A steady current of magnitude `I` and an `AC` current of peak value `I` are allowed to pass through identical resistors for the same time. The ratio of heat produced in the two resistors will be

Options

A. `2:1`
B. `1:2`
C. `1:1`
D. none of these

Explanation

To solve the problem of finding the ratio of heat produced in two identical resistors when a steady current \( I \) and an AC current of peak value \( I_0 \) are passed through them for the same time, we can follow these steps:

### Step 1: Calculate Heat Produced by Steady Current (DC)

The heat produced in a resistor due to a steady current (DC) can be calculated using the formula:
\[ H_{DC} = I^2 R t \]
where:
- \( H_{DC} \) is the heat produced in the resistor,
- \( I \) is the steady current,
- \( R \) is the resistance,
- \( t \) is the time for which the current flows.

### Step 2: Calculate Heat Produced by AC Current

For an alternating current (AC), we need to consider the root mean square (RMS) value of the current. The RMS value \( I_{RMS} \) of an AC current with peak value \( I_0 \) is given by:
\[ I_{RMS} = \frac{I_0}{\sqrt{2}} \]

The heat produced in the resistor due to the AC current can be calculated using the formula:
\[ H_{AC} = I_{RMS}^2 R t \]
Substituting the RMS value into the equation, we get:
\[ H_{AC} = \left(\frac{I_0}{\sqrt{2}}\right)^2 R t = \frac{I_0^2}{2} R t \]

### Step 3: Find the Ratio of Heat Produced

Now, we can find the ratio of heat produced in the DC circuit to that produced in the AC circuit:
\[ \text{Ratio} = \frac{H_{DC}}{H_{AC}} = \frac{I^2 R t}{\frac{I_0^2}{2} R t} \]

Notice that \( R \) and \( t \) cancel out:
\[ \text{Ratio} = \frac{I^2}{\frac{I_0^2}{2}} = \frac{2 I^2}{I_0^2} \]

### Step 4: Substitute Peak Value

Since the problem states that the peak value of the AC current is equal to the steady current, we can substitute \( I_0 = I \):
\[ \text{Ratio} = \frac{2 I^2}{I^2} = 2 \]

### Conclusion

Thus, the ratio of heat produced in the two resistors is:
\[ \text{Ratio} = 2:1 \]

### Final Answer

The ratio of heat produced in the two resistors is \( 2:1 \).

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