Consider two charged metallic spheres `S_(1)`, and `S_(2)`, of radii `R_(1)`, and `R_(2)` respectively. The electric fields `E_(1)`, (on `S_(1)`,) and `E_(2)`, (on `S_(2)`) their surfaces are such that `E_(1)//E_(2) = R_(1)//R_(2)`. Then the ratio `V_(1)`(on `S_(1)`)/`V_(2)` (on `S_(2)`) of the electrostatic potential on each sphere is

Options

A. `(R_(1)//R_(2))^(2)`
B. `((R_(1))/(R_(2)))^(3)`
C. `(R_(2)//R_(1))`
D. `R_(1)//R_(2)`

Answer

To solve the problem, we need to find the ratio of the electrostatic potentials \( V_1 \) and \( V_2 \) on the surfaces of the two charged metallic spheres \( S_1 \) and \( S_2 \) respectively. Given that the electric fields \( E_1 \) and \( E_2 \) on the surfaces of the spheres satisfy the condition \( \frac{E_1}{E_2} = \frac{R_1}{R_2} \), we can proceed with the following steps:

### Step-by-Step Solution:

1. **Understanding Electric Field and Potential**:
The electric field \( E \) at the surface of a charged metallic sphere is given by:
\[ E = \frac{kQ}{R^2} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge on the sphere, and \( R \) is the radius of the sphere.

2. **Relating Electric Field to Potential**:
The electrostatic potential \( V \) at the surface of a charged sphere is given by:
\[ V = \frac{kQ}{R} \]

3. **Expressing Electric Field in Terms of Potential**:
We can express the electric field \( E \) in terms of the potential \( V \):
\[ E = \frac{V}{R} \]
Therefore, for sphere \( S_1 \):
\[ E_1 = \frac{V_1}{R_1} \]
and for sphere \( S_2 \):
\[ E_2 = \frac{V_2}{R_2} \]

4. **Finding the Ratio of Potentials**:
From the expressions for electric fields, we can write:
\[ V_1 = E_1 R_1 \]
\[ V_2 = E_2 R_2 \]
Thus, the ratio of potentials becomes:
\[ \frac{V_1}{V_2} = \frac{E_1 R_1}{E_2 R_2} \]

5. **Substituting the Given Condition**:
We know from the problem statement that:
\[ \frac{E_1}{E_2} = \frac{R_1}{R_2} \]
Substituting this into our equation for the ratio of potentials gives:
\[ \frac{V_1}{V_2} = \frac{R_1}{R_2} \cdot \frac{R_1}{R_2} = \left(\frac{R_1}{R_2}\right)^2 \]

6. **Final Result**:
Therefore, the ratio of the electrostatic potentials on the surfaces of the two spheres is:
\[ \frac{V_1}{V_2} = \left(\frac{R_1}{R_2}\right)^2 \]

### Conclusion:
The final answer is:
\[ \frac{V_1}{V_2} = \left(\frac{R_1}{R_2}\right)^2 \]