The RMS value of AC which when passed through a resistor produces heat, which is twice that produced by a steady current of 1.414A in the same resistor is

Options

A. `2A`
B. 3.46 A
C. 2.818 A
D. 1.732 A

Answer

To solve the problem, we need to find the RMS value of the AC current that produces heat in a resistor, which is twice the heat produced by a steady current of 1.414 A.

### Step-by-Step Solution:

1. **Understanding Heat Produced by Current:**
The heat produced (H) in a resistor due to a current (I) is given by the formula:
\[ H = I^2 R t \]
where \( R \) is the resistance and \( t \) is the time duration.

2. **Heat Produced by Steady Current:**
For a steady current of \( I = 1.414 \, \text{A} \):
\[ H_{\text{steady}} = (1.414)^2 R t = 2 R t \]
(since \( 1.414^2 = 2 \)).

3. **Heat Produced by AC Current:**
Let the RMS value of the AC current be \( I_{\text{rms}} \). The heat produced by the AC current is:
\[ H_{\text{AC}} = I_{\text{rms}}^2 R t \]

4. **Setting Up the Equation:**
According to the problem, the heat produced by the AC current is twice that produced by the steady current:
\[ H_{\text{AC}} = 2 H_{\text{steady}} \]
Substituting the expressions for heat:
\[ I_{\text{rms}}^2 R t = 2 (2 R t) \]
Simplifying this gives:
\[ I_{\text{rms}}^2 R t = 4 R t \]

5. **Canceling Common Terms:**
Since \( R \) and \( t \) are common on both sides, we can cancel them out:
\[ I_{\text{rms}}^2 = 4 \]

6. **Finding RMS Current:**
Taking the square root of both sides:
\[ I_{\text{rms}} = \sqrt{4} = 2 \, \text{A} \]

### Final Answer:
The RMS value of the AC current is \( \mathbf{2 \, A} \).