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 produced in a conductor due to current is given by: \[ H = I^2 R t \] Since temperature rise \( \Delta T \propto H \), we get: \[ \Delta T \propto I^2 \] Initial: For current \( I \), temperature rise \( \Delta T_1 = 0.5^\circ \text{C} \) New current: \( I' = \sqrt{2} I \Rightarrow \left( \frac{I'}{I} \right)^2 = 2 \) So, \[ \Delta T_2 = 0.5 \cdot 2 = 1.0^\circ \text{C} \] However, since the problem says “2I current”, not “√2 I”, we recalculate: \[ \Delta T_2 = 0.5 \cdot (2)^2 = 0.5 \cdot 4 = 2.0^\circ \text{C} \] \[ \boxed{\Delta T = 2.0^\circ \text{C}} \]