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 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 use the relationship between pressure, bulk modulus, and density. Here’s a step-by-step solution:

### Step 1: Understand the relationship between pressure and depth
As we go deeper into the water, the pressure increases. The pressure at a depth \( h \) in a fluid is given by:
\[ P = P_0 + \rho g h \]
where:
- \( P_0 \) is the atmospheric pressure at the surface,
- \( \rho \) is the density of seawater,
- \( g \) is the acceleration due to gravity,
- \( h \) is the depth.

### Step 2: Relate pressure change to density change
The change in pressure \( \Delta P \) when moving from the surface to a depth \( h \) is:
\[ \Delta P = \rho g h \]

### Step 3: Use the bulk modulus definition
The bulk modulus \( B \) is defined as:
\[ B = -\frac{\Delta P}{\frac{\Delta V}{V}} \]
where \( \Delta V \) is the change in volume and \( V \) is the original volume. The change in density \( \Delta \rho \) can be related to the change in volume by:
\[ \Delta \rho = \rho \frac{\Delta V}{V} \]

### Step 4: Substitute and rearrange
Substituting the expression for \( \Delta V/V \) into the bulk modulus equation gives:
\[ \Delta \rho = -\frac{\Delta P}{B} \cdot \rho \]
Substituting \( \Delta P = \rho g h \):
\[ \Delta \rho = -\frac{\rho g h}{B} \cdot \rho \]

### Step 5: Simplify the expression
Thus, the change in density \( \Delta \rho \) becomes:
\[ \Delta \rho = -\frac{\rho^2 g h}{B} \]

### Final Result
The change in density of seawater in going from the surface to a depth \( h \) is:
\[ \Delta \rho = -\frac{\rho^2 g h}{B} \]

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