A direct current of 2 A and an alternating current having a maximum value of 2 A flow through two identical resistances. The ratio of heat produced in the two resistances will be

Options

A. 0.042361111111111
B. 0.043055555555556
C. 2
D. 0.16736111111111

Explanation

To solve the problem of finding the ratio of heat produced in two identical resistances when a direct current (DC) and an alternating current (AC) flow through them, we can follow these steps:

### Step 1: Understand the heat produced by DC
The heat produced in a resistor due to a direct current can be calculated using Joule's law, which states that the heat (H) produced is given by:
\[ H = I^2 R t \]
where:
- \( I \) is the current,
- \( R \) is the resistance,
- \( t \) is the time for which the current flows.

For the DC circuit, we have:
- \( I = 2 \, \text{A} \)

Thus, the heat produced in the DC circuit (let's call it \( H_1 \)) is:
\[ H_1 = (2)^2 R t = 4 R t \]

### Step 2: Understand the heat produced by AC
For the alternating current, we need to consider the root mean square (RMS) value of the current. The RMS value of an AC current is given by:
\[ I_{\text{rms}} = \frac{I_{\text{max}}}{\sqrt{2}} \]
where \( I_{\text{max}} \) is the maximum (peak) value of the AC current.

Given:
- \( I_{\text{max}} = 2 \, \text{A} \)

Thus, the RMS value of the AC current is:
\[ I_{\text{rms}} = \frac{2}{\sqrt{2}} = \sqrt{2} \, \text{A} \]

Now, we can calculate the heat produced in the AC circuit (let's call it \( H_2 \)):
\[ H_2 = I_{\text{rms}}^2 R t = \left(\sqrt{2}\right)^2 R t = 2 R t \]

### Step 3: Calculate the ratio of heat produced
Now, we need to find the ratio of heat produced in the two resistances:
\[ \text{Ratio} = \frac{H_1}{H_2} = \frac{4 R t}{2 R t} \]

The \( R \) and \( t \) terms cancel out:
\[ \text{Ratio} = \frac{4}{2} = 2 \]

### Conclusion
The ratio of heat produced in the two resistances is:
\[ \text{Ratio} = 2 : 1 \]