Explanation Figure 1 shows the circuit diagram. Use Kirchhoff’s junction rule to write the equation relating current in each branch. I R = I L + I C (I) Here, I R is the current passing through the resistor, I C is the current passing through the capacitor, I L is the current passing through the inductor. Use Kirchhoff’s loop to write the equation relating voltage drops around each loop. V 0 sin ω t − I R R − Q C C = 0 V 0 sin ω t − I R R − L d I L d t = 0 (II) Here, V 0 is the peak voltage, ω t is the phase angle, Q C is the charge of the capacitor, L is the inductance, C is the capacitance, R is the resistance, d I L d t is the rate of change of current in inductor with respect to time. Use equation (II) to solve for d I L d t , d I L d t = V 0 L sin ω t − I R R L (III) Write the expression for current passing through capacitor in terms of charge. I C = d Q C d t (IV) Use equation (II) to solve for Q C , Q C = C V 0 sin ω t − I R R C (V) Use equation (IV) to write the expression for differential equation of I C with respect to time. d I C d t = d 2 Q C d t 2 (VI) Use equation (V) in (VI) to find d I C d t , d I C d t = d 2 d t 2 ( C V 0 sin ω t − I R R C ) = − C V 0 ω 2 sin ω t − R C d I R 2 d t 2 (VII) Take the differential of equation (I). d I R d t = d I L d t + d I C d t (VIII) Use equation (III), (VII) in (VIII) to find d I R d t . d I R d t = V 0 L sin ω t − I R R L − C V 0 ω 2 sin ω t − R C d 2 I R d t 2 (IX) Set the current passing through the resistor I R . I R = I 0 sin ( ω t + ϕ ) (X) Here, I 0 is the peak value of the current, ϕ is the constant phase. Use sin ( A+B ) = sin A cos B + cos A sin B identity in equation (X). I R = I 0 ( sin ω t cos ϕ + cos ω t sin ϕ ) (XI) Use equation (XI) in (IX), I 0 d d t ( sin ω t cos ϕ + cos ω t sin ϕ ) = V 0 L sin ω t − ( sin ω t cos ϕ + cos ω t sin ϕ ) I 0 R L ( − C V 0 ω 2 sin ω t − R C I 0 d 2 d t 2 ( sin ω t cos ϕ + cos ω t sin ϕ ) ) (XII) Differentiate equation (XII) and it becomes, I 0 ω cos ω t cos ϕ − I 0 ω sin ω t sin ϕ = V 0 L sin ω t − I 0 R L sin ω t cos ϕ + I 0 R L cos ω t sin ϕ − C V 0 ω 2 sin ω t + R C I 0 ω 2 sin ω t cos ϕ + R C I 0 ω 2 cos ω t sin ϕ (XIII) Equate time dependent sine and cosine terms separately equal to zero to solve for the magnitude and phase of the current through the voltage source. Use equation (XIII) and equate the coefficients of cos ω t , I 0 ω cos ϕ = I 0 ω R X L sin ϕ − I 0 ω R X C sin ϕ tan ϕ = X L X C R ( X L − X C ) ϕ = tan − 1 [ X L X C R ( X L − X C ) ] (XIV) Use equation (XIII) and equate the coefficients of sin ω t , − I 0 ω sin ϕ = V 0 ω X L − I 0 ω R X L cos ϕ − V 0 ω X C + R I 0 ω X C cos ϕ I 0 = V 0 ( X C − X L ) X C X L sin ϕ + R ( X C − X L ) cos ϕ (XV) Use equation (XIV) to find sin ϕ and cos ϕ

4th Edition

ISBN: 9780134112039

Author: GIANCOLI

Publisher: PEARSON

See similar textbooks

Concept explainers

Voltage Across Inductor

An actor is a passive electrical component that consists of a coil bent in the form of wire which uses electromagnetism to produce electric current through the coil. When the electric current flows through a conductor that is the coil, a magnetic flux is…

Question

Chapter 30, Problem 100GP

To determine

The current in the resistor, capacitor, inductor and the total impedance in the circuit.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 30, Problem 99GP

Chapter 30, Problem 101GP

Students have asked these similar questions

Checkpoint 1 The figure shows the current i in a single-loop circuit with a battery B and a resistance R (and wires of neg- ligible resistance). (a) Should the emf arrow at B be drawn pointing leftward or rightward? At points a, B C R b, and c, rank (b) the magnitude of the current, (c) the electric potential, and (d) the electric potential energy of the charge carriers, greatest first.

Pls help ASAP

Pls help asap

Chapter 30 Solutions

PHYSICS FOR SCIEN & ENGNR W/MOD MAST

Ch. 30.1 - Prob. 1AE Ch. 30.1 - Prob. 1BE Ch. 30.3 - Prob. 1CE Ch. 30.4 - Show that L/R does have dimensions of lime. (See...Ch. 30.4 - Prob. 1EE Ch. 30.5 - Return to the Chapter-Opening Question, page 785,...Ch. 30.7 - At what frequency is the reactance of a 1.0-F...Ch. 30.7 - Prob. 1HE Ch. 30 - Prob. 1Q Ch. 30 - Prob. 2Q

Ch. 30 - Prob. 3Q Ch. 30 - Prob. 4Q Ch. 30 - If you are given a fixed length of wire, how would...Ch. 30 - Prob. 6Q Ch. 30 - Prob. 7Q Ch. 30 - Prob. 8Q Ch. 30 - What keeps an LC circuit oscillating even after...Ch. 30 - Is the ac current in the indicator always the same...Ch. 30 - Prob. 11Q Ch. 30 - In an ac LRC circuit, if XL XC, the circuit is...Ch. 30 - Prob. 13Q Ch. 30 - Under what conditions is the impedance in an LRC...Ch. 30 - Is it possible for the instantaneous power output...Ch. 30 - In an ac LRC circuit, does the power factor, cos,...Ch. 30 - Describe briefly how the frequency of the source...Ch. 30 - Prob. 18Q Ch. 30 - In an LRC circuit, the current and the voltage in...Ch. 30 - Compare the oscillations or an LRC circuit to the...Ch. 30 - Prob. 1P Ch. 30 - Prob. 2P Ch. 30 - Prob. 3P Ch. 30 - Prob. 4P Ch. 30 - (I) If the current in a 280-mH coil changes...Ch. 30 - Prob. 6P Ch. 30 - Prob. 7P Ch. 30 - Prob. 8P Ch. 30 - Prob. 9P Ch. 30 - (II) If the outer conductor of a coaxial cable has...Ch. 30 - Prob. 11P Ch. 30 - Prob. 12P Ch. 30 - Prob. 13P Ch. 30 - (II) Ignoring any mutual inductance, what is the...Ch. 30 - (I) The magnetic field inside an air-filled...Ch. 30 - (I) Typical large values for electric and magnetic...Ch. 30 - (II) What is the energy density at the center of a...Ch. 30 - (II) Calculate the magnetic and electric energy...Ch. 30 - Prob. 19P Ch. 30 - (II) Determine the total energy stored per unit...Ch. 30 - (II) Determine the total energy stored per unit...Ch. 30 - Prob. 22P Ch. 30 - (II) How many time constants does it take for the...Ch. 30 - (II) It takes 2.56 ms for the current in an LR...Ch. 30 - Prob. 25P Ch. 30 - (II) In the circuit of Fig. 3027, determine the...Ch. 30 - Prob. 27P Ch. 30 - Prob. 28P Ch. 30 - (II) A 12-V battery has been connected to an LR...Ch. 30 - Prob. 30P Ch. 30 - (I) The variable capacitor in the tuner of an AM...Ch. 30 - Prob. 32P Ch. 30 - (II) In some experiments, short distances are...Ch. 30 - Prob. 34P Ch. 30 - Prob. 35P Ch. 30 - Prob. 36P Ch. 30 - Prob. 37P Ch. 30 - Prob. 38P Ch. 30 - (I) At what frequency will a 32.0-mH inductor have...Ch. 30 - (I) What is the reactance of a 9.2-F capacitor at...Ch. 30 - (I) Plot a graph of the reactance of a 1.0-F...Ch. 30 - (I) Calculate the reactance of, and rms current...Ch. 30 - (II) A resistor R is in parallel with a capacitor...Ch. 30 - Prob. 44P Ch. 30 - (II) (a) What is the reactance of a 0.086-F...Ch. 30 - Prob. 46P Ch. 30 - (II) A current I = 1.80 cos 377t (I in amps, t in...Ch. 30 - (I) A 10.0-k resistor is in series with a 26.0-mH...Ch. 30 - (I) A 75- resistor and a 6.8-F capacitor are...Ch. 30 - (I) For a 120-V, 60-Hz voltage, a current of 70 mA...Ch. 30 - (II) A 2.5-k resistor in series with a 420-mH...Ch. 30 - (II) (a) What is the rms current in a series RC...Ch. 30 - (II) An ac voltage source is connected in series...Ch. 30 - (II) Determine the total impedance, phase angle,...Ch. 30 - (II) (a) What is the rms current in a series LR...Ch. 30 - (II) A 35-mH inductor with 2.0- resistance is...Ch. 30 - (II) A 25-mH coil whose resistance is 0.80 is...Ch. 30 - (II) A 75-W lightbulb is designed to operate with...Ch. 30 - (II) In the LRC circuit or Fig. 3019, suppose I =...Ch. 30 - (II) An LRC series circuit with R = 150 , L = 25...Ch. 30 - (II) An LR circuit can be used as a phase shifter....Ch. 30 - (I) A 3800-pF capacitor is connected in series to...Ch. 30 - (I) What is the resonant frequency of the LRC...Ch. 30 - (II) An LRC circuit has L = 4.15 mH and R = 3.80...Ch. 30 - (II) The frequency of the ac voltage source (peak...Ch. 30 - (II) Capacitors made from piezoelectric materials...Ch. 30 - (II) (a) Determine a formula for the average power...Ch. 30 - (II) (a) Show that oscillation of charge Q on the...Ch. 30 - (II) A resonant circuit using a 220-nF capacitor...Ch. 30 - Prob. 70P Ch. 30 - Prob. 71GP Ch. 30 - Prob. 72GP Ch. 30 - At time t = 0, the switch in the circuit shown in...Ch. 30 - Prob. 74GP Ch. 30 - Prob. 75GP Ch. 30 - Assuming the Earths magnetic field averages about...Ch. 30 - (a) For an underdamped LRC circuit, determine a...Ch. 30 - An electronic device needs to be protected against...Ch. 30 - Prob. 79GP Ch. 30 - Prob. 80GP Ch. 30 - An ac voltage source V=V0sin(t+90) is connected...Ch. 30 - A circuit contains two elements, but it is not...Ch. 30 - A 3.5-k resistor in series with a 440-mH inductor...Ch. 30 - (a) What is the rms current in on RC circuit if R...Ch. 30 - An inductance coil draws 2.5 A de when connected...Ch. 30 - The Q-value of a resonance circuit can be defined...Ch. 30 - Show that the fraction of electromagnetic energy...Ch. 30 - In a series LRC circuit, the inductance is 33mH,...Ch. 30 - Prob. 89GP Ch. 30 - A voltage V = 0.95 sin 754t is applied to an LRC...Ch. 30 - Filler circuit. Figure 3033 shows a simple filler...Ch. 30 - Show that if the inductor L in the filter circuit...Ch. 30 - A resistor R, capacitor C, and inductor L are...Ch. 30 - Suppose a series LRC circuit has two resisiors, R1...Ch. 30 - Prob. 95GP Ch. 30 - Prob. 96GP Ch. 30 - You have a small electromagnet that consumes 350 W...Ch. 30 - An inductor L in series with a resistor R, driven...Ch. 30 - In a certain LRC series circuit, when the ac...Ch. 30 - Prob. 100GP Ch. 30 - Prob. 101GP Ch. 30 - For the circuit shown in Fig. 3038, show that if...Ch. 30 - (II) The RC circuit shown in Fig. 3039 is called a...Ch. 30 - (II) The RC circuit shown in Fig. 3040 is called a...Ch. 30 - (III) Write a computer program or use a...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.

The current in the resistor, capacitor, inductor and the total impedance in the circuit.

Solution Summary: The author explains the current in the resistor, capacitor, inductor and the total impedance of the circuit.

Concept explainers

The current in the resistor, capacitor, inductor and the total impedance in the circuit.

Want to see the full answer?

Chapter 30 Solutions