Find the surface density of a charge on a conductive sphere of radius 10 cm with an electric field of 1800 N/C.

Question

At 12 cm in the radial direction from a conductive sphere of radius 10 cm, the electric field is 1800 N/C and its orientation points towards the interior of the sphere. Find the surface density of the charge on the sphere.

The answer is -7.71 x 10^-8 C/m^2 but I'm not sure how they found it.

Answer 1

The charge on a sphere produces an electric field as if the charge is located at the center of the sphere.

The electric field strength is given by E = kQ / r². As we are given the electric field and other variables are known, We can solve for Q to get 0.22² x 1800 / 8.99 x 10⁹ = 9.69 x 10^-9 coulombs.

The spere itself has a radius of 0.1 m, so the surface area is 4 x pi x r² = 4 x 3.14 x 0.1² = 1.26 x 10^-1 m².

The surface charge density is 9.69 x 10^-9 coulombs / 1.256 x 10^-1 = -7.7 1x 10^-8 C/m². The sign is negative as the direction is toward the center.