Question

Ex. Three concentric conducting spherical shells of radii R, 2R and 3R carry charges Q, -2Q and 3Q respectively. The charges at the six surfaces (starting from the innermost) are

• A. 0, Q, -Q, -Q, +Q, +2Q
• B. 0, Q, -2Q, +Q, +Q, +2Q
• C. 0, Q, 0, -2Q, 0, +3Q
• D. 0, Q, -2Q, 0, +Q, +2Q

Answer 1

Answer: (A)

A

Answer 2

The problem involves three concentric conducting spherical shells with radii R, 2R, and 3R, carrying total charges Q, -2Q, and 3Q respectively. We need to find the charges on the inner and outer surfaces of each shell, starting from the innermost shell.

Let the shells be S1, S2, and S3 with radii R, 2R, and 3R, and total charges $Q_1 = Q$, $Q_2 = -2Q$, and $Q_3 = 3Q$. There are six surfaces:

1. Inner surface of S1 (radius R, inside)
2. Outer surface of S1 (radius R, outside)
3. Inner surface of S2 (radius 2R, inside)
4. Outer surface of S2 (radius 2R, outside)
5. Inner surface of S3 (radius 3R, inside)
6. Outer surface of S3 (radius 3R, outside)

For a conducting shell in electrostatic equilibrium, the electric field inside the material of the conductor is zero. Also, any net charge on a conductor resides on its surface(s).

Shell S1 (radius R, total charge Q):

Consider a Gaussian surface within the material of S1. Since the electric field inside a conductor is zero, the net charge enclosed by this surface is zero. The region inside the inner surface of S1 is empty. Thus, the charge on the inner surface of S1 is 0.

Let $q_{1,in}$ be the charge on the inner surface of S1 and $q_{1,out}$ be the charge on the outer surface of S1.

$q_{1,in} = 0$.

The total charge on S1 is $Q_1 = q_{1,in} + q_{1,out} = Q$.

$0 + q_{1,out} = Q \implies q_{1,out} = Q$.

Charges on surfaces of S1: 0, Q.

Shell S2 (radius 2R, total charge -2Q):

Consider a Gaussian surface within the material of S2. The net charge enclosed is zero. The enclosed charge consists of the charge on the outer surface of S1 and the charge on the inner surface of S2.

Let $q_{2,in}$ be the charge on the inner surface of S2 and $q_{2,out}$ be the charge on the outer surface of S2.

Enclosed charge = $q_{1,out} + q_{2,in} = 0$.

$Q + q_{2,in} = 0 \implies q_{2,in} = -Q$.

The total charge on S2 is $Q_2 = q_{2,in} + q_{2,out} = -2Q$.

$-Q + q_{2,out} = -2Q \implies q_{2,out} = -2Q + Q = -Q$.

Charges on surfaces of S2: -Q, -Q.

Shell S3 (radius 3R, total charge 3Q):

Consider a Gaussian surface within the material of S3. The net charge enclosed is zero. The enclosed charge consists of the charge on the outer surface of S2 and the charge on the inner surface of S3.

Let $q_{3,in}$ be the charge on the inner surface of S3 and $q_{3,out}$ be the charge on the outer surface of S3.

Enclosed charge = $q_{2,out} + q_{3,in} = 0$.

$-Q + q_{3,in} = 0 \implies q_{3,in} = Q$.

The total charge on S3 is $Q_3 = q_{3,in} + q_{3,out} = 3Q$.

$Q + q_{3,out} = 3Q \implies q_{3,out} = 3Q - Q = 2Q$.

Charges on surfaces of S3: Q, 2Q.

The charges at the six surfaces starting from the innermost are: Inner S1: 0 Outer S1: Q Inner S2: -Q Outer S2: -Q Inner S3: Q Outer S3: 2Q

The sequence of charges is 0, Q, -Q, -Q, Q, 2Q. Comparing this sequence with the given options, it matches option A.