A uniformly charged solid sphere of radius \(R\) has potential ${\mathrm{}}_{}$\(V_0\) (measured with respect to \(\infty\)) on its surface. For this sphere, the equipotential surfaces with potentials \(\frac{3V_0}{2},\frac{5V_0}{4},\frac{3V_0}{4}\) and \(\frac{V_0}{4}\) have radius \(R_1,R_2,R_3\) and \(R_4\)${\mathrm{ respectively.\; then:}}_{}$

Three concentric metal shells \(A,B\) and \(C\) of respective radii \(a,b\) and \(c ~(a<b<c)\) have surface charge densities \(+\sigma,-\sigma\) and \(+\sigma\) respectively. The potential of the shell \(B\) is:
1. \( \frac{\sigma}{\varepsilon_0}\left[\frac{{a}^2-{b}^2}{{a}}+{c}\right] \)
2. \(\frac{\sigma}{\varepsilon_0}\left[\frac{{a}^2-{b}^2}{{~b}}+{c}\right] \)
3. \(\frac{\sigma}{\varepsilon_0}\left[\frac{{b}^2-{c}^2}{{~b}}+{a}\right] \)
4. \(\frac{\sigma}{\varepsilon_0}\left[\frac{{b}^2-{c}^2}{{c}}+{a}\right] \)

A point dipole \(\vec{P}=-p_0\hat{x}\) is kept at the origin. The potential and electric field due to this dipole on the \(y\)-axis at a distance \(d\) are, respectively: (Take \(V=0\) at infinity)
1. \(\frac{|\vec{P}|}{4 \pi \epsilon_0d^2},\frac{-\vec{P}}{4 \pi \epsilon_0d^3}\)
2. \(0,\frac{-\vec{P}}{4 \pi \epsilon_0d^3}\)
3. \(\frac{|\vec{P}|}{4 \pi \epsilon_0d^2},\frac{\vec{P}}{4 \pi \epsilon_0d^3}\)
4. \(0,\frac{\vec{P}}{4 \pi \epsilon_0d^3}\)

A charge \(Q\) is distributed over two concentric conducting thin spherical shells radii \(r\) and \(R\) (\(R>r\)). If the surface charge densities on the two shells are equal, the electric potential at the common centre is:

      
1. \( \frac{1}{4 \pi \varepsilon_0} \frac{(R+2 r) Q}{2\left(R^2+r^2\right)} \)
2. \( \frac{1}{4 \pi \varepsilon_0} \frac{(R+r)Q}{2\left(R^2+r^2\right)} \)
3. \( \frac{1}{4 \pi \varepsilon_0} \frac{(R+r)Q}{\left(R^2+r^2\right)} \)
4. \( \frac{1}{4 \pi \varepsilon_0} \frac{(2R+r)Q}{\left(R^2+r^2\right)} \)

Two isolated conducting spheres \(S_1\) and \(S_2\) of radius \(\frac{2R}{3}\) and \(\frac{R}{3}\) have \(12~\mathrm{\mu C}\) and \(-3~\mathrm{\mu C}\) charges, respectively, and are at a large distance from each other. They are now connected by a conducting wire. A long time after this is done the charges on \(S_1\) and \(S_2\) are respectively,
 

Ten charges are placed on the circumference of a circle of radius \(R\) with constant angular separation between successive charges. Alternate charges \(1,3,5,7,9\) have charge (\(+q\)) each, while \(2,4,6,8,10\) have charge (\(-q\)) each. The potential \(V\) and the electric field \(E\) at the centre of the circle are respectively: (Take \(V=0\) at infinity)

\(512\) identical drops of mercury are charged to a potential of \(2 \text {V}\) each. The drops are joined to
form a single drop. The potential of this drop is:
1. \(64~\mathrm{V}\)
2. \(128~\mathrm{V}\)
3. \(32~\mathrm{V}\)
4. \(16~\mathrm{V}\)

In an electrical circuit, a battery is connected to pass \(20\) C of charge through it in a certain given time. The potential difference between two plates of the battery is maintained at \(15\) V. The work done by the battery is:
1. \(100\) J
2. \(200\) J
3. \(300\) J
4. \(400\) J