Three concentric metallic spherical shells of radii R, 2R, 3R are given charges `Q_(1) Q_(2) Q_(3)`, respectively. It is found that the surface charge densities on the outer surface of the shells are equal. Then, the ratio of the charges given to the shells `Q_(1) : Q_(2) : Q_(3)` is

Options

A. `1 : 2 : 3`
B. `1 : 3 : 5`
C. `1 : 4 : 9`
D. `1 : 8 : 18`

Answer

To solve the problem, we need to find the ratio of the charges \( Q_1 : Q_2 : Q_3 \) on three concentric metallic spherical shells with radii \( R, 2R, \) and \( 3R \) respectively, given that the surface charge densities on the outer surfaces of the shells are equal.

### Step-by-Step Solution:

1. **Understanding Surface Charge Density**:
The surface charge density \( \sigma \) on a spherical shell is defined as:
\[ \sigma = \frac{Q}{A} \]
where \( Q \) is the charge on the shell and \( A \) is the surface area of the shell.

2. **Calculating Surface Areas**:
The surface area \( A \) of a sphere is given by \( 4\pi r^2 \). Therefore, the surface areas for the three shells are:
- For the first shell (radius \( R \)):
\[ A_1 = 4\pi R^2 \]
- For the second shell (radius \( 2R \)):
\[ A_2 = 4\pi (2R)^2 = 16\pi R^2 \]
- For the third shell (radius \( 3R \)):
\[ A_3 = 4\pi (3R)^2 = 36\pi R^2 \]

3. **Setting Up the Equation for Equal Surface Charge Densities**:
Since the surface charge densities are equal, we can write:
\[ \sigma_1 = \sigma_2 = \sigma_3 \]
This gives us the equations:
\[ \frac{Q_1}{4\pi R^2} = \frac{Q_2}{16\pi R^2} = \frac{Q_3}{36\pi R^2} \]

4. **Simplifying the Equations**:
We can cancel \( 4\pi R^2 \) from each term:
\[ Q_1 = k \cdot 4 \quad (1) \]
\[ Q_2 = k \cdot 16 \quad (2) \]
\[ Q_3 = k \cdot 36 \quad (3) \]
where \( k \) is a constant.

5. **Finding the Ratios**:
Now we can express the charges in terms of \( k \):
- From (1): \( Q_1 = 4k \)
- From (2): \( Q_2 = 16k \)
- From (3): \( Q_3 = 36k \)

Therefore, the ratio of the charges is:
\[ Q_1 : Q_2 : Q_3 = 4k : 16k : 36k \]
Simplifying this gives:
\[ Q_1 : Q_2 : Q_3 = 4 : 16 : 36 \]
Dividing each term by 4:
\[ Q_1 : Q_2 : Q_3 = 1 : 4 : 9 \]

### Final Answer:
The ratio of the charges given to the shells is:
\[ Q_1 : Q_2 : Q_3 = 1 : 4 : 9 \]