An air bubble of radius `r` in water is at a depth `h` below the water surface at some instant. If `P` is atmospheric pressure, `d` and `T` are density and surface tension of water respectivley . the pressure inside the bubble will be :

Options

A. `P + h dg - (4T)/(r)`
B. `P + h dg +(2T)/(r)`
C. `P + h dg - (2T)/(r)`
D. `P + h dg - (4T)/(r)`

Explanation

To find the pressure inside an air bubble of radius \( r \) located at a depth \( h \) in water, we can follow these steps:

### Step 1: Understand the pressure at a certain depth in a fluid
The pressure at a depth \( h \) below the water surface can be calculated using the formula:
\[ P_{\text{depth}} = P + h \cdot d \]
where:
- \( P \) is the atmospheric pressure,
- \( d \) is the density of water,
- \( h \) is the depth below the surface.

### Step 2: Determine the pressure inside the bubble
The pressure inside the bubble \( P_{\text{inside}} \) is affected by the surface tension of the bubble. For a bubble, the pressure difference between the inside and outside due to surface tension is given by:
\[ \Delta P = \frac{2T}{r} \]
where:
- \( T \) is the surface tension of water,
- \( r \) is the radius of the bubble.

### Step 3: Relate the internal and external pressures
The pressure inside the bubble can be expressed as:
\[ P_{\text{inside}} = P_{\text{outside}} + \Delta P \]
Substituting the expressions we have:
\[ P_{\text{inside}} = (P + h \cdot d) + \frac{2T}{r} \]

### Step 4: Final expression for the pressure inside the bubble
Combining the equations, we get:
\[ P_{\text{inside}} = P + h \cdot d + \frac{2T}{r} \]

### Conclusion
Thus, the pressure inside the bubble at a depth \( h \) is:
\[ P_{\text{inside}} = P + h \cdot d + \frac{2T}{r} \]