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You have a 200Ω resistor, a 0.400-H inductor, and a 6.00µF capacitor. Suppose you take the resistor and inductor and make a series circuit with a voltage source that has voltage amplitude 30.0 V and an angular frequency of 250 rad/s. (a) What is the impedance of the circuit? (b) What is the current amplitude? (c) What are the voltage amplitudes across the resistor and across the inductor? (d) What is the phase angle ϕ of the source voltage with respect to the current? Does the source voltage lag or lead the current? (e) Construct the phasor diagram.

Short Answer

Expert verified

a) The impedance of the circuit is 224 Ω

b) The amplitude of the current in circuit is 0.134 A

c) The voltage drop across the resistor and inductor is 26.8 V and 13.4 V respectively

d) The phase angle of the voltage source with respect to current is +26.6°

e) The phasor diagram representing the magnitude and directional relationship between current and voltage in the circuit is given by

Step by step solution

01

Concept

An inductor is a passive two-terminal device that stores energy in a magnetic field when current passes through it. When an inductor is attached to an AC supply, the resistance produced by it is called inductive reactance (XL).

Resistance is measure of opposition to the flow of current in a closed electrical circuit. It is measured in Ohm (Ω).

Impedance is defined as the effective resistance of an electric circuit to the flow of current due to the combined effect of resistance (offered by resistor) and reactance (offered by capacitor and inductor).

A phasor diagram is graphical representation of magnitude and directional relationship between two or more alternating quantities.

02

Given information

Resistance of the resistor, R = 200 Ω

The inductance of the coil, L = 0.400 H

Capacitance of capacitor, C = 6.00 µF

The amplitude of voltage source, V = 30.0 V

Angular frequency of the source, ω = 250 rad/s

03

Determination of Circuit’s Impedance

Since the inductor and resistor are connected in series, Impedance of the circuit will be given by

Z=R2+XL2

Here, XL is the inductive reactance and its value is given by

XL=ωL=250rad/s0.4H=100Ω

Putting the values in Z

Z=R2+XL2=2002+1002=224Ω

Therefore, the impedance of the circuit is 224 Ω

04

Determination of Current Amplitude

By using Ohm’s Law

V=I.ZI=VZ

It is given that voltage amplitude of the source is 30.0 V, so

I=VZ=30.0V224Ω=0.134A

Therefore, the amplitude of the current in circuit is 0.134 A

05

Determination of Voltage across Resistor and Inductor

The voltage drop across the resistor is in phase with current, so by using ohm’s law

VR=IR=0.134A200Ω=26.8V

The voltage drop across the inductor is given by

VL=IXL=0.134A100Ω=13.4V

Therefore, the voltage drop across the resistor and inductor is 26.8 V and 13.4 V respectively.

06

Determination of Phase angle

The angle between voltage and current phasors in R-L circuit is given by

tanϕ=XLRtanϕ=100Ω200Ωtanϕ=0.5ϕ=+26.6°

Therefore, the phase angle of the voltage source with respect to current is +26.6°

07

Phasor Diagram

The phasor diagram representing the magnitude and directional relationship between current and voltage in the circuit is given by

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Most popular questions from this chapter

Question: A positive point charge is placed near a very large conducting plane. A professor of physics asserted that the field caused by this configuration is the same as would be obtained by removing the plane and placing a negative point charge of equal magnitude in the mirror image position behind the initial position of the plane. Is this correct? Why or why not?

In the circuit shown in Fig. E26.47 each capacitor initially has a charge of magnitude 3.50 nC on its plates. After the switch S is closed, what will be the current in the circuit at the instant that the capacitors have lost 80.0% of their initial stored energy?

Consider the circuit of Fig. E25.30. (a)What is the total rate at which electrical energy is dissipated in the 5.0-Ω and 9.0-Ω resistors? (b) What is the power output of the 16.0-V battery? (c) At what rate is electrical energy being converted to other forms in the 8.0-V battery? (d) Show that the power output of the 16.0-V battery equals the overall rate of consumption of electrical energy in the rest of the circuit.

Fig. E25.30.

You connect a battery, resistor, and capacitor as in Fig. 26.20a, where R = 12.0 Ω and C = 5.00 x 10-6 F. The switch S is closed at t = 0. When the current in the circuit has a magnitude of 3.00 A, the charge on the capacitor is 40.0 x 10-6 C. (a) What is the emf of the battery? (b) At what time t after the switch is closed is the charge on the capacitor equal to 40.0 x 10-6 C? (c) When the current has magnitude 3.00 A, at what rate is energy being (i) stored in the capacitor, (ii) supplied by the battery

Can potential difference between the terminals of a battery ever be opposite in direction to the emf? If it can, give an example. If it cannot, explain why not.

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