The mean density of sea water is `rho`, and bulk modulus is B. The change in density of sea water in going from the surface of water to a depth of `h` is

Options

A. `(B rho^(2))/(gh)`
B. `B rho gh`
C. `(rho^(2)gh)/(B)`
D. `(rho gh)/(B)`

Answer

To find the change in density of seawater as we go from the surface to a depth \( h \), we can follow these steps:

### Step 1: Understand the relationship between bulk modulus and density change
The bulk modulus \( B \) is defined as:
\[ B = -\frac{P}{\frac{\Delta V}{V}} = \frac{P}{\frac{\Delta \rho}{\rho}} \]
where \( P \) is the change in pressure, \( \Delta V \) is the change in volume, \( V \) is the original volume, \( \Delta \rho \) is the change in density, and \( \rho \) is the original density.

### Step 2: Rearranging the formula for change in density
From the bulk modulus equation, we can rearrange to find the change in density \( \Delta \rho \):
\[ \Delta \rho = \rho \cdot \frac{P}{B} \]

### Step 3: Calculate the pressure change at depth \( h \)
The pressure at a depth \( h \) in a fluid is given by:
\[ P = \rho \cdot g \cdot h \]
where \( g \) is the acceleration due to gravity.

### Step 4: Substitute the pressure change into the density change equation
Now, substitute the expression for pressure \( P \) into the equation for \( \Delta \rho \):
\[ \Delta \rho = \rho \cdot \frac{\rho \cdot g \cdot h}{B} \]

### Step 5: Simplify the expression
This simplifies to:
\[ \Delta \rho = \frac{\rho^2 \cdot g \cdot h}{B} \]

### Conclusion
Thus, the change in density of seawater as we go from the surface to a depth \( h \) is:
\[ \Delta \rho = \frac{\rho^2 \cdot g \cdot h}{B} \]

### Final Answer
The correct option is \( \frac{\rho^2 \cdot g \cdot h}{B} \), which corresponds to option 3.

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