Two metal spheres have their surface areas in the ratio 9:16. They are put in contact with each other. A charge of `7 xx 10^(-6) `C is given to the system and now they are separated so that each exerts no influence on the other then the ratio of surface charge densities is

Options

A. `4 :3`
B. `5 :2`
C. `2 :5`
D. `7 :3`

Answer

To solve the problem, we need to find the ratio of surface charge densities of two metal spheres after they have been charged and then separated. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Relationship Between Surface Area and Radius
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

### Step 2: Set Up the Ratio of Surface Areas
We are given that the surface areas of the two spheres are in the ratio 9:16. Let the surface areas of the two spheres be \( A_1 \) and \( A_2 \):
\[ \frac{A_1}{A_2} = \frac{9}{16} \]
Using the surface area formula, we can express this in terms of their radii:
\[ \frac{4\pi r_1^2}{4\pi r_2^2} = \frac{9}{16} \]
This simplifies to:
\[ \frac{r_1^2}{r_2^2} = \frac{9}{16} \]

### Step 3: Find the Ratio of Radii
Taking the square root of both sides, we find:
\[ \frac{r_1}{r_2} = \frac{3}{4} \]

### Step 4: Charge Distribution When in Contact
Let \( q_1 \) be the charge on sphere 1 and \( q_2 \) be the charge on sphere 2. When the two spheres are in contact, they will share the total charge \( Q = q_1 + q_2 \).

### Step 5: Equal Potential Condition
When the spheres are in contact, they reach the same potential:
\[ V_1 = V_2 \]
The potential \( V \) of a sphere is given by:
\[ V = \frac{1}{4\pi \epsilon_0} \frac{q}{r} \]
Thus, we have:
\[ \frac{q_1}{r_1} = \frac{q_2}{r_2} \]
From this, we can express the ratio of charges:
\[ \frac{q_1}{q_2} = \frac{r_1}{r_2} \]

### Step 6: Substitute the Ratio of Radii
Substituting the ratio of radii we found earlier:
\[ \frac{q_1}{q_2} = \frac{3}{4} \]

### Step 7: Calculate Surface Charge Densities
The surface charge density \( \sigma \) is defined as:
\[ \sigma = \frac{q}{A} = \frac{q}{4\pi r^2} \]
Thus, for the two spheres:
\[ \sigma_1 = \frac{q_1}{4\pi r_1^2}, \quad \sigma_2 = \frac{q_2}{4\pi r_2^2} \]

### Step 8: Find the Ratio of Surface Charge Densities
Now, we can find the ratio of the surface charge densities:
\[ \frac{\sigma_1}{\sigma_2} = \frac{q_1}{q_2} \cdot \frac{r_2^2}{r_1^2} \]
Substituting the known ratios:
\[ \frac{\sigma_1}{\sigma_2} = \frac{3/4}{1} \cdot \frac{(4/3)^2}{1} = \frac{3}{4} \cdot \frac{16}{9} \]
This simplifies to:
\[ \frac{\sigma_1}{\sigma_2} = \frac{3 \cdot 16}{4 \cdot 9} = \frac{48}{36} = \frac{4}{3} \]

### Final Answer
The ratio of the surface charge densities is:
\[ \frac{\sigma_1}{\sigma_2} = \frac{4}{3} \]