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. `1:1`
B. `1:2`
C. `2:1`
D. `4:1`

Answer

To find the ratio of heat produced in two identical resistances when a direct current (DC) of 2 A and an alternating current (AC) with a maximum value of 2 A flow through them, we can follow these steps:

### Step 1: Calculate Heat Produced in DC Resistance
The heat produced in a resistance due to direct current can be calculated using the formula:
\[ H_{DC} = I^2 R T \]
Where:
- \( I \) = current (2 A for DC)
- \( R \) = resistance (same for both cases)
- \( T \) = time (same for both cases)

Substituting the values:
\[ H_{DC} = (2)^2 R T = 4 R T \]

### Step 2: Calculate RMS Value of AC Current
For the alternating current, the maximum value (\( I_0 \)) is given as 2 A. The root mean square (RMS) value of the AC current is given by:
\[ I_{rms} = \frac{I_0}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \text{ A} \]

### Step 3: Calculate Heat Produced in AC Resistance
The heat produced in a resistance due to alternating current can be calculated using the formula:
\[ H_{AC} = I_{rms}^2 R T \]
Substituting the RMS value:
\[ H_{AC} = (\sqrt{2})^2 R T = 2 R T \]

### Step 4: Calculate the Ratio of Heat Produced
Now, we can find the ratio of heat produced in the two resistances:
\[ \text{Ratio} = \frac{H_{DC}}{H_{AC}} = \frac{4 R T}{2 R T} \]
The \( R \) and \( T \) cancel out:
\[ \text{Ratio} = \frac{4}{2} = 2 \]

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