Skip to Content
OpenStax Logo
Buy book
  1. Preface
  2. Unit 1. Thermodynamics
    1. 1 Temperature and Heat
      1. Introduction
      2. 1.1 Temperature and Thermal Equilibrium
      3. 1.2 Thermometers and Temperature Scales
      4. 1.3 Thermal Expansion
      5. 1.4 Heat Transfer, Specific Heat, and Calorimetry
      6. 1.5 Phase Changes
      7. 1.6 Mechanisms of Heat Transfer
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 The Kinetic Theory of Gases
      1. Introduction
      2. 2.1 Molecular Model of an Ideal Gas
      3. 2.2 Pressure, Temperature, and RMS Speed
      4. 2.3 Heat Capacity and Equipartition of Energy
      5. 2.4 Distribution of Molecular Speeds
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 3 The First Law of Thermodynamics
      1. Introduction
      2. 3.1 Thermodynamic Systems
      3. 3.2 Work, Heat, and Internal Energy
      4. 3.3 First Law of Thermodynamics
      5. 3.4 Thermodynamic Processes
      6. 3.5 Heat Capacities of an Ideal Gas
      7. 3.6 Adiabatic Processes for an Ideal Gas
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 The Second Law of Thermodynamics
      1. Introduction
      2. 4.1 Reversible and Irreversible Processes
      3. 4.2 Heat Engines
      4. 4.3 Refrigerators and Heat Pumps
      5. 4.4 Statements of the Second Law of Thermodynamics
      6. 4.5 The Carnot Cycle
      7. 4.6 Entropy
      8. 4.7 Entropy on a Microscopic Scale
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Electricity and Magnetism
    1. 5 Electric Charges and Fields
      1. Introduction
      2. 5.1 Electric Charge
      3. 5.2 Conductors, Insulators, and Charging by Induction
      4. 5.3 Coulomb's Law
      5. 5.4 Electric Field
      6. 5.5 Calculating Electric Fields of Charge Distributions
      7. 5.6 Electric Field Lines
      8. 5.7 Electric Dipoles
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Gauss's Law
      1. Introduction
      2. 6.1 Electric Flux
      3. 6.2 Explaining Gauss’s Law
      4. 6.3 Applying Gauss’s Law
      5. 6.4 Conductors in Electrostatic Equilibrium
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 7 Electric Potential
      1. Introduction
      2. 7.1 Electric Potential Energy
      3. 7.2 Electric Potential and Potential Difference
      4. 7.3 Calculations of Electric Potential
      5. 7.4 Determining Field from Potential
      6. 7.5 Equipotential Surfaces and Conductors
      7. 7.6 Applications of Electrostatics
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Capacitance
      1. Introduction
      2. 8.1 Capacitors and Capacitance
      3. 8.2 Capacitors in Series and in Parallel
      4. 8.3 Energy Stored in a Capacitor
      5. 8.4 Capacitor with a Dielectric
      6. 8.5 Molecular Model of a Dielectric
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    5. 9 Current and Resistance
      1. Introduction
      2. 9.1 Electrical Current
      3. 9.2 Model of Conduction in Metals
      4. 9.3 Resistivity and Resistance
      5. 9.4 Ohm's Law
      6. 9.5 Electrical Energy and Power
      7. 9.6 Superconductors
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Direct-Current Circuits
      1. Introduction
      2. 10.1 Electromotive Force
      3. 10.2 Resistors in Series and Parallel
      4. 10.3 Kirchhoff's Rules
      5. 10.4 Electrical Measuring Instruments
      6. 10.5 RC Circuits
      7. 10.6 Household Wiring and Electrical Safety
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Magnetic Forces and Fields
      1. Introduction
      2. 11.1 Magnetism and Its Historical Discoveries
      3. 11.2 Magnetic Fields and Lines
      4. 11.3 Motion of a Charged Particle in a Magnetic Field
      5. 11.4 Magnetic Force on a Current-Carrying Conductor
      6. 11.5 Force and Torque on a Current Loop
      7. 11.6 The Hall Effect
      8. 11.7 Applications of Magnetic Forces and Fields
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    8. 12 Sources of Magnetic Fields
      1. Introduction
      2. 12.1 The Biot-Savart Law
      3. 12.2 Magnetic Field Due to a Thin Straight Wire
      4. 12.3 Magnetic Force between Two Parallel Currents
      5. 12.4 Magnetic Field of a Current Loop
      6. 12.5 Ampère’s Law
      7. 12.6 Solenoids and Toroids
      8. 12.7 Magnetism in Matter
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    9. 13 Electromagnetic Induction
      1. Introduction
      2. 13.1 Faraday’s Law
      3. 13.2 Lenz's Law
      4. 13.3 Motional Emf
      5. 13.4 Induced Electric Fields
      6. 13.5 Eddy Currents
      7. 13.6 Electric Generators and Back Emf
      8. 13.7 Applications of Electromagnetic Induction
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    10. 14 Inductance
      1. Introduction
      2. 14.1 Mutual Inductance
      3. 14.2 Self-Inductance and Inductors
      4. 14.3 Energy in a Magnetic Field
      5. 14.4 RL Circuits
      6. 14.5 Oscillations in an LC Circuit
      7. 14.6 RLC Series Circuits
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    11. 15 Alternating-Current Circuits
      1. Introduction
      2. 15.1 AC Sources
      3. 15.2 Simple AC Circuits
      4. 15.3 RLC Series Circuits with AC
      5. 15.4 Power in an AC Circuit
      6. 15.5 Resonance in an AC Circuit
      7. 15.6 Transformers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    12. 16 Electromagnetic Waves
      1. Introduction
      2. 16.1 Maxwell’s Equations and Electromagnetic Waves
      3. 16.2 Plane Electromagnetic Waves
      4. 16.3 Energy Carried by Electromagnetic Waves
      5. 16.4 Momentum and Radiation Pressure
      6. 16.5 The Electromagnetic Spectrum
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
  12. Index

Learning Objectives

By the end of this section, you will be able to:
  • Analyze circuits that have an inductor and resistor in series
  • Describe how current and voltage exponentially grow or decay based on the initial conditions

A circuit with resistance and self-inductance is known as an RL circuit. Figure 14.12(a) shows an RL circuit consisting of a resistor, an inductor, a constant source of emf, and switches S1S1 and S2.S2. When S1S1 is closed, the circuit is equivalent to a single-loop circuit consisting of a resistor and an inductor connected across a source of emf (Figure 14.12(b)). When S1S1 is opened and S2S2 is closed, the circuit becomes a single-loop circuit with only a resistor and an inductor (Figure 14.12(c)).

Figure a shows a resistor R and an inductor L connected in series with two switches which are parallel to each other. Both switches are currently open. Closing switch S1 would connect R and L in series with a battery, whose positive terminal is towards L. Closing switch S2 would form a closed loop of R and L, without the battery. Figure b shows a closed circuit with R, L and the battery in series. The side of L towards the battery, is at positive potential. Current flows from the positive end of L, through it, to the negative end. Figure c shows R and L connected in series. The potential across L is reversed, but the current flows in the same direction as in figure b.
Figure 14.12 (a) An RL circuit with switches S1S1 and S2.S2. (b) The equivalent circuit with S1S1 closed and S2S2 open. (c) The equivalent circuit after S1S1 is opened and S2S2 is closed.

We first consider the RL circuit of Figure 14.12(b). Once S1S1 is closed and S2S2 is open, the source of emf produces a current in the circuit. If there were no self-inductance in the circuit, the current would rise immediately to a steady value of ε/R.ε/R. However, from Faraday’s law, the increasing current produces an emf VL=L(dI/dt)VL=L(dI/dt) across the inductor. In accordance with Lenz’s law, the induced emf counteracts the increase in the current and is directed as shown in the figure. As a result, I(t) starts at zero and increases asymptotically to its final value.

Applying Kirchhoff’s loop rule to this circuit, we obtain

εLdIdtIR=0,εLdIdtIR=0,
(14.23)

which is a first-order differential equation for I(t). Notice its similarity to the equation for a capacitor and resistor in series (See RC Circuits). Similarly, the solution to Equation 14.23 can be found by making substitutions in the equations relating the capacitor to the inductor. This gives

I(t)=εR(1eRt/L)=εR(1et/τL),I(t)=εR(1eRt/L)=εR(1et/τL),
(14.24)

where

τL=L/RτL=L/R
(14.25)

is the inductive time constant of the circuit.

The current I(t) is plotted in Figure 14.13(a). It starts at zero, and as tt, I(t) approaches ε/Rε/R asymptotically. The induced emf VL(t)VL(t) is directly proportional to dI/dt, or the slope of the curve. Hence, while at its greatest immediately after the switches are thrown, the induced emf decreases to zero with time as the current approaches its final value of ε/R.ε/R. The circuit then becomes equivalent to a resistor connected across a source of emf.

Figure a shows the graph of electric current I versus time t. Current increases with time in a curve which flattens out at epsilon I R. At t equal to tau subscript L, the value of I is 0.63 epsilon I R. Figure b shows the graph of magnitude of induced voltage, mod V subscript L, versus time t. Mod V subscript L starts at value epsilon and decreases with time till the curve reaches zero. At t equal to tau subscript L, the value of I is 0.37 epsilon.
Figure 14.13 Time variation of (a) the electric current and (b) the magnitude of the induced voltage across the coil in the circuit of Figure 14.12(b).

The energy stored in the magnetic field of an inductor is

UL=12LI2.UL=12LI2.
(14.26)

Thus, as the current approaches the maximum current ε/Rε/R, the stored energy in the inductor increases from zero and asymptotically approaches a maximum of L(ε/R)2/2.L(ε/R)2/2.

The time constant τLτL tells us how rapidly the current increases to its final value. At t=τL,t=τL, the current in the circuit is, from Equation 14.24,

I(τL)=εR(1e−1)=0.63εR,I(τL)=εR(1e−1)=0.63εR,
(14.27)

which is 63%63% of the final value ε/Rε/R. The smaller the inductive time constant τL=L/R,τL=L/R, the more rapidly the current approaches ε/Rε/R.

We can find the time dependence of the induced voltage across the inductor in this circuit by using VL(t)=L(dI/dt)VL(t)=L(dI/dt) and Equation 14.24:

VL(t)=LdIdt=εet/τL.VL(t)=LdIdt=εet/τL.
(14.28)

The magnitude of this function is plotted in Figure 14.13(b). The greatest value of L(dI/dt)isε;L(dI/dt)isε; it occurs when dI/dt is greatest, which is immediately after S1S1 is closed and S2S2 is opened. In the approach to steady state, dI/dt decreases to zero. As a result, the voltage across the inductor also vanishes as t.t.

The time constant τLτL also tells us how quickly the induced voltage decays. At t=τL,t=τL, the magnitude of the induced voltage is

|VL(τL)|=εe−1=0.37ε=0.37V(0).|VL(τL)|=εe−1=0.37ε=0.37V(0).
(14.29)

The voltage across the inductor therefore drops to about 37%37% of its initial value after one time constant. The shorter the time constant τL,τL, the more rapidly the voltage decreases.

After enough time has elapsed so that the current has essentially reached its final value, the positions of the switches in Figure 14.12(a) are reversed, giving us the circuit in part (c). At t=0,t=0, the current in the circuit is I(0)=ε/R.I(0)=ε/R. With Kirchhoff’s loop rule, we obtain

IR+LdIdt=0.IR+LdIdt=0.
(14.30)

The solution to this equation is similar to the solution of the equation for a discharging capacitor, with similar substitutions. The current at time t is then

I(t)=εRet/τL.I(t)=εRet/τL.
(14.31)

The current starts at I(0)=ε/RI(0)=ε/R and decreases with time as the energy stored in the inductor is depleted (Figure 14.14).

The time dependence of the voltage across the inductor can be determined from VL=L(dI/dt):VL=L(dI/dt):

VL(t)=εet/τL.VL(t)=εet/τL.
(14.32)

This voltage is initially VL(0)=εVL(0)=ε, and it decays to zero like the current. The energy stored in the magnetic field of the inductor, LI2/2,LI2/2, also decreases exponentially with time, as it is dissipated by Joule heating in the resistance of the circuit.

The graph of I versus t. The value of I at t equal to 0 is epsilon I R. I decreases with time till the curve reaches 0. At t equal to tau subscript L, the value of I is 0.37 epsilon I R.
Figure 14.14 Time variation of electric current in the RL circuit of Figure 14.12(c). The induced voltage across the coil also decays exponentially.

Example 14.4

An RL Circuit with a Source of emf In the circuit of Figure 14.12(a), let ε=2.0V,R=4.0Ω,andL=4.0H.ε=2.0V,R=4.0Ω,andL=4.0H. With S1S1 closed and S2S2 open (Figure 14.12(b)), (a) what is the time constant of the circuit? (b) What are the current in the circuit and the magnitude of the induced emf across the inductor at t=0,att=2.0τLt=0,att=2.0τL, and as tt?

Strategy The time constant for an inductor and resistor in a series circuit is calculated using Equation 14.25. The current through and voltage across the inductor are calculated by the scenarios detailed from Equation 14.24 and Equation 14.32.

Solution

  1. The inductive time constant is
    τL=LR=4.0H4.0Ω=1.0s.τL=LR=4.0H4.0Ω=1.0s.
  2. The current in the circuit of Figure 14.12(b) increases according to Equation 14.24:
    I(t)=εR(1et/τL).I(t)=εR(1et/τL).

    At t=0,t=0,
    (1et/τL)=(11)=0;soI(0)=0.(1et/τL)=(11)=0;soI(0)=0.

    At t=2.0τLt=2.0τL and t,t, we have, respectively,
    I(2.0τL)=εR(1e−2.0)=(0.50A)(0.86)=0.43A,I(2.0τL)=εR(1e−2.0)=(0.50A)(0.86)=0.43A,

    and
    I()=εR=0.50A.I()=εR=0.50A.

    From Equation 14.32, the magnitude of the induced emf decays as
    |VL(t)|=εet/τL.|VL(t)|=εet/τL.

    Att=0,t=2.0τL,and ast,Att=0,t=2.0τL,and ast, we obtain
    |VL(0)|=ε=2.0V,|VL(2.0τL)|=(2.0V)e−2.0=0.27Vand|VL()|=0.|VL(0)|=ε=2.0V,|VL(2.0τL)|=(2.0V)e−2.0=0.27Vand|VL()|=0.

Significance If the time of the measurement were much larger than the time constant, we would not see the decay or growth of the voltage across the inductor or resistor. The circuit would quickly reach the asymptotic values for both of these. See Figure 14.15.

Figures a, b and c show the oscilloscope traces of voltage versus time of the voltage across source, the voltage across the inductor and the voltage across the resistor respectively. Figure a is a square wave varying from minus 12 volts to plus 12 volts, with a period from minus 10 ms to minus 0.001 ms. Figure b shows a square wave varying from minus 6 volts to plus 6 volts with a spike of 16 volts at the beginning of every crest and a spike of minus 16 volts at the beginning of every trough. The period is the same as that in figure a. Figure c shows a square wave varying from minus 0.3 to plus 0.3 volts, with spikes going out of the trace area in the positive direction at the beginnings of every crest and trough. The period of the wave is from minus 9.985 to plus 0.015 ms.
Figure 14.15 A generator in an RL circuit produces a square-pulse output in which the voltage oscillates between zero and some set value. These oscilloscope traces show (a) the voltage across the source; (b) the voltage across the inductor; (c) the voltage across the resistor.

Example 14.5

An RL Circuit without a Source of emf After the current in the RL circuit of Example 14.4 has reached its final value, the positions of the switches are reversed so that the circuit becomes the one shown in Figure 14.12(c). (a) How long does it take the current to drop to half its initial value? (b) How long does it take before the energy stored in the inductor is reduced to 1.0%1.0% of its maximum value?

Strategy The current in the inductor will now decrease as the resistor dissipates this energy. Therefore, the current falls as an exponential decay. We can also use that same relationship as a substitution for the energy in an inductor formula to find how the energy decreases at different time intervals.

Solution

  1. With the switches reversed, the current decreases according to
    I(t)=εRet/τL=I(0)et/τL.I(t)=εRet/τL=I(0)et/τL.

    At a time t when the current is one-half its initial value, we have
    I(t)=0.50I(0)soet/τL=0.50,I(t)=0.50I(0)soet/τL=0.50,

    and
    t=[ln(0.50)]τL=0.69(1.0s)=0.69s,t=[ln(0.50)]τL=0.69(1.0s)=0.69s,

    where we have used the inductive time constant found in Example 14.4.
  2. The energy stored in the inductor is given by
    UL(t)=12L[I(t)]2=12L(εRet/τL)2=Lε22R2e−2t/τL.UL(t)=12L[I(t)]2=12L(εRet/τL)2=Lε22R2e−2t/τL.

    If the energy drops to 1.0%1.0% of its initial value at a time t, we have
    UL(t)=(0.010)UL(0)orLε22R2e−2t/τL=(0.010)Lε22R2.UL(t)=(0.010)UL(0)orLε22R2e−2t/τL=(0.010)Lε22R2.

    Upon canceling terms and taking the natural logarithm of both sides, we obtain
    2tτL=ln(0.010),2tτL=ln(0.010),

    so
    t=12τLln(0.010).t=12τLln(0.010).

    Since τL=1.0sτL=1.0s, the time it takes for the energy stored in the inductor to decrease to 1.0%1.0% of its initial value is
    t=12(1.0s)ln(0.010)=2.3s.t=12(1.0s)ln(0.010)=2.3s.

Significance This calculation only works if the circuit is at maximum current in situation (b) prior to this new situation. Otherwise, we start with a lower initial current, which will decay by the same relationship.

Check Your Understanding 14.7

Verify that RC and L/R have the dimensions of time.

Check Your Understanding 14.8

(a) If the current in the circuit of in Figure 14.12(b) increases to 90%90% of its final value after 5.0 s, what is the inductive time constant? (b) If R=20ΩR=20Ω, what is the value of the self-inductance? (c) If the 20-Ω20-Ω resistor is replaced with a 100-Ω100-Ω resister, what is the time taken for the current to reach 90%90% of its final value?

Check Your Understanding 14.9

For the circuit of in Figure 14.12(b), show that when steady state is reached, the difference in the total energies produced by the battery and dissipated in the resistor is equal to the energy stored in the magnetic field of the coil.

Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/university-physics-volume-2/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/university-physics-volume-2/pages/1-introduction
Citation information

© Oct 6, 2016 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.