Four electric charges \(+ q,\) \(+ q,\) \(- q\) and \(- q\) are placed at the corners of a square of side \(2L\) (see figure). The electric potential at the point \(A\), mid-way between the two charges \(+ q\) and \(+ q\) is:
              
1. \(\frac{1}{4 \pi\varepsilon_{0}} \frac{2 q}{L} \left(1 + \frac{1}{\sqrt{5}}\right)\)
2. \(\frac{1}{4 \pi\varepsilon_{0}} \frac{2 q}{L} \left(1 - \frac{1}{\sqrt{5}}\right)\)
3. zero
4. \(\frac{1}{4 \pi \varepsilon_{0}} \frac{2 q}{L} \left(1 + \sqrt{5}\right)\)

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. \(12\pi \varepsilon_0 Q\times 10^{22}~\text{V/m}\)

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. \({1 \over2}CV^2\)

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. \(\dfrac{pE}{2}\)
3. \(2pE\)
4. \(pE\)

Maximum charge stored on a metal sphere of radius \(15\) cm may be \(7.5~\mu\text{C}\)$$. The potential energy of the sphere in this case is:
1. \(9.67\) J
2. \(0.25\) J
3. \(3.25\) J
4. \(1.69\) J

The electrostatic force between the metal plates of an isolated parallel plate capacitor \(C\) having a charge \(Q\) and area \(A\) is:

An electric dipole with dipole moment \(\vec{p} = \left(3 \hat{i} + 4 \hat{j}\right) \times 10^{- 30}~\text{C-m}\) is placed in an electric field \(\vec{E} = 4000 \hat{i} ~\text{N/C}\). An external agent turns the dipole slowly until its electric dipole moment becomes \(\left(- 4 \hat{i} + 3 \hat{j}\right) \times 10^{- 30}~\text{C-m}\). The work done by the external agent is equal to:
1. \(4\times 10^{-28}~\text{J}\)
2. \(-4\times 10^{-28}~\text{J}\)
3. \(2.8\times 10^{-26}~\text{J}\)
4. \(-2.8\times 10^{-26}~\text{J}\)

The variation of potential with distance \(x\) from a fixed point is shown in the figure. The electric field at \(x=13~\text m\) is:
     

In the circuit diagram shown all the capacitors are in  \(\mu \text{F} \).  The equivalent capacitance between points, \(A\) & \(B\) is (in \(\mu \text{F} \)):

 
1. \(\frac{14}{5}\)

2. \(7.5\)

3. \(\frac{3}{7}\)

4. None of these