Two conducting concentric, hollow spheres A and B have radii a and b respectively, with A inside B. Their common potentials is V. A is now given some charge such that its potential becomes zero. The potential of B will now be

Options

A. 0
B. `V(1-a//b)`
C. ` Va//b`
D. `Vb//a`

Explanation

To solve the problem, we need to analyze the situation of two concentric conducting hollow spheres, A and B, with radii a and b respectively. The potential of sphere A is made zero by giving it a certain charge. We need to find the new potential of sphere B.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions:**
- Initially, both spheres A and B have a common potential \( V \).
- Since they are conducting spheres, the electric field inside each conductor is zero, meaning the potential is constant throughout the conductor.

2. **Charge Distribution:**
- Let’s assume sphere B has an initial charge \( Q_B \) and sphere A has an initial charge of zero.
- The potential \( V \) of sphere B can be expressed as:
\[ V = \frac{k Q_B}{b} \]
where \( k \) is Coulomb's constant.

3. **Applying Charge to Sphere A:**
- Sphere A is given a charge \( Q_A \) such that its potential becomes zero.
- The potential \( V_A \) of sphere A is given by:
\[ V_A = \frac{k Q_A}{a} + \frac{k Q_B}{b} = 0 \]
This means:
\[ \frac{k Q_A}{a} = -\frac{k Q_B}{b} \]
Simplifying this gives:
\[ Q_A = -\frac{a}{b} Q_B \]

4. **Calculating the New Potential of Sphere B:**
- The potential of sphere B after the charge is added to sphere A can be calculated using the new charge distribution.
- The potential \( V_B \) of sphere B is now influenced by both its own charge and the charge on sphere A:
\[ V_B = \frac{k Q_B}{b} + \frac{k Q_A}{b} \]
Substituting \( Q_A \):
\[ V_B = \frac{k Q_B}{b} - \frac{k \left( \frac{a}{b} Q_B \right)}{b} \]
Simplifying this:
\[ V_B = \frac{k Q_B}{b} - \frac{k a Q_B}{b^2} \]
Factoring out \( \frac{k Q_B}{b} \):
\[ V_B = \frac{k Q_B}{b} \left( 1 - \frac{a}{b} \right) = \frac{k Q_B}{b} \cdot \frac{b - a}{b} \]
Thus, we can express it as:
\[ V_B = V \left( 1 - \frac{a}{b} \right) \]

5. **Final Expression:**
- Therefore, the potential of sphere B after sphere A is given a charge such that its potential becomes zero is:
\[ V_B = V \left( 1 - \frac{a}{b} \right) \]