Question

A metal sphere of radius R = 0.4 m, carrying charge q 10⁻¹⁰ C, is surrounded by a thick concentric metal shell (inner radius a 0.8 m, outer radius 6 1.0 m). The shell carries no net charge.

Now the outer surface is touched to a grounding wire, which drains off charge and lowers its potential to zero (same as at infinity).
(e) What is the surface charge density atra now (for the grounded setup)? Give the answer in units of pC/m² = 10⁻¹²C/m² to three significant figures.
(f) Find the potential at the center, using infinity as a reference point. Give the answer in SI units of V to three significant figures.
Give all answers to three significant figures. If the answer is known numerically exactly, state this instead.

Answer 1

The surface charge density σ at the inner radius of a grounded metallic shell that encloses a charged metal sphere is -3.973 pC/m^2. The potential at the center of the sphere, with infinity as the reference point, is approximately 0.225 V.

Step-by-step explanation:

To find the surface charge density σ at radius 'a' after grounding, we must remember that a grounded conductor will have a potential of zero. Since the metal shell originally carries no net charge and the sphere inside has a charge q, when the shell is grounded, it will acquire charge -q on its inner surface to cancel the potential due to the sphere charge q, thus ensuring a net potential of zero. Given the charge q = -10^-10 C, and the inner radius of the shell a = 0.8 m, the surface charge density σ is calculated as the charge per unit area:

σ = q / (4πa^2)

σ = (-10^-10 C) / (4π(0.8 m)^2)

σ = -3.973 x 10^-12 C/m^2

Expressed in pC/m^2, it is σ = -3.973 pC/m^2.

To find the potential at the center, considering infinity as the reference point where the potential is zero, we use the formula for the potential V due to a point charge q at a distance r:

V = k * q / r

For the center of the sphere (r = 0), since we consider the effects of only external charges:

V = k * q / R

V = (8.988 x 10^9 N*m^2/C^2) * (10^-10 C) / 0.4 m

V = 2.247 x 10^-1 V

This is the potential at the center considering only the contribution from the sphere's charge and ignoring the induced charge on the metal shell.