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.

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moya1316 moya1316 · Answer 1 · 2020-03-22T03:35:39+01:00

Answer:

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))