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)/(2l))`
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 find the amplitude of the amount of charge on the capacitor as a function of time \( t \) in an LCR circuit behaving like a damped harmonic oscillator, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Damped Harmonic Oscillator**:
In a damped harmonic oscillator, the motion can be described by the second-order differential equation:
\[ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0 \]
where \( m \) is the mass, \( b \) is the damping constant, and \( k \) is the spring constant.

2. **Amplitude Decay in Damped Oscillation**:
The general solution for the amplitude \( A(t) \) of a damped oscillator is given by:
\[ A(t) = A_0 e^{-\frac{b}{2m} t} \]
where \( A_0 \) is the initial amplitude.

3. **Apply to the LCR Circuit**:
In the context of an LCR circuit, we can relate the parameters:
- Replace \( m \) with \( L \) (inductance),
- Replace \( b \) with \( R \) (resistance),
- Replace \( k \) with \( \frac{1}{C} \) (capacitance).

4. **Write the Equation for Charge**:
The charge \( Q(t) \) on the capacitor as a function of time can be expressed as:
\[ Q(t) = Q_0 e^{-\frac{R}{2L} t} \]
where \( Q_0 \) is the initial charge on the capacitor.

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

### Final Result:
The amplitude of the amount of charge on the capacitor as a function of time \( t \) is:
\[ Q(t) = Q_0 e^{-\frac{R}{2L} t} \]