sqrt2~kq}{d}\) |
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