Question

The r.m.s. value of A.C. which when passed through a resistor produces heat, which is twice that produced by a steady current of $1.414$ A in the same resistor is

• A. $2$ A
• B. $3.46$ A
• C. $2.818$ A
• D. $1.732$ A
• E. $1$ A

Hint:

Heat $\propto I^2$ → doubling heat means current increases by $\sqrt{2}$ factor.

Answer 1

The Correct Option is A

Concept: Heating effect (Joule’s Law)
Heat produced: \[ H = I^2 R t \] For AC: \[ H = I_{\text{rms}}^2 R t \] ---

Step 1: Given condition
Heat by AC is twice heat by DC: \[ I_{\text{rms}}^2 R t = 2 \times (1.414)^2 R t \] ---

Step 2: Cancel common terms \[ I_{\text{rms}}^2 = 2 \times (1.414)^2 \] ---

Step 3: Calculate \[ (1.414)^2 \approx 2 \] \[ I_{\text{rms}}^2 = 2 \times 2 = 4 \] \[ I_{\text{rms}} = 2 \text{ A} \] --- Physical Interpretation:
• Heating depends on square of current
• Doubling heat → current increases by $\sqrt{2}$ factor --- Final Answer: \[ \boxed{2 \text{ A}} \]