A series RLC circuit, with R = 200ohms, L = 0.1H and C = 13.33µF, has an initial charge on the capacitor of Q = 2.67mC. A switch is closed at t=0 allowing the capacitor to discharge through the R and L mentioned earlier.The circuit is1. Critically damped2. Underdamped3. Overdamped

To determine whether the circuit is critically damped, underdamped, or overdamped, we need to calculate the damping ratio. The damping ratio (ζ) is given by the formula:

ζ = R / (2 * sqrt(L/C))

where R is the resistance, L is the inductance, and C is the capacitance.

Let's plug in the given values:

ζ = 200 / (2 * sqrt(0.1/13.33e-6))

After calculating the above expression, we get ζ ≈ 1.

If ζ = 1, the system is critically damped. If ζ < 1, the system is underdamped. If ζ > 1, the system is overdamped.

Since ζ ≈ 1, the RLC circuit is critically damped.

To determine the damping characteristics of the series RLC circuit, we can calculate the damping ratio (ζ) using the formula:

ζ = R / (2 * √(L / C))

Given that R = 200 ohms, L = 0.1H, and C = 13.33µF, we can substitute these values into the formula:

ζ = 200 / (2 * √(0.1 / 13.33 * 10^-6))

Simplifying the expression:

ζ = 200 / (2 * √(0.1 / 0.01333))

ζ = 200 / (2 * √7.5)

ζ = 200 / (2 * 2.7386)

ζ ≈ 36.6

Now, we can determine the damping characteristics based on the value of ζ:

1. If ζ = 1, the circuit is critically damped.
2. If ζ < 1, the circuit is underdamped.
3. If ζ > 1, the circuit is overdamped.

In this case, since ζ ≈ 36.6, which is greater than 1, the circuit is overdamped.

The behavior of an RLC circuit (whether it's critically damped, underdamped, or overdamped) is determined by the damping factor, which is a function of resistance (R), inductance (L), and capacitance (C).

The damping factor (ζ) is given by the formula:

ζ = R / (2 * sqrt(L/C))

Let's calculate the damping factor for the given RLC circuit:

R = 200 ohms L = 0.1 H C = 13.33 µF = 13.33 * 10^-6 F

ζ = 200 / (2 * sqrt(0.1/(13.33 * 10^-6))) ζ = 200 / (2 * sqrt(7518.8)) ζ = 200 / (2 * 86.72) ζ = 200 / 173.44 ζ = 1.15

The damping factor is greater than 1, which means the RLC circuit is overdamped.