Question

Three uniformly charged concentric shells are kept as shown in the diagram. Charges on individual shells are as shown. Find the final potential on each shell :

• A. $V_{A} = \frac{KQ_{1}}{a} + \frac{KQ_{2}}{b} + \frac{KQ_{3}}{c}$, $V_{B} = \frac{K(Q_{1}+Q_{2}+Q_{3})}{c}$, $V_{C} = \frac{KQ_{1}}{b} + \frac{KQ_{2}}{b} + \frac{KQ_{3}}{c}$
• B. $V_{A} = \frac{KQ_{1}}{b} + \frac{KQ_{2}}{b} + \frac{KQ_{3}}{c}$, $V_{B} = \frac{KQ_{1}}{a} + \frac{KQ_{2}}{b} + \frac{KQ_{3}}{c}$, $V_{C} = \frac{K(Q_{1}+Q_{2}+Q_{3})}{c}$
• C. $V_{A} = \frac{K(Q_{1}+Q_{2}+Q_{3})}{c}$, $V_{B} = \frac{KQ_{1}}{b} + \frac{KQ_{2}}{b} + \frac{KQ_{3}}{c}$, $V_{C} = \frac{KQ_{1}}{a} + \frac{KQ_{2}}{b} + \frac{KQ_{3}}{c}$
• D. $V_{A} = \frac{KQ_{1}}{a} + \frac{KQ_{2}}{b} + \frac{KQ_{3}}{c}$, $V_{B} = \frac{KQ_{1}}{b} + \frac{KQ_{2}}{b} + \frac{KQ_{3}}{c}$, $V_{C} = \frac{K(Q_{1}+Q_{2}+Q_{3})}{c}$

Hint:

To quickly find the potential at any concentric shell, remember this rule: For charges on shells *smaller* than the point of interest, use the distance to the point. For charges on shells *larger* than the point, use the radius of that larger shell. This is because the potential inside a charged shell is uniform and equal to its surface potential.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We have three concentric spherical conducting shells A, B, and C with radii $a, b,$ and $c$ respectively ($a<b<c$). Charges $Q_{1}, Q_{2},$ and $Q_{3}$ are given to shells A, B, and C. We need to find the total electric potential at the surface of each shell.
Step 2: Key Formula or Approach:
The potential at any point is the scalar sum of the potentials created by each charge individually.
For a thin spherical shell of radius $R$ with charge $Q$:
1. Potential inside the shell ($r \le R$): $V = \frac{KQ}{R}$ (constant and same as on the surface).
2. Potential outside the shell ($r>R$): $V = \frac{KQ}{r}$ (as if all charge is concentrated at the center).
Step 3: Detailed Explanation:
Let's find the potential for each shell by summing the contributions from all three charges.
Potential of shell A (radius a):
- Contribution from its own charge $Q_{1}$: $\frac{KQ_{1}}{a}$
- Contribution from shell B (since A is inside B): $\frac{KQ_{2}}{b}$
- Contribution from shell C (since A is inside C): $\frac{KQ_{3}}{c}$
$V_{A} = \frac{KQ_{1}}{a} + \frac{KQ_{2}}{b} + \frac{KQ_{3}}{c}$
Potential of shell B (radius b):
- Contribution from shell A (since B is outside A): $\frac{KQ_{1}}{b}$
- Contribution from its own charge $Q_{2}$: $\frac{KQ_{2}}{b}$
- Contribution from shell C (since B is inside C): $\frac{KQ_{3}}{c}$
$V_{B} = \frac{KQ_{1}}{b} + \frac{KQ_{2}}{b} + \frac{KQ_{3}}{c} = \frac{K(Q_{1} + Q_{2})}{b} + \frac{KQ_{3}}{c}$
Potential of shell C (radius c):
- Contribution from shell A (since C is outside A): $\frac{KQ_{1}}{c}$
- Contribution from shell B (since C is outside B): $\frac{KQ_{2}}{c}$
- Contribution from its own charge $Q_{3}$: $\frac{KQ_{3}}{c}$
$V_{C} = \frac{KQ_{1}}{c} + \frac{KQ_{2}}{c} + \frac{KQ_{3}}{c} = \frac{K(Q_{1} + Q_{2} + Q_{3})}{c}$
Comparing this with the given options, Option (D) correctly lists these three potentials.
Step 4: Final Answer:
The correct set of potentials is given by option (D).