Question

When a large bubble rises from bottom of a water lake to its surface, then its radius doubles. If the atmospheric pressure is equal to the pressure of height \(H\) of a certain water column then the depth of lake will be

• A. \(2H\)
• B. \(H\)
• C. \(7H\)
• D. \(4H\)

Hint:

Apply Boyle's law between the bottom and the surface. Volume becomes 8 times when the radius doubles.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The temperature is constant, so Boyle's law applies: \(P_1V_1=P_2V_2\).

Step 2: Pressures
Atmospheric pressure is \(P_0=\rho gH\). At depth \(h\), \(P_1=\rho g(H+h)\) and at the surface \(P_2=\rho gH\).

Step 3: Volumes
The radius doubles, so \(V_2=8V_1\).

Step 4: Solve
\((H+h)V_1=H\cdot8V_1\) gives \(H+h=8H\), so \(h=7H\).

Final Answer:
The depth of the lake is 7H. \[ \boxed{\text{(C) }7H} \]