In a parallel L-C-R circuit spring constant K is analogous to :-

Options

A. L
B. C
C. 1/L
D. 1/C

Answer

To solve the problem of finding what the spring constant \( K \) is analogous to in a parallel L-C-R circuit, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Mechanical-Electrical Analogy**:
- In mechanics, the spring constant \( K \) relates to the force exerted by a spring, which is given by Hooke's Law: \( F = -Kx \), where \( x \) is the displacement from the equilibrium position.
- In electrical circuits, we can draw analogies between mechanical quantities and electrical quantities.

2. **Identify the Analogous Quantities**:
- In the mechanical system, the displacement \( x \) is analogous to the charge \( Q \) in the electrical system.
- The force \( F \) is analogous to the electromotive force (EMF) in the electrical circuit.

3. **Energy Stored in the Systems**:
- The energy stored in a spring is given by the formula:
\[ U_{\text{spring}} = \frac{1}{2} K x^2 \]
- The energy stored in a capacitor is given by:
\[ U_{\text{capacitor}} = \frac{Q^2}{2C} \]

4. **Relate the Energies**:
- To find the relationship between \( K \) and \( C \), we can set the two energy expressions equal to each other, as they represent the same concept in their respective systems.
- From the energy expressions, we can see:
\[ \frac{1}{2} K x^2 \quad \text{and} \quad \frac{Q^2}{2C} \]
- Since \( x \) is analogous to \( Q \), we can substitute \( Q \) for \( x \) in the energy equation:
\[ \frac{1}{2} K Q^2 = \frac{Q^2}{2C} \]

5. **Derive the Relationship**:
- By simplifying the equation, we can cancel \( \frac{Q^2}{2} \) from both sides (assuming \( Q \neq 0 \)):
\[ K = \frac{1}{C} \]
- This shows that the spring constant \( K \) is inversely proportional to the capacitance \( C \).

6. **Conclusion**:
- Therefore, in a parallel L-C-R circuit, the spring constant \( K \) is analogous to \( \frac{1}{C} \).

### Final Answer:
The spring constant \( K \) is analogous to \( \frac{1}{C} \) in a parallel L-C-R circuit.

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