Resistances `R_(1)` and `R_(2)` are joined in parallel and a current is passed so that the amount of heat liberated is `H_(1)` and `H_(2)` respectively. The ratio `(H_(1))/(H_(2))` has the value

Options

A. `(R_(2))/(R_(1))`
B. `(R_(1))/(R_(2))`
C. `(R_(1)^(2))/(R_(2)^(2))`
D. `(R_(2)^(2))/(R_(1)^(2))`

Answer

To solve the problem of finding the ratio of heat liberated in two resistances connected in parallel, we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Setup**:
- We have two resistances, \( R_1 \) and \( R_2 \), connected in parallel.
- When a current passes through these resistances, they produce heat \( H_1 \) and \( H_2 \) respectively.

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

3. **Current in Parallel Resistors**:
- In a parallel circuit, the voltage across each resistor is the same. Therefore, the current through each resistor can be expressed as:
\[ I_1 = \frac{V}{R_1} \quad \text{and} \quad I_2 = \frac{V}{R_2} \]
- This means that the current is inversely proportional to the resistance.

4. **Finding the Ratio of Currents**:
- The ratio of the currents can be derived as follows:
\[ \frac{I_1}{I_2} = \frac{R_2}{R_1} \]

5. **Substituting into the Heat Formula**:
- Now, substituting \( I_1 \) and \( I_2 \) into the heat formula, we get:
\[ H_1 = I_1^2 R_1 T \quad \text{and} \quad H_2 = I_2^2 R_2 T \]

6. **Calculating the Ratio of Heat**:
- The ratio of heat produced in the two resistors is:
\[ \frac{H_1}{H_2} = \frac{I_1^2 R_1 T}{I_2^2 R_2 T} \]
- The time \( T \) cancels out:
\[ \frac{H_1}{H_2} = \frac{I_1^2 R_1}{I_2^2 R_2} \]

7. **Substituting the Current Ratio**:
- We can substitute \( \frac{I_1}{I_2} = \frac{R_2}{R_1} \):
\[ \frac{H_1}{H_2} = \frac{(I_1/I_2)^2 R_1}{R_2} = \frac{\left(\frac{R_2}{R_1}\right)^2 R_1}{R_2} \]

8. **Simplifying the Expression**:
- Simplifying gives:
\[ \frac{H_1}{H_2} = \frac{R_2}{R_1} \]

9. **Final Result**:
- Therefore, the ratio of heat liberated is:
\[ \frac{H_1}{H_2} = \frac{R_2}{R_1} \]

### Conclusion
The ratio of heat liberated \( \frac{H_1}{H_2} \) is \( \frac{R_2}{R_1} \).