A rise of temperature of `4^(@)C` is observed in a conductor by passing a current. If the current is tripled, the rise of temperature will be

Options

A. `8^(@)C`
B. `12^(@)C`
C. `16^(@)C`
D. `36^(@)C`

Explanation

To solve the problem, we need to understand how the temperature rise in a conductor is related to the current flowing through it. The heat produced in the conductor can be expressed using Joule's law, which states:

\[ H = I^2 R t \]

Where:
- \( H \) is the heat produced,
- \( I \) is the current,
- \( R \) is the resistance of the conductor,
- \( t \) is the time for which the current flows.

The rise in temperature (\( \Delta T \)) of the conductor is also related to the heat produced by the equation:

\[ H = m S \Delta T \]

Where:
- \( m \) is the mass of the conductor,
- \( S \) is the specific heat capacity of the material,
- \( \Delta T \) is the change in temperature.

Since both expressions for heat \( H \) are equal, we can set them equal to each other:

\[ I^2 R t = m S \Delta T \]

From this, we can derive the relationship between the change in temperature and the current:

1. **Initial Condition**: Let the initial current be \( I_1 \) and the initial temperature rise be \( \Delta T_1 = 4^\circ C \).

\[ I_1^2 R t = m S \Delta T_1 \]

2. **New Condition**: If the current is tripled, the new current \( I_2 = 3I_1 \). We need to find the new temperature rise \( \Delta T_2 \).

\[ I_2^2 R t = m S \Delta T_2 \]

3. **Substituting the New Current**:

\[ (3I_1)^2 R t = m S \Delta T_2 \]

This simplifies to:

\[ 9 I_1^2 R t = m S \Delta T_2 \]

4. **Setting the Equations Equal**:

Since both expressions equal \( m S \):

\[ I_1^2 R t = m S \Delta T_1 \]
\[ 9 I_1^2 R t = m S \Delta T_2 \]

Dividing the second equation by the first:

\[ \frac{9 I_1^2 R t}{I_1^2 R t} = \frac{m S \Delta T_2}{m S \Delta T_1} \]

This simplifies to:

\[ 9 = \frac{\Delta T_2}{\Delta T_1} \]

5. **Finding \( \Delta T_2 \)**:

Now, substituting \( \Delta T_1 = 4^\circ C \):

\[ \Delta T_2 = 9 \times \Delta T_1 = 9 \times 4^\circ C = 36^\circ C \]

Thus, the rise of temperature when the current is tripled will be \( 36^\circ C \).