A LCR circuit behaves like a damped harmonic oscillator. Comparing it with a physical spring mass damped oscillator having damping constant ‘b’. If the amount of initial charge on the capacitor be `Q_(0).` then the amplitude of the amount of charge on the capacitor as a function of time t will be:

Options

A. `Q = Q_(0)e ^(-(Rt)/(2I))`
B. `Q = Q_(0)(1-e ^((2Rt)/(L )))`
C. `Q = Q_(0)e ^(-(Rt)/(L ))`
D. `Q = Q _(0)(1-e ^(-(Rt)/(L )))`

Explanation

To solve the problem of finding the amplitude of the charge on the capacitor in an LCR circuit as a function of time, we can follow these steps:

### Step 1: Understand the LCR Circuit Dynamics
An LCR circuit consists of an inductor (L), a capacitor (C), and a resistor (R). The behavior of this circuit can be modeled similarly to a damped harmonic oscillator, where the charge on the capacitor behaves like the displacement in a mechanical oscillator.

### Step 2: Write the Differential Equation
The governing equation for the charge \( Q \) on the capacitor in the LCR circuit can be derived from Kirchhoff's laws. The equation is:
\[ L \frac{d^2Q}{dt^2} + R \frac{dQ}{dt} + \frac{Q}{C} = 0 \]
This is a second-order linear differential equation.

### Step 3: Compare with Damped Harmonic Motion
The standard form of the damped harmonic oscillator is:
\[ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0 \]
By comparing the two equations, we can identify:
- \( m = L \)
- \( b = R \)
- \( k = \frac{1}{C} \)

### Step 4: Determine the Damping Ratio
For a damped harmonic oscillator, the amplitude \( A(t) \) decreases exponentially over time due to damping. The amplitude of the charge on the capacitor can be expressed as:
\[ Q(t) = Q_0 e^{-\frac{b}{2m} t} \]
Substituting \( b = R \) and \( m = L \):
\[ Q(t) = Q_0 e^{-\frac{R}{2L} t} \]

### Step 5: Final Expression for Amplitude
Thus, the amplitude of the charge on the capacitor as a function of time \( t \) is:
\[ Q(t) = Q_0 e^{-\frac{R}{2L} t} \]