Question

Coaxial Cylinders. A long metal cylinder with radius a is supported on an insulating stand on the axis of a long, hollow, metal tube with radius b. The positive charge per unit length on the inner cylinder is$\mathit{\lambda }$, and there is an equal negative charge per unit length on the outer cylinder. (a) Calculate the potential V (2) for (i) r<a; (ii) a<r<b; (iii) r>b. (Hint: The net potential is the sum of the potentials due to the individual conductors.) Take V = 0 at r = b. (b) Show that the potential of the inner cylinder with respect to the outer is

(c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the cylinders has magnitude

(d) What is the potential difference between the two cylinders if the outer cylinder has no net charge?

Short answer

Potential

Potential of the inner cylinder is

Electric field

Potential difference between two cylinders

Step-by-step solution

Step 1:

The potential of a very long charge with a linear charge density is at a distance (r) is here $r_0$is the distance from the axis.

So, the potential of the metal tube of radius (b) and charge per unit length is$\left(-\lambda \right)$

And the potential of the metal tube of radius (a) and charge per unit length is

Therefore, the net potential is

Step 2:

The potential of the inner cylinder with respect to the outer cylinder at (r=b):

Step 3:

From the solution of part (a)

Hence, the Electric field

Step 4:

Therefore, the Potential difference between two the cylinders is due to the inner cylinder;