To analyze the given RLC series circuit with no voltage source, we start with the original equation:

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

Since $V(t) = 0$, this simplifies to:

$RI + L \frac{dI}{dt} + \frac{q}{C} = 0$

Now, substituting the relationship between current $I$ and charge $q$:

$I = \frac{dq}{dt}$

Substituting this into the equation gives us:

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

Rearranging this, we obtain the standard form of a second-order linear differential equation:

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

Characteristic Equation

To solve this differential equation, we assume a solution of the form $q(t) = e^{rt}$. Substituting this into the equation leads to the characteristic equation:

$L r^2 + R r + \frac{1}{C} = 0$

This is a quadratic equation in $r$. We can use the quadratic formula to find the roots:

$r = \frac{-R \pm \sqrt{R^2 - 4L\left(\frac{1}{C}\right)}}{2L}$

Types of Damping

The nature of the roots (real and distinct, real and repeated, or complex) will depend on the discriminant $D = R^2 - \frac{4L}{C}$:

1. Overdamped: If $D > 0$, there are two distinct real roots.
2. Critically damped: If $D = 0$, there is one repeated real root.
3. Underdamped: If $D < 0$, the roots are complex conjugates.

Conclusion

The characteristic equation provides insight into the behavior of the charge $q(t)$ in the circuit over time, depending on the parameters $R$, $L$, and $C$.

Would you like more details on solving this equation or the implications of different damping scenarios? Here are some questions to consider:

1. What are the physical interpretations of the different damping cases in an RLC circuit?
2. How would the response of the circuit change with varying values of $R$, $L$, and $C$?
3. Can you describe how to derive the time-domain solution for each damping case?
4. What practical applications are there for RLC circuits in engineering?
5. How do you measure the parameters $R$, $L$, and $C$ in an experimental setup?

Tip: When analyzing RLC circuits, always check the damping factor to understand how the circuit will respond over time.