A constant voltage is applied between the two ends of a metallic wire. If both the length and the radius of the wire are doubled, the rate of heat developed in the wire

Options

A. will be doubled
B. will be halved
C. will remain the same
D. will be quadrupled

Answer

To solve the problem, we need to determine how the rate of heat developed in a metallic wire changes when both its length and radius are doubled. The rate of heat developed in the wire can be expressed in terms of power, which is given by the formula:

\[ P = \frac{V^2}{R} \]

where \( P \) is the power (rate of heat developed), \( V \) is the voltage applied, and \( R \) is the resistance of the wire.

### Step-by-Step Solution:

1. **Identify the Original Resistance**:
The resistance \( R \) of a wire is given by the formula:
\[ R = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity of the material, \( L \) is the length of the wire, and \( A \) is the cross-sectional area of the wire.

2. **Calculate the Original Area**:
The cross-sectional area \( A \) of the wire, which has a circular cross-section, is given by:
\[ A = \pi r^2 \]
where \( r \) is the radius of the wire.

3. **Determine the New Length and Radius**:
If both the length and the radius are doubled:
- New length \( L' = 2L \)
- New radius \( r' = 2r \)

4. **Calculate the New Area**:
The new cross-sectional area \( A' \) becomes:
\[ A' = \pi (r')^2 = \pi (2r)^2 = \pi \cdot 4r^2 = 4A \]

5. **Calculate the New Resistance**:
The new resistance \( R' \) can now be calculated using the new length and area:
\[ R' = \frac{\rho L'}{A'} = \frac{\rho (2L)}{4A} = \frac{2\rho L}{4A} = \frac{\rho L}{2A} = \frac{R}{2} \]
Thus, the new resistance is half of the original resistance.

6. **Determine the New Power**:
The new power \( P' \) can be calculated using the new resistance:
\[ P' = \frac{V^2}{R'} \]
Substituting \( R' = \frac{R}{2} \):
\[ P' = \frac{V^2}{\frac{R}{2}} = \frac{2V^2}{R} = 2P \]
Therefore, the new power is double the original power.

### Conclusion:
The rate of heat developed in the wire when both the length and radius are doubled is **doubled**.