Question

A steady current \( I \) is passed through a conductor at room temperature for time \( t \). It is observed that its temperature rises by \( 0.5^\circ\text{C} \). If \( 2I \) current is passed through the conductor (at room temperature) for the same duration, the rise in its temperature will be approximately:

• A. \( 1.0^\circ\text{C} \)
• B. \( 1.5^\circ\text{C} \)
• C. \( 2.0^\circ\text{C} \)
• D. \( 4.0^\circ\text{C} \)

Hint:

Temperature rise due to resistive heating is proportional to the square of current: If current doubles, temperature rise becomes 4 times.

Answer 1

The Correct Option is C

The heat ($H$) generated in a conductor by current ($I$) is given by $H = I^2 R t$. Since the temperature rise ($\Delta T$) is proportional to the heat ($H$), we have $\Delta T \propto H$. Therefore, $\Delta T \propto I^2$. Initial Condition: For a current $I$, the temperature rise is $\Delta T_1 = 0.5^\circ \text{C}$. Scenario 1: Current is $\sqrt{2} I$ If the new current is $I' = \sqrt{2} I$, then $\left( \frac{I'}{I} \right)^2 = 2$. The new temperature rise, $\Delta T_2$, would be $0.5 \cdot 2 = 1.0^\circ \text{C}$. Scenario 2: Current is $2I$ However, the problem states the current is "2I". The new temperature rise is calculated as $\Delta T_2 = 0.5 \cdot (2)^2 = 0.5 \cdot 4 = 2.0^\circ \text{C}$. The final temperature rise is $\boxed{\Delta T = 2.0^\circ \text{C}}$.