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))

the arctic is a region of the planet, north of the arctic circle, and includes the arctic ocean, greenland, baffin island, other smaller northern islands, and the far northern parts of europe, russia (siberia), alaska and canada.

a straight line sloped