What is the r.m.s. value of an alternating current which when passed through a resistor produces heat which is thrice of that produced by a direct current of `2` amperes in the same resistor?

Options

A. `6` amp
B. `2` amp
C. `3.46` amp
D. `0.66` amp

Explanation

To find the r.m.s. value of the alternating current (AC) that produces heat three times that of a direct current (DC) of 2 amperes in the same resistor, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Heat Produced by DC Current:**
The heat produced (H) in a resistor by a direct current (I) is given by the formula:
\[ H = I^2 R t \]
For a DC current of 2 A, the heat produced can be expressed as:
\[ H_{DC} = (2)^2 R t = 4Rt \]

2. **Heat Produced by AC Current:**
For an alternating current, the r.m.s. value of the current (I_rms) is used in the same formula:
\[ H_{AC} = I_{rms}^2 R t \]

3. **Relate the Heat Produced by AC to DC:**
According to the problem, the heat produced by the AC current is three times that produced by the DC current:
\[ H_{AC} = 3 H_{DC} \]
Substituting the expressions for heat:
\[ I_{rms}^2 R t = 3 (4Rt) \]

4. **Simplify the Equation:**
We can cancel \(R\) and \(t\) from both sides (assuming they are non-zero):
\[ I_{rms}^2 = 12 \]

5. **Calculate the r.m.s. Value of AC Current:**
Taking the square root of both sides gives:
\[ I_{rms} = \sqrt{12} = 2\sqrt{3} \]

### Final Answer:
The r.m.s. value of the alternating current is:
\[ I_{rms} = 2\sqrt{3} \text{ A} \]