The normal density of gold is `rho` and its bulk modulus is K. The increase is density of a lump of gold when a pressure P is applied uniformly on all sides is

Options

A. `rho` P/K
B. `rho` K/P
C. `P//rho` K
D. `K//rho P`

Explanation

To solve the problem of finding the increase in density of a lump of gold when a pressure \( P \) is applied uniformly on all sides, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Density and Volume**:
Density (\( \rho \)) is defined as mass (\( m \)) divided by volume (\( V \)):
\[ \rho = \frac{m}{V} \]
When pressure is applied, the volume of the gold will change, but the mass will remain constant.

2. **Define the Change in Volume**:
Let \( \Delta V \) be the change in volume due to the applied pressure. The new volume \( V' \) can be expressed as:
\[ V' = V - \Delta V \]

3. **Express the New Density**:
The new density \( \rho' \) after the change in volume can be expressed as:
\[ \rho' = \frac{m}{V - \Delta V} \]

4. **Relate the Change in Density to the Original Density**:
The change in density \( \Delta \rho \) can be defined as:
\[ \Delta \rho = \rho' - \rho \]
Substituting the expression for \( \rho' \):
\[ \Delta \rho = \frac{m}{V - \Delta V} - \frac{m}{V} \]

5. **Use Bulk Modulus Definition**:
The bulk modulus \( K \) is defined as:
\[ K = -\frac{P}{\frac{\Delta V}{V}} \]
Rearranging gives:
\[ \frac{\Delta V}{V} = -\frac{P}{K} \]

6. **Relate Change in Volume to Change in Density**:
From the definition of density, we can relate the change in volume to the change in density:
\[ \Delta \rho = \rho \cdot \left(-\frac{\Delta V}{V}\right) \]
Substituting for \( \frac{\Delta V}{V} \):
\[ \Delta \rho = \rho \cdot \left(\frac{P}{K}\right) \]

7. **Final Expression for Increase in Density**:
Thus, the increase in density \( \Delta \rho \) can be expressed as:
\[ \Delta \rho = \frac{P \cdot \rho}{K} \]

### Conclusion:
The increase in density of the lump of gold when a pressure \( P \) is applied uniformly on all sides is given by:
\[ \Delta \rho = \frac{P \cdot \rho}{K} \]