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 \( Q_1 : Q_2 : Q_3 \) given to three concentric metallic spherical shells with radii \( R, 2R, \) and \( 3R \) respectively, under the condition that the surface charge densities on their outer surfaces 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. For a sphere, the surface area \( A \) is given by \( 4\pi r^2 \).

2. **Surface Charge Densities for Each Shell**:
- For the inner shell (radius \( R \)):
\[ \sigma_1 = \frac{Q_1}{4\pi R^2} \]
- For the middle shell (radius \( 2R \)):
\[ \sigma_2 = \frac{Q_2}{4\pi (2R)^2} = \frac{Q_2}{16\pi R^2} \]
- For the outer shell (radius \( 3R \)):
\[ \sigma_3 = \frac{Q_3}{4\pi (3R)^2} = \frac{Q_3}{36\pi R^2} \]

3. **Setting the Surface Charge Densities Equal**:
Since it is given that the surface charge densities on the outer surfaces are equal, we can set:
\[ \sigma_1 = \sigma_2 = \sigma_3 \]
This gives us the following equations:
\[ \frac{Q_1}{4\pi R^2} = \frac{Q_2}{16\pi R^2} \]
\[ \frac{Q_2}{16\pi R^2} = \frac{Q_3}{36\pi R^2} \]

4. **Solving for Ratios**:
From the first equation:
\[ Q_1 = \frac{1}{4} Q_2 \quad \text{(1)} \]
From the second equation:
\[ Q_2 = \frac{16}{36} Q_3 = \frac{4}{9} Q_3 \quad \text{(2)} \]

5. **Substituting Equation (2) into Equation (1)**:
Substitute \( Q_2 \) from equation (2) into equation (1):
\[ Q_1 = \frac{1}{4} \left( \frac{4}{9} Q_3 \right) = \frac{1}{9} Q_3 \]

6. **Expressing All Charges in Terms of \( Q_3 \)**:
Now we can express \( Q_1 \) and \( Q_2 \) in terms of \( Q_3 \):
- From \( Q_1 = \frac{1}{9} Q_3 \)
- From \( Q_2 = \frac{4}{9} Q_3 \)

7. **Finding the Ratio**:
The charges can now be expressed as:
\[ Q_1 : Q_2 : Q_3 = \frac{1}{9} Q_3 : \frac{4}{9} Q_3 : Q_3 \]
Simplifying this gives:
\[ Q_1 : Q_2 : Q_3 = 1 : 4 : 9 \]

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