Two resistances R and 2R are connected in parallel 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 resistances R and 2R connected in parallel, we can follow these steps:

### Step 1: Understand the Circuit Configuration
- Two resistances, R and 2R, are connected in parallel across a voltage source V. In a parallel circuit, the voltage across each resistor is the same.

### Step 2: Calculate Power Developed in Each Resistor
- The power (P) developed in a resistor can be calculated using the formula:
\[ P = \frac{V^2}{R} \]
- For resistance R:
\[ P_R = \frac{V^2}{R} \]
- For resistance 2R:
\[ P_{2R} = \frac{V^2}{2R} \]

### Step 3: Calculate Thermal Energy Developed
- The thermal energy (E) developed in a resistor over time (t) can be calculated using the formula:
\[ E = P \cdot t \]
- For resistance R:
\[ E_R = P_R \cdot t = \left(\frac{V^2}{R}\right) \cdot t = \frac{V^2 t}{R} \]
- For resistance 2R:
\[ E_{2R} = P_{2R} \cdot t = \left(\frac{V^2}{2R}\right) \cdot t = \frac{V^2 t}{2R} \]

### Step 4: Find the Ratio of Thermal Energies
- Now, we can find the ratio of thermal energies developed in R and 2R:
\[ \text{Ratio} = \frac{E_R}{E_{2R}} = \frac{\frac{V^2 t}{R}}{\frac{V^2 t}{2R}} \]
- Simplifying this gives:
\[ \text{Ratio} = \frac{V^2 t}{R} \cdot \frac{2R}{V^2 t} = \frac{2R}{R} = 2 \]

### Step 5: Final Ratio
- Therefore, the ratio of thermal energy developed in R to that in 2R is:
\[ \text{Ratio} = 2:1 \]

### Conclusion
The thermal energy developed in resistances R and 2R are in the ratio of **2:1**.

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