We know that the current through an inductor $L$ cannot be turned on or off instantaneously. The change in current changes flux, inducing an emf opposing the change (Lenz’s law). How long does the opposition last? Current * will* flow and

*be turned off, but how long does it take? [link] shows a switching circuit that can be used to examine current through an inductor as a function of time.*

*can*When the switch is first moved to position 1 (at *$t=0$*), the current is zero and it eventually rises to ${I}_{0}=\text{V/R}$, where $R$ is the total resistance of the circuit. The opposition of the inductor $L$ is greatest at the beginning, because the amount of change is greatest. The opposition it poses is in the form of an induced emf, which decreases to zero as the current approaches its final value. The opposing emf is proportional to the amount of change left. This is the hallmark of an exponential behavior, and it can be shown with calculus that

is the current in an *RL* circuit when switched on (Note the similarity to the exponential behavior of the voltage on a charging capacitor). The initial current is zero and approaches ${I}_{0}=\text{V/R}$ with a characteristic time constant
$\tau $
for an *RL* circuit, given by

where $\tau $ has units of seconds, since $\text{1}\phantom{\rule{0.25em}{0ex}}\text{H}\text{=}\text{1}\phantom{\rule{0.25em}{0ex}}\text{\Omega}\text{\xb7}\text{s}$. In the first period of time $\tau $, the current rises from zero to $0\text{.}\text{632}{I}_{0}$, since $I={I}_{0}(1-{e}^{-1})={I}_{0}(1-0\text{.}\text{368})=0\text{.}\text{632}{I}_{0}$. The current will go 0.632 of the remainder in the next time $\tau $. A well-known property of the exponential is that the final value is never exactly reached, but 0.632 of the remainder to that value is achieved in every characteristic time $\tau $. In just a few multiples of the time $\tau $, the final value is very nearly achieved, as the graph in [link](b) illustrates.

The characteristic time $\tau $ depends on only two factors, the inductance $L$ and the resistance $R$. The greater the inductance $L$, the greater $\tau $ is, which makes sense since a large inductance is very effective in opposing change. The smaller the resistance $R$, the greater $\tau $ is. Again this makes sense, since a small resistance means a large final current and a greater change to get there. In both cases—large $L$ and small $R$ —more energy is stored in the inductor and more time is required to get it in and out.

When the switch in [link](a) is moved to position 2 and cuts the battery out of the circuit, the current drops because of energy dissipation by the resistor. But this is also not instantaneous, since the inductor opposes the decrease in current by inducing an emf in the same direction as the battery that drove the current. Furthermore, there is a certain amount of energy, $(\text{1/2}){\text{LI}}_{0}^{2}$, stored in the inductor, and it is dissipated at a finite rate. As the current approaches zero, the rate of decrease slows, since the energy dissipation rate is ${I}^{2}R$. Once again the behavior is exponential, and $I$ is found to be

(See [link](c).) In the first period of time $\tau =L/R$ after the switch is closed, the current falls to 0.368 of its initial value, since $I={I}_{0}{e}^{-1}=0\text{.}\text{368}{I}_{0}$. In each successive time $\tau $, the current falls to 0.368 of the preceding value, and in a few multiples of $\tau $, the current becomes very close to zero, as seen in the graph in [link](c).

*RL*Circuit

(a) What is the characteristic time constant for a 7.50 mH inductor in series with a $\text{3.00 \Omega}$ resistor? (b) Find the current 5.00 ms after the switch is moved to position 2 to disconnect the battery, if it is initially 10.0 A.

**Strategy for (a)**

The time constant for an *RL* circuit is defined by $\tau =L/R$.

**Solution for (a)**

Entering known values into the expression for $\tau $ given in $\tau =L/R$ yields

**Discussion for (a)**

This is a small but definitely finite time. The coil will be very close to its full current in about ten time constants, or about 25 ms.

**Strategy for (b)**

We can find the current by using $I={I}_{0}{e}^{-t/\tau}$, or by considering the decline in steps. Since the time is twice the characteristic time, we consider the process in steps.

**Solution for (b)**

In the first 2.50 ms, the current declines to 0.368 of its initial value, which is

After another 2.50 ms, or a total of 5.00 ms, the current declines to 0.368 of the value just found. That is,

**Discussion for (b)**

After another 5.00 ms has passed, the current will be 0.183 A (see [link]); so, although it does die out, the current certainly does not go to zero instantaneously.

In summary, when the voltage applied to an inductor is changed, the current also changes, * but the change in current lags the change in voltage in an RL circuit*. In Reactance, Inductive and Capacitive, we explore how an

*RL*circuit behaves when a sinusoidal AC voltage is applied.

# Section Summary

- When a series connection of a resistor and an inductor—an
*RL*circuit—is connected to a voltage source, the time variation of the current is$I={I}_{0}(1-{e}^{-t/\tau})\text{(turning on).}$where ${I}_{0}=V/R$ is the final current. - The characteristic time constant $\tau $ is $\tau =\frac{L}{R}$ , where $L$ is the inductance and $R$ is the resistance.
- In the first time constant $\tau $, the current rises from zero to $0\text{.}\text{632}{I}_{0}$, and 0.632 of the remainder in every subsequent time interval $\tau $.
- When the inductor is shorted through a resistor, current decreases as
$I={I}_{0}{e}^{-t/\tau}\text{(turning off).}$Here ${I}_{0}$ is the initial current.
- Current falls to $0\text{.}\text{368}{I}_{0}$ in the first time interval $\tau $, and 0.368 of the remainder toward zero in each subsequent time $\tau $.

# Problem Exercises

If you want a characteristic *RL* time constant of 1.00 s, and you have a $\text{500 \Omega}$ resistor, what value of self-inductance is needed?

500 H

Your *RL* circuit has a characteristic time constant of 20.0 ns, and a resistance of $\text{5.00 M\Omega}$. (a) What is the inductance of the circuit? (b) What resistance would give you a 1.00 ns time constant, perhaps needed for quick response in an oscilloscope?

A large superconducting magnet, used for magnetic resonance imaging, has a 50.0 H inductance. If you want current through it to be adjustable with a 1.00 s characteristic time constant, what is the minimum resistance of system?

$\text{50.0 \Omega}$

Verify that after a time of 10.0 ms, the current for the situation considered in [link] will be 0.183 A as stated.

Suppose you have a supply of inductors ranging from 1.00 nH to 10.0 H, and resistors ranging from
$\text{0.100 \Omega}$ to
$\text{1.00 M\Omega}$. What is the range of characteristic *RL* time constants you can produce by connecting a single resistor to a single inductor?

$1\text{.}\text{00}\times {\text{10}}^{\text{\u201318}}\phantom{\rule{0.25em}{0ex}}\text{s}$ to 0.100 s

(a) What is the characteristic time constant of a 25.0 mH inductor that has a resistance of $\text{4.00 \Omega}$? (b) If it is connected to a 12.0 V battery, what is the current after 12.5 ms?

What percentage of the final current ${I}_{\text{0}}$ flows through an inductor $L$ in series with a resistor $R$, three time constants after the circuit is completed?

95.0%

The 5.00 A current through a 1.50 H inductor is dissipated by a $\text{2.00 \Omega}$ resistor in a circuit like that in [link] with the switch in position 2. (a) What is the initial energy in the inductor? (b) How long will it take the current to decline to 5.00% of its initial value? (c) Calculate the average power dissipated, and compare it with the initial power dissipated by the resistor.

(a) Use the exact exponential treatment to find how much time is required to bring the current through an 80.0 mH inductor in series with a $\text{15.0 \Omega}$ resistor to 99.0% of its final value, starting from zero. (b) Compare your answer to the approximate treatment using integral numbers of $\tau $. (c) Discuss how significant the difference is.

(a) 24.6 ms

(b) 26.7 ms

(c) 9% difference, which is greater than the inherent uncertainty in the given parameters.

(a) Using the exact exponential treatment, find the time required for the current through a 2.00 H inductor in series with a $\text{0.500 \Omega}$ resistor to be reduced to 0.100% of its original value. (b) Compare your answer to the approximate treatment using integral numbers of $\tau $. (c) Discuss how significant the difference is.

- College Physics
- Preface
- Introduction: The Nature of Science and Physics
- Kinematics
- Introduction to One-Dimensional Kinematics
- Displacement
- Vectors, Scalars, and Coordinate Systems
- Time, Velocity, and Speed
- Acceleration
- Motion Equations for Constant Acceleration in One Dimension
- Problem-Solving Basics for One-Dimensional Kinematics
- Falling Objects
- Graphical Analysis of One-Dimensional Motion

- Two-Dimensional Kinematics
- Dynamics: Force and Newton's Laws of Motion
- Introduction to Dynamics: Newton’s Laws of Motion
- Development of Force Concept
- Newton’s First Law of Motion: Inertia
- Newton’s Second Law of Motion: Concept of a System
- Newton’s Third Law of Motion: Symmetry in Forces
- Normal, Tension, and Other Examples of Forces
- Problem-Solving Strategies
- Further Applications of Newton’s Laws of Motion
- Extended Topic: The Four Basic Forces—An Introduction

- Further Applications of Newton's Laws: Friction, Drag, and Elasticity
- Uniform Circular Motion and Gravitation
- Work, Energy, and Energy Resources
- Introduction to Work, Energy, and Energy Resources
- Work: The Scientific Definition
- Kinetic Energy and the Work-Energy Theorem
- Gravitational Potential Energy
- Conservative Forces and Potential Energy
- Nonconservative Forces
- Conservation of Energy
- Power
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- World Energy Use

- Linear Momentum and Collisions
- Statics and Torque
- Rotational Motion and Angular Momentum
- Introduction to Rotational Motion and Angular Momentum
- Angular Acceleration
- Kinematics of Rotational Motion
- Dynamics of Rotational Motion: Rotational Inertia
- Rotational Kinetic Energy: Work and Energy Revisited
- Angular Momentum and Its Conservation
- Collisions of Extended Bodies in Two Dimensions
- Gyroscopic Effects: Vector Aspects of Angular Momentum

- Fluid Statics
- Introduction to Fluid Statics
- What Is a Fluid?
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- Pressure
- Variation of Pressure with Depth in a Fluid
- Pascal’s Principle
- Gauge Pressure, Absolute Pressure, and Pressure Measurement
- Archimedes’ Principle
- Cohesion and Adhesion in Liquids: Surface Tension and Capillary Action
- Pressures in the Body

- Fluid Dynamics and Its Biological and Medical Applications
- Introduction to Fluid Dynamics and Its Biological and Medical Applications
- Flow Rate and Its Relation to Velocity
- Bernoulli’s Equation
- The Most General Applications of Bernoulli’s Equation
- Viscosity and Laminar Flow; Poiseuille’s Law
- The Onset of Turbulence
- Motion of an Object in a Viscous Fluid
- Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

- Temperature, Kinetic Theory, and the Gas Laws
- Heat and Heat Transfer Methods
- Thermodynamics
- Introduction to Thermodynamics
- The First Law of Thermodynamics
- The First Law of Thermodynamics and Some Simple Processes
- Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency
- Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated
- Applications of Thermodynamics: Heat Pumps and Refrigerators
- Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy
- Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

- Oscillatory Motion and Waves
- Introduction to Oscillatory Motion and Waves
- Hooke’s Law: Stress and Strain Revisited
- Period and Frequency in Oscillations
- Simple Harmonic Motion: A Special Periodic Motion
- The Simple Pendulum
- Energy and the Simple Harmonic Oscillator
- Uniform Circular Motion and Simple Harmonic Motion
- Damped Harmonic Motion
- Forced Oscillations and Resonance
- Waves
- Superposition and Interference
- Energy in Waves: Intensity

- Physics of Hearing
- Electric Charge and Electric Field
- Introduction to Electric Charge and Electric Field
- Static Electricity and Charge: Conservation of Charge
- Conductors and Insulators
- Coulomb’s Law
- Electric Field: Concept of a Field Revisited
- Electric Field Lines: Multiple Charges
- Electric Forces in Biology
- Conductors and Electric Fields in Static Equilibrium
- Applications of Electrostatics

- Electric Potential and Electric Field
- Introduction to Electric Potential and Electric Energy
- Electric Potential Energy: Potential Difference
- Electric Potential in a Uniform Electric Field
- Electrical Potential Due to a Point Charge
- Equipotential Lines
- Capacitors and Dielectrics
- Capacitors in Series and Parallel
- Energy Stored in Capacitors

- Electric Current, Resistance, and Ohm's Law
- Circuits, Bioelectricity, and DC Instruments
- Magnetism
- Introduction to Magnetism
- Magnets
- Ferromagnets and Electromagnets
- Magnetic Fields and Magnetic Field Lines
- Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field
- Force on a Moving Charge in a Magnetic Field: Examples and Applications
- The Hall Effect
- Magnetic Force on a Current-Carrying Conductor
- Torque on a Current Loop: Motors and Meters
- Magnetic Fields Produced by Currents: Ampere’s Law
- Magnetic Force between Two Parallel Conductors
- More Applications of Magnetism

- Electromagnetic Induction, AC Circuits, and Electrical Technologies
- Introduction to Electromagnetic Induction, AC Circuits and Electrical Technologies
- Induced Emf and Magnetic Flux
- Faraday’s Law of Induction: Lenz’s Law
- Motional Emf
- Eddy Currents and Magnetic Damping
- Electric Generators
- Back Emf
- Transformers
- Electrical Safety: Systems and Devices
- Inductance
- RL Circuits
- Reactance, Inductive and Capacitive
- RLC Series AC Circuits

- Electromagnetic Waves
- Geometric Optics
- Vision and Optical Instruments
- Wave Optics
- Introduction to Wave Optics
- The Wave Aspect of Light: Interference
- Huygens's Principle: Diffraction
- Young’s Double Slit Experiment
- Multiple Slit Diffraction
- Single Slit Diffraction
- Limits of Resolution: The Rayleigh Criterion
- Thin Film Interference
- Polarization
- *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light

- Special Relativity
- Introduction to Quantum Physics
- Atomic Physics
- Introduction to Atomic Physics
- Discovery of the Atom
- Discovery of the Parts of the Atom: Electrons and Nuclei
- Bohr’s Theory of the Hydrogen Atom
- X Rays: Atomic Origins and Applications
- Applications of Atomic Excitations and De-Excitations
- The Wave Nature of Matter Causes Quantization
- Patterns in Spectra Reveal More Quantization
- Quantum Numbers and Rules
- The Pauli Exclusion Principle

- Radioactivity and Nuclear Physics
- Medical Applications of Nuclear Physics
- Particle Physics
- Frontiers of Physics
- Atomic Masses
- Selected Radioactive Isotopes
- Useful Information
- Glossary of Key Symbols and Notation