A disk ring of inner radius a and outer radius b lying on the y − z plane has a uniform surface electric charge density +σ(> 0) on it. (a) Find the electric potential at the point P on the axis of the ring and of a distance x from the center of the ring, taking the zero point of the electric potential at infinity. (b) Find the electric field (both direction and magnitude) at the point P.

a)  V = k 2π σ (√(b² + x²) - √ (a² + x²)) ,  

b)  E = - k 2π σ x (1 /√(b² + x²) - 1 /√(a² + x²))

Explanation:

a) The expression for the electric potential is

        V = k ∫ dq / r

For this case, consider the disk formed by a series of concentric rings of radius r and width dr, the distance of each ring to point P

         R = √(x² + r²)

The charge on a ring is

        σ = dq / dA

The area of a ring is

        A = π r

        dA = 2π r dr

So the charge is

        dq = σ  2π r dr

We substitute

       V = k σ 2pi ∫ r dr / √(r² + x²)

We integrate

       V = k 2π σ √(r² + x²)

We evaluate from the lower limit r = a to the upper limit r = b

      V = k 2π σ (√(b² + x²) - √ (a² + x²))

b) the electric field and the potential are related

        E = - dV / dx

        E = - k 2π σ (1/2 2x /√(b² + x²) - ½ 2x /√(a² + x²))

        E = - k 2π σ x (1 /√(b² + x²) - 1 /√(a² + x²))

search it up!

explanation: