Let's look at an alternating current or RLC circuit where are connected in series
resistor (resistance R), capacitor (capacitance C) and coil (inductance
dance L). This is described by the equation: RI+L*(dI/dt)+q/C=V(t)

where V is the voltage of the voltage source and q is the electric charge of the capacitor. Current can also be represented by charge, because I = dq/dt. Place this in the equation of the RLC circuit, and you get a second-order diﬀerential equation for the charge and solve the characteristic equation corresponding to this equation in the case where there is no voltage source connected to the circuit (i.e. V (t) = 0).

Under what conditions is it a) underdamped b) over-
damped c) critically damped vibration? d) Prove that it is a harmonic oscillation if R = 0 and solve for the angular frequency of the harmonic oscillation.

To analyze the given RLC series circuit and derive the conditions for underdamped, overdamped, and critically damped vibrations, let's start by rewriting the equation you've provided and deriving the corresponding second-order differential equation.

### 1. Formulation of the Differential Equation

The given equation for the RLC circuit is:

\[
RI + L \frac{dI}{dt} + \frac{q}{C} = V(t)
\]

Using the relationship \(I = \frac{dq}{dt}\), we can rewrite \(I\):

\[
R \frac{dq}{dt} + L \frac{d^2q}{dt^2} + \frac{q}{C} = V(t)
\]

For the case where \(V(t) = 0\) (no voltage source), the equation becomes:

\[
L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{q}{C} = 0
\]

### 2. Characteristic Equation

Assuming a solution of the form \(q(t) = e^{\lambda t}\), we can substitute this into the differential equation:

\[
L \lambda^2 e^{\lambda t} + R \lambda e^{\lambda t} + \frac{1}{C} e^{\lambda t} = 0
\]

Factoring out \(e^{\lambda t}\) (which is never zero) gives us the characteristic equation:

\[
L \lambda^2 + R \lambda + \frac{1}{C} = 0
\]

### 3. Solutions of the Characteristic Equation

The characteristic equation is a quadratic equation in the form \(a\lambda^2 + b\lambda + c = 0\). The solutions for \(\lambda\) can be found using the quadratic formula:

\[
\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = L\), \(b = R\), and \(c = \frac{1}{C}\):

\[
\lambda = \frac{-R \pm \sqrt{R^2 - 4\frac{L}{C}}}{2L}
\]

### 4. Conditions for Damping

1. **Underdamped**: The system is underdamped if the discriminant \(D = R^2 - 4\frac{L}{C} < 0\). This leads to complex roots, which result in oscillatory behavior.

2. **Critically damped**: The system is critically damped if \(D = R^2 - 4\frac{L}{C} = 0\). This results in a repeated real root, leading to the fastest return to equilibrium without oscillation.

3. **Overdamped**: The system is overdamped if \(D = R^2 - 4\frac{L}{C} > 0\). This results in two distinct real roots and the system returns to equilibrium slowly without oscillation.

### 5. Harmonic Oscillation

To prove that the system exhibits harmonic oscillation when \(R = 0\):

The differential equation reduces to:

\[
L \frac{d^2q}{dt^2} + \frac{q}{C} = 0
\]

This is a second-order linear differential equation with constant coefficients, which can be rewritten as:

\[
\frac{d^2q}{dt^2} + \frac{1}{LC} q = 0
\]

The characteristic equation is:

\[
\lambda^2 + \frac{1}{LC} = 0
\]

This has solutions:

\[
\lambda = \pm j\sqrt{\frac{1}{LC}}
\]

The solution for \(q(t)\) will be:

\[
q(t) = A \cos\left(\sqrt{\frac{1}{LC}} t + \phi\right)
\]

where \(A\) is the amplitude and \(\phi\) is the phase constant. The angular frequency \(\omega\) of the harmonic oscillation is given by:

\[
\omega = \sqrt{\frac{1}{LC}}
\]

### Summary

- **Underdamped**: \(R^2 - 4\frac{L}{C} < 0\)
- **Critically damped**: \(R^2 - 4\frac{L}{C} = 0\)
- **Overdamped**: \(R^2 - 4\frac{L}{C} > 0\)
- **Harmonic Oscillation (if \(R=0\))**: \(\omega = \sqrt{\frac{1}{LC}}\)

If you have any questions or need further details, feel free to ask! Here are some related questions you might consider:

1. How does the value of resistance \(R\) affect the transient response in an RLC circuit?
2. What practical applications utilize the concepts of damping in electrical circuits?
3. How would you analyze an RLC circuit with a constant voltage source?
4. What is the impact of varying capacitance \(C\) and inductance \(L\) on the system's natural frequency?
5. Can you describe a physical system that exhibits similar behavior to the RLC circuit in terms of damping? 

**Tip**: When analyzing circuits, always pay attention to the signs and units of parameters to avoid mistakes in calculations!

Let's look at an alternating current or RLC circuit where are connected in series resistor (resistance R), capacitor (capacitance C) and coil (inductance L). This is described by the equation: RI + L*(dI/dt) + q/C = V(t) where V is the voltage of the voltage source and q is the electric charge of the capacitor. Current can also be represented by charge, because I = dq/dt. Place this in the equation of the RLC circuit, and you get a second-order differential equation for the charge and solve the characteristic equation corresponding to this equation in the case where there is no voltage source connected to the circuit (i.e. V(t) = 0). Under what conditions is it a) underdamped b) over-damped c) critically damped vibration? d) Prove that it is a harmonic oscillation if R = 0 and solve for the angular frequency of the harmonic oscillation.

Explore the behavior of an RLC circuit through its differential equation, determining conditions for underdamped, overdamped, and critically damped vibrations. Learn how to derive the conditions for harmonic oscillation when resistance is zero, along with the corresponding angular frequency. Suitable for advanced high school students.

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