Two conductors carrying equal and opposite charge produce a non uniform electric field along X – axis given by…

Q: Two conductors carrying equal and opposite charge produce a non uniform electric field along X – axis given by $E = \frac{Q}{\epsilon_0 A} (1 + B x^2)$  where A and B are constants. Separation between the conductors along X=axis is X. Find the capacitance of the capacitor formed.

Sol. Potential difference between the conductors is given by

$\large V = V_+ – V_- = \int_{0}^{x} E dx $

$\large V = \int_{0}^{x} \frac{Q}{\epsilon_0 A} (1 + B x^2) dx $

$\large V = \frac{Q}{\epsilon_0 A} [x + \frac{B x^3}{3}]_{0}^{x}$ 

$\large V = \frac{Q}{\epsilon_0 A} [x + \frac{B x^3}{3}] $

$\large C = \frac{Q}{V} = \frac{\epsilon_0 A}{x + \frac{B x^3}{3}}$