A material has normal density `rho` and bulk modulus `K`. The increase in the density of the material when it is subjected to an external pressure `P` from all sides is

Options

A. `P//rhoK`
B. `K//rhoP`
C. `rhoP//K`
D. `rhoK//P`

Explanation

To solve the problem of finding the increase in the density of a material when subjected to an external pressure \( P \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Bulk Modulus**:
The bulk modulus \( K \) of a material is defined as the ratio of the change in pressure \( P \) to the relative change in volume \( \frac{\Delta V}{V} \). Mathematically, it is expressed as:
\[ K = -\frac{P}{\frac{\Delta V}{V}} \]
where \( \Delta V \) is the change in volume and \( V \) is the original volume.

2. **Relating Volume Change to Density Change**:
The mass \( M \) of the material is given by:
\[ M = \rho V \]
where \( \rho \) is the density. If we differentiate this equation, we have:
\[ dM = \rho dV + V d\rho \]
Since mass is conserved when pressure is applied, \( dM = 0 \). Thus, we can set:
\[ 0 = \rho dV + V d\rho \]
Rearranging gives:
\[ \rho dV = -V d\rho \]
This can be rewritten as:
\[ \frac{dV}{V} = -\frac{d\rho}{\rho} \]

3. **Substituting in the Bulk Modulus Equation**:
From the definition of bulk modulus, we have:
\[ K = -\frac{P}{\frac{dV}{V}} \]
Substituting \( \frac{dV}{V} = -\frac{d\rho}{\rho} \) into the bulk modulus equation gives:
\[ K = \frac{P}{\frac{d\rho}{\rho}} \]

4. **Rearranging to Find Change in Density**:
Rearranging the equation for \( d\rho \):
\[ K \cdot \frac{d\rho}{\rho} = P \]
This leads to:
\[ d\rho = \frac{\rho P}{K} \]

5. **Conclusion**:
The increase in density \( d\rho \) of the material when subjected to an external pressure \( P \) is:
\[ d\rho = \frac{\rho P}{K} \]

### Final Answer:
The increase in the density of the material when it is subjected to an external pressure \( P \) from all sides is given by:
\[ d\rho = \frac{\rho P}{K} \]