When a large bubble rises from the bottom of a lake to the surface its radius doubles. If atmospheric pressure is equal to that of column of water height H then the depth of lake is

Options

A. `H`
B. `2H`
C. `7H`
D. `8H`

Answer

To solve the problem of a large bubble rising from the bottom of a lake and doubling its radius, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A bubble rises from the bottom of a lake, and its radius doubles from \( r \) to \( 2r \).
- We need to find the depth of the lake, given that atmospheric pressure is equal to the pressure of a column of water of height \( H \).

2. **Pressure at the Surface**:
- At the surface of the lake, the pressure \( P_0 \) is equal to atmospheric pressure, which can be expressed as:
\[ P_0 = H \rho g \]
where \( \rho \) is the density of water and \( g \) is the acceleration due to gravity.

3. **Pressure at Depth \( x \)**:
- At a depth \( x \) in the lake, the pressure \( P_1 \) can be expressed as:
\[ P_1 = P_0 + x \rho g = H \rho g + x \rho g \]

4. **Volume of the Bubble**:
- The volume \( V \) of a sphere (the bubble) is given by:
\[ V = \frac{4}{3} \pi r^3 \]
- Initially, at depth \( x \), the volume of the bubble is:
\[ V_1 = \frac{4}{3} \pi r^3 \]
- When the bubble rises to the surface and its radius doubles, the new volume \( V_2 \) is:
\[ V_2 = \frac{4}{3} \pi (2r)^3 = \frac{4}{3} \pi (8r^3) = \frac{32}{3} \pi r^3 \]

5. **Applying Boyle's Law (Isothermal Conditions)**:
- According to Boyle's Law for isothermal conditions, we have:
\[ P_1 V_1 = P_2 V_2 \]
- Substituting the pressures and volumes:
\[ (H \rho g + x \rho g) \left(\frac{4}{3} \pi r^3\right) = (H \rho g) \left(\frac{32}{3} \pi r^3\right) \]

6. **Canceling Common Terms**:
- We can cancel \( \frac{4}{3} \pi r^3 \) from both sides:
\[ H \rho g + x \rho g = 8H \rho g \]

7. **Solving for \( x \)**:
- Rearranging the equation gives:
\[ x \rho g = 8H \rho g - H \rho g \]
\[ x \rho g = 7H \rho g \]
- Dividing both sides by \( \rho g \):
\[ x = 7H \]

8. **Conclusion**:
- The depth of the lake is \( 7H \).

### Final Answer:
The depth of the lake is \( 7H \).