Question

Consider three metal spherical shells A, B, and C, each of radius \( R \). Each shell has a concentric metal ball of radius \( R/10 \). The spherical shells A, B, and C are given charges \( +6q, -4q, \) and \( 14q \) respectively. Their inner metal balls are also given charges \( -2q, +8q \) and \( -10q \) respectively. Compare the magnitude of the electric fields due to shells A, B, and C at a distance \( 3R \) from their centers.

Hint:

The electric field outside a spherical shell depends on the total charge on the shell. It is treated as if the charge is concentrated at the center of the shell.

Answer 1

For a spherical shell, the electric field outside the shell at a distance \( r \) from the center is given by the formula: \[ E = \frac{kQ}{r^2} \] where \( Q \) is the total charge on the shell, and \( k \) is Coulomb's constant. At a distance \( 3R \) from the center, the electric field due to each shell is the same as if the entire charge were concentrated at the center of the shell. Therefore, the electric fields due to shells A, B, and C at a distance \( 3R \) are given by: \[ E_A = \frac{k(6q)}{(3R)^2} = \frac{k(6q)}{9R^2} \] \[ E_B = \frac{k(-4q)}{(3R)^2} = \frac{k(-4q)}{9R^2} \] \[ E_C = \frac{k(14q)}{(3R)^2} = \frac{k(14q)}{9R^2} \] Thus, the magnitude of the electric fields is: \[ |E_A| = \frac{6kq}{9R^2}, \quad |E_B| = \frac{4kq}{9R^2}, \quad |E_C| = \frac{14kq}{9R^2} \] The electric field due to shell C is the largest, followed by shell A, and then shell B.

Answer 2

1. Gauss's Law:

According to Gauss's Law, the electric field outside a spherical shell (where the radius is larger than the shell's radius) depends only on the net charge enclosed within the shell. The electric field at a distance \( r \) from the center of a spherical shell with total charge \( Q \) is given by:

\[ E = \frac{kQ}{r^2} \]

Where:

• \( E \) is the electric field at distance \( r \),
• \( k \) is Coulomb’s constant \((9 \times 10^9 \, \text{N.m}^2/\text{C}^2)\),
• \( Q \) is the total charge enclosed by the spherical shell,
• \( r \) is the radial distance from the center of the shell (which is \( 3R \) in this case).

 

2. Charge Distribution:

Each shell consists of two charges:

• The charge on the outer spherical shell,
• The charge on the inner concentric metal ball.

 

To calculate the electric field outside each spherical shell at a distance \( 3R \), we will only consider the total charge enclosed by the outer shell. The charge on the inner concentric metal ball does not contribute to the electric field outside the shell because the electric field inside a conducting shell is zero.

3. Total Charge on Each Shell:

• Shell A: Total charge \( Q_A = (+6q) + (-2q) = 4q \),
• Shell B: Total charge \( Q_B = (-4q) + (+8q) = 4q \),
• Shell C: Total charge \( Q_C = (+14q) + (-10q) = 4q \).

Thus, the net charge on all three shells is \( 4q \).

4. Electric Field at a Distance \( 3R \):

The electric field at a distance \( 3R \) from the center of each shell is given by:

\[ E = \frac{k \cdot 4q}{(3R)^2} = \frac{k \cdot 4q}{9R^2} \]

5. Conclusion:

Since the net charge on each shell is the same and the electric field depends only on the net charge enclosed within the spherical shell, the electric field at a distance \( 3R \) from the center of all three shells (A, B, and C) will be the same.

• The magnitudes of the electric fields at a distance \( 3R \) from their centers are identical for all three shells.