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 how much the density of a material increases when subjected to an external pressure \( P \), we can follow these steps:

### Step-by-Step Solution:

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

2. **Relating Density and Volume**:
The mass \( m \) of the material can be expressed in terms of density \( \rho \) and volume \( V \):
\[ m = \rho V \]
Since mass is constant, any change in volume will affect the density.

3. **Differentiating the Mass Equation**:
Differentiating the equation \( m = \rho V \) gives:
\[ 0 = \rho dV + V d\rho \]
Rearranging this, we find:
\[ \rho dV = -V d\rho \]
Dividing both sides by \( V \):
\[ \frac{dV}{V} = -\frac{d\rho}{\rho} \]

4. **Substituting into the Bulk Modulus Equation**:
From the bulk modulus definition, we can substitute \( \frac{dV}{V} \):
\[ K = -\frac{P}{\frac{dV}{V}} \implies \frac{dV}{V} = -\frac{P}{K} \]
Thus, we have:
\[ -\frac{d\rho}{\rho} = -\frac{P}{K} \]
This simplifies to:
\[ \frac{d\rho}{\rho} = \frac{P}{K} \]

5. **Finding the Change in Density**:
Rearranging gives us the change in density \( d\rho \):
\[ d\rho = \frac{P}{K} \rho \]

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

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