Two uniformly charged solid spheres are such that `E_1` is electric field at surface of `1^(st)` sphere due to itself. `E_2` is electric field at surface of `2^(nd)` sphere due to itself. `r_1, r_2` are radius of `1^(st)` and `2^(nd)` sphere respectively. If `E_1/E_2=r_1/r_2` then ratio of potential at the surface of spheres `1^(st)` and `2^(nd)` due to their self charges is :

Options

A. `r_1/r_2`
B. `(r_1/r_2)^2`
C. `r_2/r_1`
D. `(r_2/r_1)^2`

Explanation

To solve the problem, we need to find the ratio of the electric potentials at the surfaces of two uniformly charged solid spheres based on the given relationship between their electric fields and radii.

### Step-by-Step Solution:

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

2. **Understanding the Potential**:
The electric potential \( V \) at the surface of a uniformly charged solid sphere is given by:
\[ V = \frac{kQ}{R} \]

3. **Relating Electric Field and Potential**:
We can relate the electric field and potential using the formula:
\[ E = \frac{V}{R} \quad \Rightarrow \quad V = E \cdot R \]

4. **Calculating Potentials for Both Spheres**:
For the first sphere:
\[ V_1 = E_1 \cdot r_1 \]
For the second sphere:
\[ V_2 = E_2 \cdot r_2 \]

5. **Finding the Ratio of Potentials**:
The ratio of the potentials \( \frac{V_1}{V_2} \) can be expressed as:
\[ \frac{V_1}{V_2} = \frac{E_1 \cdot r_1}{E_2 \cdot r_2} \]

6. **Using the Given Relationship**:
We are given that:
\[ \frac{E_1}{E_2} = \frac{r_1}{r_2} \]
Substituting this into our ratio of potentials:
\[ \frac{V_1}{V_2} = \frac{E_1}{E_2} \cdot \frac{r_1}{r_2} = \left(\frac{r_1}{r_2}\right) \cdot \frac{r_1}{r_2} = \left(\frac{r_1}{r_2}\right)^2 \]

7. **Final Result**:
Thus, the ratio of the potentials at the surfaces of the two spheres is:
\[ \frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^2 \]

### Conclusion:
The ratio of the potential at the surface of the first sphere to the potential at the surface of the second sphere is given by:
\[ \frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^2 \]