COLLEGE PHYSICS
COLLEGE PHYSICS
2nd Edition
ISBN: 9781464196393
Author: Freedman
Publisher: MAC HIGHER
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Chapter 21, Problem 69QAP
To determine

(a)

The impedance of a LRC circuit with 500 O resistor, 0.20 H inductor and 2.0 μF capacitor connected to a 100 V peak, 500 Hz ac source.

Expert Solution
Check Mark

Answer to Problem 69QAP

The impedance of a LRC circuit with 500 O resistor, 0.20 H inductor and 2.0 μF capacitor connected to a 100 V peak, 500 Hz ac source is found to be 690 Ω.

Explanation of Solution

Given:

L=0.20 HC=2.0 μF=2.0×106FR=500 Ωf=500 HzV0=100 V

Formula used:

The impedance Z of an ac circuit is given by the expression,

  Z=( 1 2πfC2πfL)2+R2

Calculation:

Substitute the values of the variables in the formula and calculate the value of impedance of the circuit.

  Z=( 1 2πfC 2πfL)2+R2=( 1 2( 3.14 )( 500 Hz )( 2.0× 10 6 F ) 2( 3.14 )( 500 Hz )( 0.20 H ))2+( 500 Ω)2=685.5 Ω=690 Ω

Conclusion:

Thus, the impedance of a LRC circuit with 500 O resistor, 0.20 H inductor and 2.0 μF capacitor connected to a 100 V peak, 500 Hz ac source is found to be 690 Ω.

To determine

(b)

The amplitude of the current from the source.

Expert Solution
Check Mark

Answer to Problem 69QAP

The amplitude of the current from the source is found to be 0.15 A.

Explanation of Solution

Given:

  V0=100 VZ=685.5 Ω

Formula used:

The amplitude of the current from the source is equal to the peak value of current. This is given by,

  i0=V0Z

Calculation:

Substitute the values of the variables in the formula and calculate the value of the peak current.

  i0=V0Z=100 V685.5 Ω Ω=0.146 A=0.15 A

Conclusion;Thus, the amplitude of the current from the source is found to be 0.15 A.

To determine

(c)

The equation which shows the variation of current with time.

Expert Solution
Check Mark

Answer to Problem 69QAP

The current in the given circuit obeys the following equation:

  i(t)=(0.15 A)sin(1000πt0.754 rad).

Explanation of Solution

Given:

  L=0.20 HC=2.0 μF=2.0×106FR=500 Ωf=500 Hzi0=0.15 A

The equation showing the variation of voltage with time

  V(t)=(100 V)sin(1000πt)

Formula used:

The current in the circuit varies with time according to the equation:

  i(t)=i0sin(1000πt+ϕ)......(1)

The phase angle is given by the expression,

  ϕ=arctan[1R(12πfC2πfL)]......(2)

Calculation:

Calculate the phase angle between current and voltage by substituting the values of the variables in equation (2).

  ϕ=arctan[1R(1 2πfC2πfL)]=arctan[1( 500 Ω)(1 2( 3.14 )( 500 Hz )( 2.0× 10 6 F )2( 3.14)( 500 Hz)( 0.20 H))]=arctan(469 Ω500 Ω)=43.2°

Express the angle in radians.

  ϕ=(43.2°)(π rad180°)=43.2°×3.14 rad180°=0.754 rad

Substitute the value of i0 and ϕ in equation (1).

  i(t)=i0sin(1000πt+ϕ)=(0.15 A)sin(1000πt0.754 rad)

Conclusion:

Thus the current in the given circuit obeys the following equation:

  i(t)=(0.15 A)sin(1000πt0.754 rad).

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