Consider two charged metallic spheres `S_(1) and S_(2)` of radii 3R and R, respectively. The electric potential `V_(1) (on S_(1) ) and (on S_(2))` on their surfaces are such that `(V_(1))/(V_(2)) = (4)/(1)`. Then the ratio `E_(1)(on S_(1)) / E_(2) (on S_(2))` of the electric fields on their surfaces is:

Options

A. `((5)/(4))`
B. `((4)/(3))`
C. `((4)/(5))`
D. `((3)/(4))`

Explanation

To solve the problem, we need to find the ratio of the electric fields \( E_1 \) and \( E_2 \) on the surfaces of two charged metallic spheres \( S_1 \) and \( S_2 \) with given radii and potentials.

### Step-by-step Solution:

1. **Understand the relationship between electric potential and electric field:**
The electric potential \( V \) at the surface of a charged sphere is given by the formula:
\[ V = \frac{kQ}{r} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( r \) is the radius of the sphere.

The electric field \( E \) at the surface of a charged sphere is given by:
\[ E = \frac{kQ}{r^2} \]

2. **Define the variables for the two spheres:**
For sphere \( S_1 \) (radius \( 3R \)):
- Electric potential:
\[ V_1 = \frac{kQ_1}{3R} \]
- Electric field:
\[ E_1 = \frac{kQ_1}{(3R)^2} = \frac{kQ_1}{9R^2} \]

For sphere \( S_2 \) (radius \( R \)):
- Electric potential:
\[ V_2 = \frac{kQ_2}{R} \]
- Electric field:
\[ E_2 = \frac{kQ_2}{R^2} \]

3. **Use the given ratio of potentials:**
We know that:
\[ \frac{V_1}{V_2} = \frac{4}{1} \]
This implies:
\[ V_1 = 4V_2 \]

4. **Express \( V_1 \) in terms of \( V_2 \):**
Substituting the expressions for \( V_1 \) and \( V_2 \):
\[ \frac{kQ_1}{3R} = 4 \cdot \frac{kQ_2}{R} \]
Simplifying this gives:
\[ Q_1 = 12Q_2 \]

5. **Find the ratio of electric fields \( E_1 \) and \( E_2 \):**
Now we can find the ratio:
\[ \frac{E_1}{E_2} = \frac{\frac{kQ_1}{9R^2}}{\frac{kQ_2}{R^2}} = \frac{Q_1}{9Q_2} \]
Substituting \( Q_1 = 12Q_2 \):
\[ \frac{E_1}{E_2} = \frac{12Q_2}{9Q_2} = \frac{12}{9} = \frac{4}{3} \]

### Final Answer:
The ratio of the electric fields on the surfaces of the spheres \( S_1 \) and \( S_2 \) is:
\[ \frac{E_1}{E_2} = \frac{4}{3} \]