The gravitational potential of two homogeneous spherical shells `A` and `B` of same surface density at their respective centres are in the ratio `3:4`. If the two shells collapse into a single one such that surface charge density remains the same, then the ratio of potential at an internal point of the new shell to shell `A` is equal to

Options

A. `3 : 5`
B. `4 : 5`
C. `5 : 3`
D. `5 : 4`

Answer

To solve the problem, we need to find the ratio of the gravitational potential at an internal point of the new shell formed by the combination of two spherical shells A and B, compared to the potential at shell A.

### Step-by-Step Solution:

1. **Understanding Gravitational Potential**:
The gravitational potential \( V \) at the center of a homogeneous spherical shell of radius \( R \) and surface mass density \( \sigma \) is given by:
\[ V = \frac{G M}{R} \]
where \( M \) is the mass of the shell.

2. **Calculating Mass of Shells**:
The mass \( M \) of a spherical shell can be expressed as:
\[ M = 4 \pi R^2 \sigma \]
Therefore, the gravitational potential at the center of shell A (with radius \( R_A \)) is:
\[ V_A = \frac{G (4 \pi R_A^2 \sigma)}{R_A} = 4 \pi G \sigma R_A \]
Similarly, for shell B (with radius \( R_B \)):
\[ V_B = 4 \pi G \sigma R_B \]

3. **Using the Given Ratio of Potentials**:
We are given that the ratio of the potentials at the centers of shells A and B is:
\[ \frac{V_A}{V_B} = \frac{3}{4} \]
Substituting the expressions for \( V_A \) and \( V_B \):
\[ \frac{4 \pi G \sigma R_A}{4 \pi G \sigma R_B} = \frac{3}{4} \]
This simplifies to:
\[ \frac{R_A}{R_B} = \frac{3}{4} \]

4. **Finding the Radius of the Combined Shell**:
When shells A and B collapse into a single shell while maintaining the same surface density \( \sigma \), the total mass is conserved. The mass of the combined shell \( M_C \) is:
\[ M_C = M_A + M_B = 4 \pi R_A^2 \sigma + 4 \pi R_B^2 \sigma = 4 \pi \sigma (R_A^2 + R_B^2) \]
The radius \( R \) of the new shell can be expressed as:
\[ M_C = 4 \pi R^2 \sigma \implies R^2 = R_A^2 + R_B^2 \]

5. **Calculating the Ratio of the New Potential to Potential of Shell A**:
The potential at the center of the new shell is:
\[ V_C = 4 \pi G \sigma R \]
To find the ratio \( \frac{V_C}{V_A} \):
\[ \frac{V_C}{V_A} = \frac{4 \pi G \sigma R}{4 \pi G \sigma R_A} = \frac{R}{R_A} \]

6. **Finding \( R \) in Terms of \( R_A \)**:
Since \( R_B = \frac{4}{3} R_A \) (from the ratio derived earlier):
\[ R^2 = R_A^2 + R_B^2 = R_A^2 + \left(\frac{4}{3} R_A\right)^2 = R_A^2 + \frac{16}{9} R_A^2 = \frac{25}{9} R_A^2 \]
Thus,
\[ R = \frac{5}{3} R_A \]

7. **Final Ratio Calculation**:
Now substituting back into the ratio of potentials:
\[ \frac{V_C}{V_A} = \frac{R}{R_A} = \frac{\frac{5}{3} R_A}{R_A} = \frac{5}{3} \]

### Conclusion:
The ratio of the gravitational potential at an internal point of the new shell to that of shell A is:
\[ \frac{V_C}{V_A} = \frac{5}{3} \]