A `5^(@)C` rise in the temperature is observed in a conductor by passing some current. When the current is doubled, then rise in temperature will be equal to

Options

A. `16^(@)C`
B. `10^(@)C`
C. `20^(@)C`
D. `12^(@)C`

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Understand the relationship between power and temperature rise
The power \( P \) dissipated in a conductor due to current \( I \) is given by the formula:
\[ P = I^2 R \]
where \( R \) is the resistance of the conductor. The temperature rise \( \Delta T \) in the conductor is directly proportional to the power dissipated.

### Step 2: Identify the initial conditions
We know that with an initial current \( I \), the temperature rise is:
\[ \Delta T_1 = 5^\circ C \]

### Step 3: Determine the new conditions when current is doubled
When the current is doubled, the new current becomes:
\[ I' = 2I \]

### Step 4: Calculate the new power
Substituting the new current into the power formula gives:
\[ P' = (2I)^2 R = 4I^2 R = 4P \]
This shows that the power has increased by a factor of 4.

### Step 5: Relate the new power to the temperature rise
Since the temperature rise is proportional to the power, we can express the new temperature rise \( \Delta T_2 \) as:
\[ \Delta T_2 = k \cdot P' = k \cdot (4P) = 4 \cdot (k \cdot P) = 4 \Delta T_1 \]
where \( k \) is the proportionality constant.

### Step 6: Substitute the known temperature rise
Now substituting \( \Delta T_1 = 5^\circ C \):
\[ \Delta T_2 = 4 \cdot 5^\circ C = 20^\circ C \]

### Conclusion
Thus, the rise in temperature when the current is doubled is:
\[ \Delta T_2 = 20^\circ C \]

### Final Answer
The rise in temperature will be equal to \( 20^\circ C \).

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