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 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 formula for heat produced
The heat produced (H) in a resistor is given by the formula:
\[ H = I^2 R T \]
where:
- \( I \) is the current,
- \( R \) is the resistance,
- \( T \) is the time for which the current flows.

### Step 2: Calculate heat produced by the direct current (DC)
For the direct current of 2 A:
- The current \( I_{DC} = 2 \, \text{A} \).
- The heat produced in the DC circuit is:
\[ H_{DC} = I_{DC}^2 R T = (2)^2 R T = 4 R T \]

### Step 3: Calculate heat produced by the alternating current (AC)
For the alternating current, we are given the maximum value of 2 A. The effective (RMS) value of the alternating current is given by:
\[ I_{RMS} = \frac{I_{max}}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \, \text{A} \]

Now, we can calculate the heat produced in the AC circuit:
\[ H_{AC} = I_{RMS}^2 R T = (\sqrt{2})^2 R T = 2 R T \]

### Step 4: Find the ratio of heat produced in DC to AC
Now we can find the ratio of heat produced in the two resistances:
\[ \text{Ratio} = \frac{H_{DC}}{H_{AC}} = \frac{4 R T}{2 R T} = \frac{4}{2} = 2 \]

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

### Final Answer
The ratio of heat produced in the two resistances is \( 2 : 1 \).

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