Question

Two coaxial conducting cylindrical shells have equal and opposite charges. The inner shell has charge +q and an outer radius a, and the outer shell has charge -q and an inner radius b. The length of each cylindrical shell is L, and L is very long compared with b. Find the potential difference Va − Vb between the shells. (Use any variable or symbol stated above along with the following as necessary: k for the Coulomb's constant.)

Answer 1

Answer:

$\Delta V = \frac{q ln(\frac{b}{a})}{2\pi \epsilon_0 L}$

Explanation:

As we know that the charge per unit length of the long cylinder is given as

$\lambda = \frac{q}{L}$

here we know that the electric field between two cylinders is given by

$E = \frac{2k\lambda}{r}$

now we know that electric potential and electric field is related to each other as

$\Delta V = - \int E.dr$

$\Delta V = -\int_a^b (\frac{2k\lambda}{r})dr$

$\Delta V = -2k \lambda ln(\frac{b}{a})$

$\Delta V = \frac{\lambda ln(\frac{b}{a})}{2\pi \epsilon_0}$

$\Delta V = \frac{q ln(\frac{b}{a})}{2\pi \epsilon_0 L}$