Ratio of thermal energy released in two resistors R and 3R connected in parallel in an electric circuit is :

Options

A. ` 1 : 3`
B. ` 3 : 1`
C. `1 : 1`
D. `1 : 27`

Answer

To find the ratio of thermal energy released in two resistors \( R \) and \( 3R \) connected in parallel, we can follow these steps:

### Step 1: Understand the Concept
When resistors are connected in parallel, the potential difference (voltage) across each resistor is the same. The thermal energy \( E \) released in a resistor can be calculated using the formula:
\[ E = \frac{V^2}{R} \times t \]
where \( V \) is the voltage across the resistor, \( R \) is the resistance, and \( t \) is the time for which the current flows.

### Step 2: Define the Resistors
Let:
- \( R_1 = R \) (the first resistor)
- \( R_2 = 3R \) (the second resistor)

### Step 3: Calculate Thermal Energy for Each Resistor
For the first resistor \( R_1 \):
\[ E_1 = \frac{V^2}{R_1} \times t = \frac{V^2}{R} \times t \]

For the second resistor \( R_2 \):
\[ E_2 = \frac{V^2}{R_2} \times t = \frac{V^2}{3R} \times t \]

### Step 4: Find the Ratio of Thermal Energies
Now, we can find the ratio of the thermal energies released in the two resistors:
\[ \frac{E_1}{E_2} = \frac{\frac{V^2}{R} \times t}{\frac{V^2}{3R} \times t} \]

### Step 5: Simplify the Ratio
Notice that \( V^2 \) and \( t \) cancel out:
\[ \frac{E_1}{E_2} = \frac{1/R}{1/(3R)} = \frac{3R}{R} = 3 \]

### Final Answer
Thus, the ratio of thermal energy released in resistors \( R \) and \( 3R \) is:
\[ \frac{E_1}{E_2} = 3:1 \]