General Physics (calculus based) Class Notes

Dr. Rakesh Kapoor, M.Sc., Ph.D.

Former Faculty-University of Alabama at Birmingham, Birmingham, AL 35294

Induction and Inductance


Faraday’s law of induction

Lenz’s rule

Induction and Energy Transfer

Electric field induced by a changing magnetic field

Inductance and mutual inductance

RL circuits

Energy stored in a magnetic field

Faraday's law of Induction

Experiment - 1

Let us do a simulation experiment based on the experiments done by Michael Faraday and Joseph Henry.

Consider a conducting wire loop connected to a sensitive Ammeter (or Galvanometer). There is no battery in the circuit, current in the loop should be zero.

Observe the ammeter and see what happens when we move a magnet towards the loop?

Now move the magnet back and again observe the ammeter.

Increase the speed of magnet movement from 0.5 to 1 and repeat the experiment.

Reverse the polarity of the magnet and repeat the experiment.

What were our observations?

Movement of a magnet produces current in the loop.

Faster motion produces a greater current.

If moving the magnet's north pole toward the loop causes, say, clockwise current, then moving the north pole away causes counterclockwise current.

Moving the south pole toward or away from the loop also causes currents, but in the reversed directions.

What is the conclusion?

Current appears in the loop, only if there is relative motion between the loop and the magnet (one must move relative to the other); the current disappears when the relative motion between them ceases.

Such a current produced in the loop is called an "induced current".

The emf producing this induced current is called an induced emf ( Work done per unit charge to produce that current).

The process of producing the current and emf is called induction.

Experiment - 2

Let us conduct an other experiment. Consider two conducting loops as shown in the simulation figure.(for proper working first select 1 for Ammeter testing)

An ammeter is connected in one of the loop but no battery is there to produce current in that loop.

An emf source (battery) is connected in the second loop with an open switch.

Let us close the switch of second loop and observe the ammeter in first loop.

Now again open the switch of second loop and observe the ammeter in first loop.

What were our observations?

If we close switch S, to turn on a current in the right-hand loop, the meter suddenly and briefly registers a current—an induced current—in the left-hand loop.

When we open the switch, another sudden and brief induced current appears in the left-hand loop, but in the opposite direction.

What is the conclusion?

We get an induced current (and thus an induced emf) only when the current in the right-hand loop is changing (either turning on or turning off) and not when it is constant (even if current is large).

Summary of these Experiments :

Faraday summarized the results of these experiments in what is known as Faraday's law of Induction.

An emf is induced in a loop when the magnitude or strength of magnetic field that pass through the loop is changing.

Faraday equated the “amount of magnetic field” in terms of the magnetic field lines passing through the loop. Now the Faraday' s law of induction, can also be stated as:

An emf is induced in a loop when the number of magnetic field lines that pass through the loop is changing.

Faraday' s law is not an explanation of Induction but merely a description of what induction is.

A Quantitative Treatment

Magnetic field Flux Induction Inductance_3.gif.

First we need to define a quantity called Magnetic field flux Induction Inductance_4.gif.

We are well aware of the electric field flux Induction Inductance_5.gif of an electric field Induction Inductance_6.gif passing through a loop of surface are Induction Inductance_7.gif. It is defined as

Induction Inductance_8.gif

In a similar fashion the magnetic field flux Induction Inductance_9.gif through a loop of area Induction Inductance_10.gif, placed in a magnetic field Induction Inductance_11.gif is defined as

Induction Inductance_12.gif


Induction Inductance_13.gif

Where φ is the angle between magnetic field Induction Inductance_14.gif and area vector Induction Inductance_15.gif.

Induction Inductance_16.gif

When Induction Inductance_17.gif is uniform and area vector Induction Inductance_18.gif of the loop is parallel to the magnetic field Induction Inductance_19.gif, the magnetic flux Induction Inductance_20.gif through the loop is simply given as

Induction Inductance_21.gif

SI unit for magnetic flux is the Tesla - square meter, which is called the Weber (abbreviated Wb) :

Induction Inductance_22.gif

Faraday's law can be restated in quantitative form as

The magnitude of the emf E induced in a conducting loop is equal to the rate at which the magnetic flux Induction Inductance_23.gif changes through that loop with time.

Induction Inductance_24.gif

Will will learn about the negative sign in front of Induction Inductance_25.gif  in the next section.

Induced emf in a coil :

If there are N turns in a coil, the change in magnetic flux through the coil will induce emf in each turns.

The total emf induced in the coil will be the sum of these individual induced emfs.

Induction Inductance_26.gif

Here we are assuming that the coil is tightly wound (closely packed), so that the same magnetic flux  passes through all the turns.

Checkpoint 1

The graph gives the magnitude B(t) of a uniform magnetic field that exists throughout a conducting loop, with the direction of the field perpendicular to the plane of the loop. Rank the five regions of the graph according to the magnitude of the emf induced in the loop, greatest first.

Induction Inductance_27.gif

Hint : Induction Inductance_28.gif is equal to the slope of the line in above graph.

Method's of Changing Magnetic flux through a loop:

Change the magnitude B of the magnetic field within the coil.

Change the angle between the direction of the magnetic field and the plane of the coil (for example, by rotating the coil so that field is first perpendicular to the plane of the coil and then is along that plane).

Change the total area of the coil or the portion of that area that lies within the magnetic field (for example, by expanding the coil or sliding it into or out of the field).

Problem/Demonstration :

A rectangular coil of N turns and of length a and width b is rotated at frequency f in a uniform magnetic field. The coil is connected to co - rotating cylinders, against which metal brushes slide to make contact.
(a) Show that the emf induced in the coil is given as a function of time t
This is the principle of the commercial alternating - current generator.

Induction Inductance_32.gif

(b) What value of N ab gives an emf with Induction Inductance_33.gif, when the loop is rotated at 60.0 rev/s in a uniform magnetic field of 0.500 T?

Solution :

(a) The magnetic flux Induction Inductance_34.gif through  N loops of identical area A placed in a uniform magnetic field B, is given as

Induction Inductance_35.gif

Where φ is the angle between area vector Induction Inductance_36.gif of the loop and magnetic field Induction Inductance_37.gif. Since the loop is rotating say with frequency f, the angle φ is changing with time as

Induction Inductance_38.gif

Now the magnetic flux Induction Inductance_39.gif as a function of time is given as

Induction Inductance_40.gif

Rate of change of the magnetic flux Induction Inductance_41.gif is computed as

Induction Inductance_42.gif

According to Faraday's law induced emf E is given as

Induction Inductance_43.gif


Induction Inductance_44.gif


Induction Inductance_45.gif

(b) Now once the value of Induction Inductance_46.gif , B and f is known we can find the value of N ab.

Induction Inductance_47.gif

Induction Inductance_48.gif

Induction Inductance_49.gif

Lenz's Law

With Lenz's rule or law we can find out the direction of induced current or the reason for the negative sign in front of rate of change of magnetic flux Induction Inductance_50.gif in the expression for induced emf E.

An induced current has a direction such that the magnetic field due to the current opposes the change in the magnetic flux that induces the current.

To understand it further let us look at the following simulation.

We know that any current in a loop produces a dipole magnet.

The induced current in the loop will also produce an induced magnet (magnetic dipole). In the simulation it appears inside the loop as induced current flows through it.

According to Lenz' s the direction of this induced magnet will always try to oppose the motion of the magnet responsible for the induction.

If the magnet is coming closed to the loop, the induced magnet will try to repel it.

If the magnet is moving away from the loop, the induced magnet will try to attract it.

It will always resist the change in magnetic flux.

If flux is increasing, the induced magnet will try to decrease it.

If flux is decreasing, the induced magnet will try to increase it.

Once we know the direction of induced magnetic field, we can figure out the direction of induced current by using the right hand rule.

Checkpoint 2

The figure shows three situations in which identical circular conducting loops are in uniform magnetic fields that are either increasing (Inc) or decreasing (Dec) in magnitude at identical rates. In each, the dashed line coincides with a diameter. Rank the situations according to the magnitude of the current induced in the loops, greatest first.

Induction Inductance_52.gif

Hint: Induced current is proportional to the net change in magnetic field flux through the loop.

How Electric guitar works?

Look at the following figure

Induction Inductance_53.gif

Presence of a permanent magnet makes the guitar wire a magnet.

When the magnetic guitar wire vibrates, it changes the magnetic flux in the coil.

This flux induces a current in the coil which exactly follow the vibrations of the guitar wire.

This current is fed to the amplifier and we can hear the sound of vibration of the guitar wire.

Quiz- 1
Consider the drawing in which someone is pulling a rectangular conducting loop out of a region containing a magnetic field at a constant velocity.  Which one of the following is an expression of  the rate of work (the power) applied by the person?
Induction Inductance_54.gif

P = i2 R

P = i v

P = Fapp v

P = i B L v

P = Fapp i B/L

Induction and Energy Transfer.

Consider a rectangular conducting wire loop placed in a uniform magnetic field (into the page).

Induction Inductance_59.gif

Since magnetic field is normal to the plane of the loop, the flux through the loop will be given as

Induction Inductance_60.gif

Let us pull the loop with a constant velocity Induction Inductance_61.gif in positive x-direction.

The change in magnetic flux due to this motion will induced emf E in the loop.

Induction Inductance_62.gif

E  is proportional to the length of the edge perpendicular to the direction of motion.

Relation between the power dissipated in loop and power delivered by applied force.

Let us draw an equivalent circuit of the loop to compute dissipated power Induction Inductance_63.gif in the loop.

Induction Inductance_64.gif

The current i through the loop can be computed from the induced emf E and the resistance R (any conductor always has some resistance) an.

Induction Inductance_65.gif

Therefore power dissipated Induction Inductance_66.gif in the resistance R is given as

Induction Inductance_67.gif

Power delivered by applied force

Magnetic force Induction Inductance_68.gif on any wire carrying current i is given as

Induction Inductance_69.gif

We can see that only three segments of the loop are in the magnetic field.

Induction Inductance_70.gif

By right hand rule the direction of Induction Inductance_71.gif and Induction Inductance_72.gif are opposite to each other but the magnitude are same. These forces cancel each other.

Net force acting on loop will be sum of applied force Induction Inductance_73.gif and magnetic force Induction Inductance_74.gif.

If the loop is moving with constant velocity, net force on it should be zero.

Induction Inductance_75.gif

Therefore the applied force by us Induction Inductance_76.gif must be equal in magnitude to the magnetic force (which resists the motion of loop) acting on the loop.

Induction Inductance_77.gif

By substituting the value of induced current i in above equation we get

Induction Inductance_78.gif

Power delivered by the applied force Induction Inductance_79.gif in moving the loop is given as

Induction Inductance_80.gif

We can see that the power delivered Induction Inductance_81.gif in moving the loop is equal to power dissipated Induction Inductance_82.gif.

The work that you do in pulling the loop through the magnetic field appears as thermal energy in the loop.

Eddy Currents

Above statement is not only true for the loop but for any conductor.

When ever a conductor is experiencing changing magnetic flux, the energy causing the change in flux appears as thermal energy in the conductor.

The induced currents generated in the conductor are called Eddy Currents.

Induction Inductance_83.gif Induction Inductance_84.gif
Quiz- 2
When a metal sheet is pulled from a region containing a magnetic field, currents are induced in the metal sheet.  What is the name given to these currents?

andy currents

betty currents

curie currents

drew currents

eddy currents

Checkpoint 3

The figure shows four wire loops, with edge lengths of either L or 2L. All four loops will move through a region of uniform magnetic field Induction Inductance_85.gif (directed out of the page) at the same constant velocity. Rank the four loops according to the maximum magnitude of the emf induced as they move through the field, greatest first.

Induction Inductance_86.gif

Hint: Induced current is proportional to the length of the edge perpendicular to the direction of motion.

Electric field Induced by a Changing Magnetic Field

Presence of induced current in a conducting ring due to changing flux of magnetic field implies that an induced electric field Induction Inductance_87.gif is present.

Using above argument we can reformulate Faraday's las as follow:  

A changing magnetic field produces an electric field.

This induced field is present even in the absence of the conducting ring.

Consider an imaginary circular path in a changing magnetic field.

Induction Inductance_88.gif

Let us consider that the charge completes one revolution around this path due to induced emf E.

Emf E in a circuit is defined as work done per unit charge in moving the charge from its initial to final position and in this case is given as

Induction Inductance_89.gif

As per Faraday's law emf E in this situation is given as

Induction Inductance_90.gif

By comparing these two equations we can find the relation between the induced electric field Induction Inductance_91.gif and change in magnetic flux Induction Inductance_92.gif.

Induction Inductance_93.gif

In a uniform changing magnetic field, the line integral around a circular path of radius r is given as

Induction Inductance_94.gif

Similarly change in magnetic flux Induction Inductance_95.gif is given as

Induction Inductance_96.gif

Comparing these two equations give us the magnitude of induced electric field around the circular path of radius r as

Induction Inductance_97.gif

Induction Inductance_98.gif

Quiz- 3
Complete the following statement: Faraday’s law indicates that a changing magnetic field produces

an electric field.

an induced magnetic field.

a force field.


global warming.



Any conductor capable of producing magnetic field is called an inductor.

Flow of current through any conductor can produce magnetic field, therefore every conductor is an inductor.

Simple example of an inductor is a solenoid. The symbol of an inductor is also a solenoid (Induction Inductance_99.gif).

A capacitor produce electric field and an inductor produce magnetic field.


Consider a solenoid of length ℓ with N loops each of area A.

The number of loops per unit length n=N/ℓ.

Induction Inductance_100.gif

If i current is flowing through it, the magnetic field in the solenoid is given as

Induction Inductance_101.gif

The magnetic flux through each loop is B A, so total magnetic flux Induction Inductance_102.gif through the solenoid is

Induction Inductance_103.gif

By substituting the value of N we get

Induction Inductance_104.gif

The term Induction Inductance_105.gifℓA) is called inductance of the solenoid.

The symbol for inductance is L and , like capacitance, inductance also depends only on the geometrical parameters of a conductor.

Induction Inductance_106.gif

The magnetic flux through the solenoid due to current i is given as

Induction Inductance_107.gif

This relation is true for any shaped inductor.

SI Units for Inductance L is Henry (symbol H).

Induction Inductance_108.gif

Self induced emf (self Induction):

Change in current in one loop can induced emf in a neighboring loop. Similarly when current changes in a loop or a solenoid, it induces emf in it self.

If current is changing in a solenoid or an inductor, the change in magnetic flux is given as

Induction Inductance_109.gif

According the Faraday's law self induced emf E will be given as

Induction Inductance_110.gif

Induction Inductance_111.gif Induction Inductance_112.gif

Alternative definition of 1 H

When a change of 1 A/s induce 1 V of emf in an inductor, its inductance is said to be 1 H.

Checkpoint 4

The figure shows an emf Induction Inductance_113.gif induced in a coil. Which of the following can describe the current through the coil: (a) constant and rightward, (b) constant and leftward, (c) increasing and rightward, (d) decreasing and right-ward, (e) increasing and leftward, (f) decreasing and leftward?

Induction Inductance_114.gif

Hint: Induced emf always opposes the change in current. If current is increasing, it will decrease it if current is decreasing it will increase it.

Mutual Inductance :

We know that when two coils are placed side by side, current Induction Inductance_115.gif in coil-1 will produce magnetic flux Induction Inductance_116.gif through the coil-2. Change in flux Induction Inductance_117.gif will induce emf Induction Inductance_118.gif in coil-2.

Induction Inductance_119.gif

Induction Inductance_120.gif is a constant and depends on the geometry of two coils and their relative position.

Induction Inductance_121.gif Induction Inductance_122.gif

Similarly current Induction Inductance_123.gif in coil-2 will produce magnetic flux Induction Inductance_124.gif through the coil-1. Change in flux Induction Inductance_125.gif will induce emf Induction Inductance_126.gif in coil-1.

Induction Inductance_127.gif

Induction Inductance_128.gif is a again a constant and depends on the geometry of two coils and their relative position. It can be shown that for any two coils.

Induction Inductance_129.gif

M is called "Mutual Inductance" of two coils. SI Units for Mutual Inductance M is Henry (symbol H).

When a change of 1 A/s in one coil induce 1 V of emf in the other coil, their mutual inductance M is said to be 1 H.

Quiz- 4
What unit is used for inductance?

Weber (Wb)

Henry (H)

ampere (A)

morgan (M)

volt (V)

RL Circuits

Rise of Current :

Consider a circuit as shown below.

Induction Inductance_130.gif

Suppose at time t=0 the switch is thrown in a position. The current will start increasing in the circuit and this increase will induce an emf Induction Inductance_131.gif across the inductor L. Now the equivalent circuit will be as shown below.

Induction Inductance_132.gif

According to Kirchhoff rule

Induction Inductance_133.gif

Where Induction Inductance_134.gif, above equation can be rewritten as

Induction Inductance_135.gif

Solution of this equation gives us the current through the circuit at any time t.

Induction Inductance_136.gif

This equation can be re written as

Induction Inductance_137.gif

Here Induction Inductance_138.gif, the induced time constant, is given by

Induction Inductance_139.gif

Plot of current as a function of time is given as

Induction Inductance_140.gif

It can be seen that it takes a while (about 5τ time) for the current to reach its constant value.

Once the current is constant, the induced emf Induction Inductance_141.gif. Change of Induction Inductance_142.gif as a function of time is given as

Induction Inductance_143.gif

Decay of current :

Once a steady current is flowing through the inductor and now if we throw the switch in b position. The current will not go to zero instantaneously.

Induction Inductance_144.gif

Suppose at time t=0 the switch is thrown in b position. The current will start decreasing in the circuit and this decrease will induce an emf Induction Inductance_145.gif across the inductor L. Now according to Kirchhoff's rule

Induction Inductance_146.gif

Where Induction Inductance_147.gif, above equation can be rewritten as

Induction Inductance_148.gif

Solution of this equation gives us the current through the circuit at any time t.

Induction Inductance_149.gif

Induction Inductance_150.gif

Quiz- 5
When a battery, a resistor, a switch, and an inductor form a circuit and the switch is closed, the inductor acts to oppose the change in the current.  How is the time constant of the inductor affected by doubling the resistance in the circuit?

The time constant would increase to four times its original value.

The time constant would increase to twice its original value.

The time constant would remain the same.

The time constant would decrease to one-half its original value.

The time constant would decrease to one-fourth its original value.

Energy stored in an Inductor :

We have seen that energy is stored in electric field of a capacitor. Similarly energy is stored in the magnetic field of an inductor.

Induction Inductance_151.gif

In the above circuit when current was building up the Kirchhoff's law gave following equation

Induction Inductance_152.gif

If we multiply both sides by i we get

Induction Inductance_153.gif

Left hand side term E i is the power delivered by the battery.

Right hand side term Induction Inductance_154.gif is the power dissipated in resistance R similarly Induction Inductance_155.gif should be the (power) rate at which energy Induction Inductance_156.gif is being stored in inductor L.

Induction Inductance_157.gif

Current changes from zero to steady state value i, therefore total energy stored in the inductor is given as

Induction Inductance_158.gif

Energy Stored in a Magnetic Field

Consider a solenoid of length ℓ , cross sectional area A and n number of loops per unit length. The energy stored in such an inductor will be

Induction Inductance_159.gif

Volume of the solenoid is V=A ℓ, the energy density Induction Inductance_160.gif due to magnetic field in the solenoid is given as

Induction Inductance_161.gif

Magnetic field B produced in the solenoid is given as

Induction Inductance_162.gif

By substituting the value of B in the expression for Induction Inductance_163.gif, we get

Induction Inductance_164.gif

This expression is valid for any magnetic field B.

Compare it with the expression of electric energy density Induction Inductance_165.gif associated with electric field

Induction Inductance_166.gif

Quiz- 6
Which one of the following terms is used for the effect in which a changing current in one circuit induces an emf in another circuit?




Lenz effect

mutual induction

Quiz- 7
In a circuit containing an emf, a resistor, and an inductor, where is the magnetic potential energy stored?

in the inductance of the inductor

in the resistor

in the magnetic field

in the magnetic flux

in the current