Two resistors R and 2R are connected in series in an electric circuit. The thermal energy developed in R and 2R are in the ratio

Options

A. (a)1:2
B. (b)2:1
C. (c )1:4
D. (d)4:1

Answer

To find the ratio of thermal energy developed in resistors R and 2R connected in series, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Resistors and Their Values**:
- Let the first resistor be \( R_1 = R \).
- Let the second resistor be \( R_2 = 2R \).

2. **Understand the Circuit Configuration**:
- The resistors are connected in series across a potential difference \( V \).
- In a series circuit, the same current \( I \) flows through both resistors.

3. **Use the Formula for Heat Dissipation**:
- The heat (thermal energy) developed in a resistor can be calculated using the formula:
\[ H = I^2 R T \]
- Where \( H \) is the heat developed, \( I \) is the current, \( R \) is the resistance, and \( T \) is the time for which the current flows.

4. **Calculate Heat Developed in Each Resistor**:
- For resistor \( R_1 \) (which is \( R \)):
\[ H_1 = I^2 R T \]
- For resistor \( R_2 \) (which is \( 2R \)):
\[ H_2 = I^2 (2R) T = 2 I^2 R T \]

5. **Find the Ratio of Heat Developed**:
- To find the ratio \( \frac{H_1}{H_2} \):
\[ \frac{H_1}{H_2} = \frac{I^2 R T}{2 I^2 R T} \]
- Simplifying this gives:
\[ \frac{H_1}{H_2} = \frac{1}{2} \]

6. **Conclusion**:
- The ratio of thermal energy developed in resistors \( R \) and \( 2R \) is \( 1:2 \).

### Final Answer:
The thermal energy developed in \( R \) and \( 2R \) are in the ratio \( 1:2 \).

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