Consider two charged metallic spheres `S_1` and `S_2` of radii R and 4R, respectively. The electric fields `E_1` (on `S_1` ) and `E_2` (on `S_2`) on their surfaces are such that `E_1/E_2 = 5/1` . Then the ratio `V_1` (on `S_1` ) / `V_1` (on `S_2` ) of the electrostatic potentials on each sphere is :

Options

A. `(5/4)`
B. `(3/2)^3`
C. `(4/5)`
D. `(16/9)`

Explanation

To solve the problem, we need to find the ratio of the electrostatic potentials \( V_1 \) and \( V_2 \) on the surfaces of two charged metallic spheres \( S_1 \) and \( S_2 \) with given radii and electric fields.

### Step-by-Step Solution

1. **Understanding the Electric Field and Potential Relation**:
The electric field \( E \) on the surface of a charged sphere is given by:
\[ E = \frac{Q}{4\pi \epsilon_0 R^2} \]
where \( Q \) is the charge on the sphere, \( R \) is the radius, and \( \epsilon_0 \) is the permittivity of free space.

The potential \( V \) on the surface of the sphere is given by:
\[ V = \frac{Q}{4\pi \epsilon_0 R} \]

2. **Finding the Ratios**:
For sphere \( S_1 \) with radius \( R \):
\[ E_1 = \frac{Q_1}{4\pi \epsilon_0 R^2}, \quad V_1 = \frac{Q_1}{4\pi \epsilon_0 R} \]

For sphere \( S_2 \) with radius \( 4R \):
\[ E_2 = \frac{Q_2}{4\pi \epsilon_0 (4R)^2} = \frac{Q_2}{64\pi \epsilon_0 R^2}, \quad V_2 = \frac{Q_2}{4\pi \epsilon_0 (4R)} = \frac{Q_2}{16\pi \epsilon_0 R} \]

3. **Using the Given Ratio of Electric Fields**:
We know from the problem statement that:
\[ \frac{E_1}{E_2} = \frac{5}{1} \]
Substituting the expressions for \( E_1 \) and \( E_2 \):
\[ \frac{\frac{Q_1}{4\pi \epsilon_0 R^2}}{\frac{Q_2}{64\pi \epsilon_0 R^2}} = 5 \]
Simplifying this gives:
\[ \frac{Q_1}{Q_2} \cdot 16 = 5 \quad \Rightarrow \quad \frac{Q_1}{Q_2} = \frac{5}{16} \]

4. **Finding the Ratio of Potentials**:
Now we can find the ratio of the potentials:
\[ \frac{V_1}{V_2} = \frac{\frac{Q_1}{4\pi \epsilon_0 R}}{\frac{Q_2}{16\pi \epsilon_0 R}} = \frac{Q_1}{Q_2} \cdot 4 \]
Substituting \( \frac{Q_1}{Q_2} = \frac{5}{16} \):
\[ \frac{V_1}{V_2} = \frac{5}{16} \cdot 4 = \frac{5}{4} \]

5. **Final Answer**:
The ratio of the electrostatic potentials on each sphere is:
\[ \frac{V_1}{V_2} = \frac{5}{4} \]