metal object, the inductance of this circuit increase by 1.000%, while that the

Question

metal object, the inductance of this circuit increase by 1.000%, while that the second is unchanged. Determine the beat frequency that would be detected in this situation. A series circuit contains a 3.00-H inductor, a 3.00-muF capacitor, and a 30.0-Ohms resistor connected to a 120-V (rms) source of variable frequency. Find the power delivered to the circuit when the frequency of the source is the resonance frequency, one-half the resonance frequency, one-fourth the resonance frequency, two times the resonance frequency. From your calculations, can you draw a conclusion about the frequency at which the maximum power is delivered to the circuit?

Explanation / Answer

L = 3.00 H     C = 3.00 x 10-6 F      R = 30.0

First let's find the resonance frequence fo = 1/2(LC) = 53.0516 Hz

to find the power delivered we must first find I(rms), which is equal to Vrms/Z, where Z = (R2 + [Xl - Xc]2)

(a) at the resonance frequency Z = R and and XL = XC

Irms = 120V/30 = 4 A

Power = 120V(4A) = 480 W

(b) When not at the resonance frequency, XL = 2fL and XC = 1/2fC

1/2 fo = 26.5258 Hz

XL = 500 and XC = 2000

Z = 1500.29997

Irms = Vrms/Z = 0.079984004 A

P = Vrms(Irms) = 9.60 W

(c) 1/4 fo = 13.2629 hz

XL = 249.999 and XC = 4000

Z = 3750

Irms = 0.031998 A

P = 3.84 W

(d) 2fo = 106.1032 hz

XL = 2000 and XC = 500

Z = 1500.29997

Irms = .07998 A

P = 9.60 W

(e) 4fo = 2122.064 hz

XL = 39999.96 and XC = 25

Z = 39974.97

Irms = 0.003001878333 A

P = 0.360 W

the maximum power is delivered to the circuit when we are at the resonance frequency fo

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