A metal sphere of radius R, carrying charge q, is surrounded...

A metal sphere of radius $R$, carrying charge $q$, is surrounded by a thick concentric metal shell (inner radius $a$, outer radius $b$, as in Fig. 2.48 ). The shell carries no net charge. (a) Find the surface charge density $\sigma$ at $R,$ at $a,$ and at $b$. (b) Find the potential at the center, using infinity as the reference point. (c) Now the outer surface is touched to a grounding wire, which lowers its potential to zero (same as at infinity). How do your answers to (a) and (b) change?

A metal sphere of radius $R$, carrying charge $q$, is surrounded by a thick concentric metal shell (inner radius $a$, outer radius $b$, as in Fig. 2.48 ). The shell carries no net charge.
(a) Find the surface charge density $\sigma$ at $R,$ at $a,$ and at $b$.
(b) Find the potential at the center, using infinity as the reference point.
(c) Now the outer surface is touched to a grounding wire, which lowers its potential to zero (same as at infinity). How do your answers to (a) and (b) change?

Key Concepts

Grounding

Grounding a conductor connects it to the Earth, effectively setting its potential to zero. This process alters the charge distribution on the conductor by allowing charges to flow between the conductor and the ground, and it affects measurements of both surface charge density and electrostatic potential in the system.

Electrostatic Potential

Electrostatic potential represents the potential energy per unit charge at each point in an electric field, typically calculated relative to a reference point such as infinity. In spherical configurations, the potential at different points is determined by the contributions from distributed surface charges and must satisfy continuity conditions at the surfaces of conductors.

Surface Charge Distribution

Surface charge distribution deals with how excess electric charges are arranged on the surface of conductors. In problems involving concentric spherical conductors, symmetry and boundary conditions dictate that the induced charges arrange in specific patterns on different surfaces, influenced by the nearby charges and the conductor's geometry.

Conductors in Electrostatic Equilibrium

In electrostatics, conductors are in equilibrium when the electric field within their bulk is zero. This condition causes excess charges to reside on the surface and ensures that the potential throughout the conductor is constant. This concept is crucial when determining surface charge densities and understanding the behavior of charges on spherical metal bodies.

Gauss's Law

Gauss's Law relates the net electric flux through a closed surface to the charge enclosed by that surface. This principle is essential for analyzing electric fields and determining charge distributions in systems with symmetry, such as spherical geometries, by simplifying the computation of fields and potentials.

Transcript

00:01 Hi there, so for this problem, we are told that a metal sphere of radius capital r carrying a charge q is surrounded by a thin concentric metal shield.

00:12 Within a radius a and outer radius b, as is shown in this figure.

00:18 Now, the shell carries a no net charge.

00:22 So for part a of this problem, we need to find the surface charge density at the radius r, at the radius l, and at the radius b.

00:33 So to obtain that, we know that the charge density at the radius art is going to be equal to the total charge q divided by four times pi times the radius to the square, because we are just enclosing the sphere with radius r, capital r.

01:07 Now, the charge density at a is going to be minus q because we know that this is the indus charge because of the inner sphere.

01:21 That's why it is minus.

01:24 And this divided by 4 times pi times a square.

01:28 And finally, the charge density at b, the surface charge density, in that case, is q.

01:36 And this divided by 4 times pi times b square.

01:41 So that's a solution for part a of this problem.

01:45 Now for part b, we are asked to find the potential at the center using infinity as the reference point.

01:55 So in this case, we use the definition of the potential.

02:00 Remember that in this case, we need to determine the potential at zero at the center.

02:05 And we know that the potential is defined as the integral, in this case from minus from infinity to zero, of the product between the electric field and the differential in the length.

02:19 That length is in the separation.

02:23 So this is going, we can separate this into four integrals.

02:32 The first one is minus from infinity to b.

02:37 Which is the outer radius...

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