A conducting sphere of radius \(R\) is given a charge \(Q\). The electric potential and field at the center of the sphere respectively are:

1.	Zero and \(\mathrm{Q} / 4 \pi \varepsilon_{\mathrm{o}} \mathrm{R}^2\)
2.	\(\mathrm{Q} / 4 \pi \varepsilon_{\mathrm{O}} \mathrm{R}\) and zero
3.	\(\mathrm{Q} / 4 \pi \varepsilon_{\mathrm{O}} \mathrm{R}\) and \(\mathrm{Q} / 4 \pi \varepsilon_{\mathrm{o}} \mathrm{R}^2\)
4.	Both are zero

Four point charges \(-Q, -q,2q~\text{and}~2Q\)$$ are placed, one at each corner of the square. The relation between \(Q\) and \(q\) for which the potential at the center of the square is zero, is:

Two metallic spheres of radii 1 cm and 3 cm

are given charges of -1$\times {10}^{-2}\mathrm{C}$ and $5\times {10}^{-2}\mathrm{C}$,

respectively. If these are connected by a conducting

wire, the final charge on the bigger sphere is

(a) $2\times {10}^{-2}\mathrm{C}$

(b) $3\times {10}^{-2}\mathrm{C}$

(c) $4\times {10}^{-2}\mathrm{C}$

(d) $1\times {10}^{-2}\mathrm{C}$

A parallel plate condenser has a uniform electric

field E(V/m) in the space between the plates. If

the distance between the plates is d(m) and area

of each plate is $\mathrm{A}\left({\mathrm{m}}^2\right)$, the energy (joule) stored

in the condenser is

(a) $\frac{1}{2}{\mathrm{\epsilon }}_0{\mathrm{E}}^2$

(b) ${\mathrm{\epsilon }}_0\mathrm{EAd}$

(c) $\frac{1}{2}{\mathrm{\epsilon }}_0{\mathrm{E}}^2\mathrm{Ad}$

(d) ${\mathrm{E}}^2\mathrm{Ad}/{\mathrm{\epsilon }}_0$

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 point A, mid-way between the two charges +q and +q, is

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

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

(c)    Zero

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

Three charges, each +q, are placed at the corners of an isosceles triangle ABC of sides BC and AC. D and E are the mid-points of BC and CA. The work done in taking a charge Q from D to E is:
(Given, BC=AC=2a)

1. $\frac{qQ}{8{\mathrm{\pi \epsilon }}_{0}a}$                                 
2. $\frac{qQ}{4{\mathrm{\pi \epsilon }}_0\mathrm{a}}$
3. zero                           
4. $\frac{3qQ}{4{\mathrm{\pi \epsilon }}_0\mathrm{a}}$

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. \(3C, 3V\)

Five identical plates each of area \(A\) are joined as shown in the figure. The distance between the plates is \(d\). The plates are connected to a potential difference of \(V\) volts. The charge on plates \(1\) and \(4\) will be:

1. \(-\frac{\varepsilon_{0} A V}{d} ,  \frac{2\varepsilon_{0} A V}{d}\)
2. \(\frac{\varepsilon_{0} A V}{d} ,  \frac{2\varepsilon_{0} A V}{d}\)
3. \(\frac{\varepsilon_{0} A V}{d} , -\frac{2\varepsilon_{0} A V}{d}\)
4. \(-\frac{\varepsilon_{0} A V}{d} ,  -\frac{2\varepsilon_{0} A V}{d}\)

A network of four capacitors of capacity equal to \(C_1 = C, C_2 = 2C, C_3 = 3C\) and \(C_4 = 3C\) are connected in a battery as shown in the figure. The ratio of the charges on \(C_2\) and \(C_4\) is:
       
1. \(\frac{22}{3}\)
2. \(\frac{3}{22}\)
3. \(\frac{7}{4}\)
4. \(\frac{4}{7}\)

An electric field is given by ${\mathrm{E}}_{\mathrm{x}} = – 2{\mathrm{x}}^3 \mathrm{kN}/\mathrm{C}$. The potential of the point (1, –2), if potential of the point (2, 4) is taken as zero, is –

1.  $-7.5\times {10}^3 \mathrm{V}$                     

2.  $7.5\times {10}^3 \mathrm{V}$

3.  $-15\times {10}^3 \mathrm{V}$                     

4.  $15\times {10}^3 \mathrm{V}$