Four electric charges \(+\mathrm q,\) \(+\mathrm q,\) \(-\mathrm q\) and \(-\mathrm q\) are placed at the corners of a square of side \(2\mathrm{L}\) (see figure). The electric potential at point A, mid-way between the two charges \(+\mathrm q\) and \(+\mathrm q\) is:
              

1.  $\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{2\mathrm{q}}{\mathrm{L}}\left(1+\frac{1}{\sqrt{5}}\right)$

2.  $\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{2\mathrm{q}}{\mathrm{L}}\left(1-\frac{1}{\sqrt{5}}\right)$

3.  zero

4.  $\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{2\mathrm{q}}{\mathrm{L}}\left(1+\sqrt{5}\right)$

A series combination of n₁ capacitors, each of value C₁, is charged by a source of potential difference 4V. When another parallel combination of n₂ capacitors, each of value C₂, is charged by a source of potential difference V, it has the same (total) energy stored in it, as the first combination has. The value of C₂, in terms of C₁, is then:

1. $\frac{2C_1}{n_1{n}_2}$

2. $16\frac{n_2}{n_1}{C}_1$

3. $2\frac{n_2}{n_1}{C}_1$

4. $\frac{16C_1}{n_1{n}_2}$

Three concentric spherical shells have radii a, b, and c (a<b<c) and have surface charge densities $\mathrm{\sigma },$ $-\mathrm{\sigma }$, and $\mathrm{\sigma }$ respectively. If ${\mathrm{V}}_{\mathrm{A}},$ ${\mathrm{V}}_{\mathrm{B}}$, and ${\mathrm{V}}_{\mathrm{C}}$ denote the potential of the three shells, and c=a+b, it can be concluded that:

Three capacitors each of capacitance \(C\) and of breakdown voltage \(V\) are joined in series. The capacitance and breakdown voltage of the combination will be:
1. $\frac{C}{3},$ $\frac{V}{3}$

2. $3C,$ $\frac{V}{3}$

3. $\frac{C}{3},$ $3V$

4. \(3C,~3V\)

The electric potential at a point in free space due to a charge \(Q\) coulomb is \(Q\times10^{11}~\text{V}\). The electric field at that point is:
1. \(4\pi \varepsilon_0 Q\times 10^{22}~\text{V/m}\)
2. \(12\pi \varepsilon_0 Q\times 10^{20}~\text{V/m}\)
3. \(4\pi \varepsilon_0 Q\times 10^{20}~\text{V/m}\)
4. \(12\pi \varepsilon_0 Q\times 10^{22}~\text{V/m}\)

The energy required to charge a parallel plate condenser of plate separation, \(d\) and plate area of cross-section, \(A\) such that the uniform electric field between the plates is \(E,\) is:
1. $\frac{1}{2}$ ${\mathrm{\epsilon }}_0{\mathrm{E}}^2/\mathrm{Ad}$

2. ${\mathrm{\epsilon }}_0{\mathrm{E}}^2/\mathrm{Ad}$

3. ${\mathrm{\epsilon }}_0{\mathrm{E}}^2\mathrm{Ad}$

4. $\frac{1}{2}$ ${\mathrm{\epsilon }}_0{\mathrm{E}}^2\mathrm{Ad}$

Two condensers, one of capacity \(C\) and the other of capacity \(\frac{C}2\) are connected to a \(V\) volt battery, as shown in the figure. 
          
The energy stored in the capacitors when both condensers are fully charged will be:
1. \(2CV^2\)
2. \({1 \over4}CV^2\)
3. \({3 \over4}CV^2\)
4. \({1 \over2}CV^2\)

Charges +q and –q are placed at points A and B, respectively; which are at a distance 2L apart. C is the midpoint between A and B. The work done in moving a charge +Q along the semicircle CRD is:
   
1. $\frac{qQ}{4{\mathrm{\pi \epsilon }}_0\mathrm{L}}$
2. $\frac{qQ}{2{\mathrm{\pi \epsilon }}_0\mathrm{L}}$
3. $\frac{qQ}{6{\mathrm{\pi \epsilon }}_0\mathrm{L}}$
4. $-\frac{qQ}{6{\mathrm{\pi \epsilon }}_0\mathrm{L}}$

An electric dipole of moment \(\vec {p} \) is lying along a uniform electric field \(\vec{E}\). The work done in rotating the dipole by \(90^{\circ}\) is:
1. \(\sqrt{2}pE\)
2. \(\frac{pE}{2}\)
3. \(2pE\)
4. \(pE\)