An LCR series circuit behaves like a damped harmonic oscillator. Comparing it with a physical spring-mass damped oscillator having damping constant ‘b’, the correct equivalence would be (Take, i = current, V = voltage, x = displacement, q = charge, F = force, v = velocity)

Options

A. `qharrx,vltharrgti,FharrV`
B. `qharrv,iharrx,FharrV`
C. `qharrx,vharrV,Ftoi`
D. `qharrF,iharrx,Vtov`

Explanation

To solve the problem of comparing an LCR series circuit with a damped harmonic oscillator, we will derive the equations for both systems and establish the equivalences.

### Step-by-Step Solution:

1. **Write the Equation for a Damped Harmonic Oscillator:**
The equation for a damped harmonic oscillator (spring-mass system) is given by:
\[ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0 \]
Here, \(m\) is the mass, \(b\) is the damping constant, \(k\) is the spring constant, \(x\) is the displacement, and \(\frac{dx}{dt}\) is the velocity.

**Hint:** Identify the terms in the equation: mass, damping, and restoring force.

2. **Write the Equation for an LCR Series Circuit:**
For an LCR circuit, applying Kirchhoff's voltage law gives:
\[ L \frac{di}{dt} + Ri + \frac{q}{C} = 0 \]
where \(L\) is the inductance, \(R\) is the resistance, \(C\) is the capacitance, \(i\) is the current, and \(q\) is the charge on the capacitor.

**Hint:** Recognize that the terms represent inductive, resistive, and capacitive effects.

3. **Relate Current and Charge:**
The current \(i\) is the rate of change of charge \(q\):
\[ i = \frac{dq}{dt} \]
Substitute this into the LCR equation:
\[ L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{q}{C} = 0 \]

**Hint:** Use the relationship between current and charge to express the equation in terms of charge.

4. **Compare the Two Equations:**
Now we have two equations:
- Damped oscillator: \(m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0\)
- LCR circuit: \(L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{q}{C} = 0\)

By comparing coefficients, we can establish the following equivalences:
- \(m \leftrightarrow L\) (mass corresponds to inductance)
- \(b \leftrightarrow R\) (damping constant corresponds to resistance)
- \(k \leftrightarrow \frac{1}{C}\) (spring constant corresponds to the reciprocal of capacitance)

**Hint:** Look for the corresponding terms in both equations to establish equivalences.

5. **Establish the Final Equivalences:**
From the comparisons, we can summarize:
- Charge \(q\) is analogous to displacement \(x\)
- Current \(i\) is analogous to velocity \(\frac{dx}{dt}\)
- Voltage \(V\) is analogous to force \(F\)

**Hint:** Identify how each physical quantity in the LCR circuit corresponds to the quantities in the damped oscillator.

### Conclusion:
The correct equivalences based on the analysis are:
- \(q \leftrightarrow x\)
- \(i \leftrightarrow \frac{dx}{dt}\)
- \(V \leftrightarrow F\)
- \(b \leftrightarrow R\)
- \(m \leftrightarrow L\)

Thus, the correct answer is the first option that matches these equivalences.