Capacitance of a Random Conductor

Q. The capacitance of a cylindrical capacitor of length $72\text{mm}$ is $2\text{pF}$. If the ratio of radii of its outer cylindrical conductor to that of its inner cylindrical conductor is $e^x$, then $x$ is

1. $4$
2. $2$
3. $3$
4. $5$

Q. The charge per unit volume inside an insulating sphere of radius $R$ is given as a function of distance from centre $\rho \left(r\right)=cr$ , Where $0<r<R$. The magnitude of electric field at a distance ${}^{\prime }{b}^{\prime }$ from the centre of sphere will be

1. $\frac{c}{\pi b^4}$
2. $\frac{c{b}^4}{{ϵ}_0}$
3. $\frac{c{b}^2}{4{ϵ}_0}$
4. $\frac{cb}{\pi {ϵ}_0}$

Q. A spherical drop of capacitance $12\mu F$ is broken into eight drops of equal radius. Then, what is the capacitance of each small drop in $\mu F$?

Q. The charge on a conductor is positive Q and its potential is V
(i.e. the electric potential on its surface)
If some more positive charge is added to it, what are the possible values of its new potential?

1. $\frac{3V}{2}$
2. $\frac{2V}{3}$
3. $\frac{5V}{6}$
4. $\frac{4V}{3}$

Q. Two long cylindrical wires of radius 'a' have linear charge densities $+\lambda$ and $-\lambda$ respectively. The distance between them is na (n > 1). Find the capacitance per unit length of the system.

1. $\pi {ϵ}_0$
   
2. $\frac{\pi {ϵ}_0}{n}$
3. $\frac{\pi {ϵ}_0}{ln\left(n\right)}$
4. $\frac{2\pi {ϵ}_0}{ln\left(n\right)}$