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Chapter 30: Problem 31

A 2.00 - \(\mu\) F capacitor was fully charged by being connected to a 12.0 - \(V\) battery. The fully charged capacitor is then connected in series with a resistor and an inductor: \(R=50.0 \Omega\) and \(L=0.200 \mathrm{H}\). Calculate the damped frequency of the resulting circuit.

Short Answer

Expert verified
Answer: To find the damped frequency (ω_d) of the RLC circuit, first calculate the angular frequency (ω_0) using the formula $$\omega_0 = \frac{1}{\sqrt{LC}}$$, where L = 0.200 H and C = 2.00 × 10^{-6} F. Next, use the formula for damped frequency, $$\omega_d = \sqrt{\omega_0^2 - (\frac{R}{2L})^2}$$, and plug in the values for R = 50.0 Ω, L = 0.200 H, and the calculated value of ω_0. Finally, calculate the damped frequency (ω_d) in rad/s.

Step by step solution

01

Write down the formula for damped frequency

The formula for damped frequency (ω_d) in an RLC circuit is given by: $$ \omega_d = \sqrt{\omega_0^2 - (\frac{R}{2L})^2} $$ where - ω_0 is the angular frequency of the circuit; - R is the resistance of the resistor; - L is the inductance of the inductor. Now, we need to calculate the angular frequency (ω_0).
02

Calculate the angular frequency (ω_0)

The angular frequency (ω_0) of an RLC circuit can be found using the formula: $$ \omega_0 = \frac{1}{\sqrt{LC}} $$ where - L is the inductance of the inductor; - C is the capacitance of the capacitor. We are given the values for L and C: L = 0.200 H; C = 2.00 × 10^{-6} F. Plug in the values into the formula and calculate ω_0: $$ \omega_0 = \frac{1}{\sqrt{(0.200)(2.00 × 10^{-6})}}\thinspace \mathrm{rad/s} $$
03

Calculate the damped frequency (ω_d)

Now that we have the value of ω_0, we can now calculate the damped frequency (ω_d) using the formula from Step 1: $$ \omega_d = \sqrt{\omega_0^2 - (\frac{R}{2L})^2} $$ We are given the values for R and L: R = 50.0 Ω; L = 0.200 H. Plug in the values and the calculated value of ω_0 into the formula and find ω_d: $$ \omega_d = \sqrt{(\frac{1}{\sqrt{(0.200)(2.00 × 10^{-6})}})^2 - (\frac{50.0}{2(0.200)})^2}\thinspace \mathrm{rad/s} $$ Calculate the damped frequency using the values provided. The result will be the damped frequency of the RLC circuit.

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