Electromagnetic Induction - Michigan State University · 2014. 3. 12. · • Covers Chapters 27, 28, 29 in the book • Magnetism, Magnetic Fields, Electromagnetic Induction •

March 12, 2014 Chapter 29 1

Electromagnetic Induction!

Notes !  Exam 4 next Tuesday

•  Covers Chapters 27, 28, 29 in the book •  Magnetism, Magnetic Fields, Electromagnetic Induction •  Material from the week before spring break and this week •  Homework sets 7 and 8

!  I will be out of town next week •  Prof. Joey Huston will teach lectures



Faraday’s Law of Induction !  The magnitude of the potential difference, ΔVind,

induced in a conducting loop is equal to the time rate of change of the magnetic flux through the loop.

!  We can change the magnetic flux in several ways:

!  Where A is the area of the loop, B is the magnetic field strength, and θ is the angle between the surface normal and the B field vector

ΔVind = −

dΦB

dt

A,θ constant: ΔVind = −Acosθ dB

dt

B,θ constant: ΔVind = −Bcosθ dA

dt

A,B constant: ΔVind =ωABsinθ


Law of Induction !  This potential difference means that the changing magnetic field creates an electric field around the loop!

!  Thus, there are two ways of producing an electric field !  If the electric field arises from a charge, the resulting electric

force on a test charge is conservative !  Conservative forces do no work when they act on an object

whose path starts and ends at the same point in space !  In contrast, electric fields generated by changing magnetic fields give rise to electric forces that are not conservative

!  Thus, a test particle that moves around a circular loop once will have work done on it by this electric field

!  In fact, the amount of the work done is the induced potential difference times the charge of the test particle


Lenz’s Law !  Lenz’s Law defines a rule for determining the direction of an

induced current in a loop !  An induced current will have a direction such that the

magnetic field due to the induced current opposes the change in the magnetic flux that induces the current

!  The direction of the induced current corresponds to the direction of the induced emf

!  We can apply Lenz’s Law to the situations described in yesterday’s lecture


!  The physical situation shown here involves moving a magnet toward a loop with the north pole pointed toward the loop

!  The magnetic field lines point away from the north pole of the magnet

!  As the magnet moves toward the loop, the magnitude of the magnetic field within the loop, in the direction pointing toward the loop, increases

!  Lenz’s law states that a current is induced in the loop that tends to oppose the change in magnetic flux

!  This induced magnetic field then points in the opposite direction from the field from the magnet

Lenz’s Law


!  The physical situation shown here involves moving a magnet toward a loop with the south pole pointed toward the loop

!  The magnetic field lines point toward the south pole of the magnet

!  As the magnet moves toward the loop, the magnitude of the field increases in the direction pointing toward the south pole

!  Lenz’s law states that a current is induced in the loop that tends to oppose the change in magnetic flux

!  This induced magnetic field then points in the opposite direction from the field from the magnet

Lenz’s Law


Lenz’s Law - Four Cases

!  (a) An increasing magnetic field pointing to the right induces a current that creates a magnetic field to the left

!  (b) An increasing magnetic field pointing to the left induces a current that creates a magnetic field to the right

!  (c) A decreasing magnetic field pointing to the right induces a current that creates a magnetic field to the right

!  (d) A decreasing magnetic field pointing to the left induces a current that creates a magnetic field to the left


Eddy Currents !  Let’s consider two pendulums, each with a non-magnetic

conducting metal plate at the end that is designed to pass through the gap of a strong permanent magnet

!  One metal plate is solid and the other has slots cut in it

!  We pull back both pendulums and release them


Eddy Currents !  The pendulum with the solid metal sheet stops in the

gap of the magnet while the grooved sheet passes through the magnetic field, only slowing slightly

!  This demonstrates the importance of induced eddy currents !  As the pendulum with the solid plate enters the magnetic field,

Lenz’s law tells us that the changing magnetic flux will induce currents that tend to oppose the change in flux

!  Just like eddy currents, swirls we see in turbulent water flow !  These currents interact with the magnetic field to stop the

pendulum •  Kinetic energy is converted into heat

!  Often, eddy currents have to be eliminated through segmenting •  Pulsed magnets •  Transformers

!  Induced eddy currents can also be employed in practical situations such as braking railroad cars


Induced Potential Difference on a Moving Wire in a Magnetic Field

!  Consider a conducting wire moving with constant velocity perpendicular to a constant magnetic field

!  The wire is oriented such that it is perpendicular to the velocity and to the magnetic field

!  The magnetic field exerts a force on the conducting electrons in the wire, causing them to move downward

!  This motion of the electrons produces a net negative charge at the bottom of the wire and a net positive charge at the top of the wire

!  This charge separation produces an electric field that exerts a force on the conduction electrons that tends to cancel the magnetic force


Induced Potential Difference on a Moving Wire in a Magnetic Field

!  After some time the forces on the electrons are in equilibrium and we can write

!  Thus we can express the induced electric field as

!  This electric field produces a potential between the two ends of the wire given by

!  We can then write the voltage between the ends of the wire as

FB = qvB = FE = qE

E = vB

E = ΔVind

= vB

ΔVind = vB

Satellite Tethered to a Space Shuttle !  In 1996, the Space Shuttle Columbia

deployed a tethered satellite on a wire out to a distance of 20 km

!  The wire was oriented perpendicular to the Earth’s magnetic field at that point, and the magnitude of the field was B = 5.1·10–5 T

!  Columbia was traveling at a speed of 7.6 km/s PROBLEM

!  What was the potential difference induced between the ends of the wire?


Satellite Tethered to a Space Shuttle SOLUTION

!  The length of the wire is 20 km and the speed of the wire through the magnetic field of the Earth is the same as the speed of the Space Shuttle, which is 7.6 km/s

!  The induced potential difference is

!  The astronauts on the Space Shuttle measured a current of about 0.5 A at a voltage of 3500 V

!  The circuit consisted of the deployed wire and ionized atoms in space as the return path for the current

!  The wire broke just as the deployment length reached 20 km March 12, 2014 Chapter 29 17

ΔVind = vLB = 7.6 ⋅103 m/s( ) 20 ⋅103 m( ) 5.1⋅10−5 T( )ΔVind = 7800 V

Pulled Connecting Rod !  A conducting rod is pulled

horizontally by a constant force of F = 5.00 N along a set of conducting rails separated by a distance a = 0.500 m

!  The two rails are connected, and no friction occurs between the rod and the rails

!  A uniform magnetic field with magnitude B = 0.500 T is directed into the page

!  The rod moves at constant speed, v = 5.00 m/s PROBLEM

!  What is the magnitude of the induced potential difference in the loop created by the connected rails and the moving rod?


Pulled Connecting Rod SOLUTION

!  The induced potential difference is

!  The area of the loop is increasing with time

!  The change in the area as a function of time is

!  The magnitude of the induced potential difference is


ΔVind = −Bcosθ dA

dt

A = A0 +a vt( ) = A0 + vta

dAdt

= ddt

A0 + vta( ) = va

ΔVind = −Bcosθ dA

dt= vab = 5.00 m/s( ) 0.500 m( ) 0.500 T( ) =1.25 V


Generators and Motors

!  The third special case of our simple induction processes is technologically by far the most interesting

!  It is the variation of the angle between the loop and the magnetic field with time, while keeping the area of the loop as well as the magnetic field strength constant in time

!  In this way, Faraday’s Law of Induction can be applied to the generation and use of electric current

!  A device that produces electric current from mechanical motion is called an electric generator

!  A device that produces mechanical motion from electric current is called an electric motor


Generators and Motors !  A simple generator consists of a loop forced to rotate in a fixed magnetic field

!  The driving force that causes the loop to rotate can be supplied by hot steam running over a turbine

!  Or the loop can be made to rotate by water or wind in a pollution-free way of generating electrical power

!  In a direct-current generator, the rotating loop is connected to an external circuit through a split commutator ring

!  As the loop turns, the connection is reversed twice per revolution, so the induced potential difference always has the same sign


Generators and Motors !  A similar arrangement is used to produce an alternating

current !  An alternating current is a current that varies in time

between positive and negative values, with the variation often showing a sinusoidal form

!  Each end of the loop is connected to the external circuit through its own solid slip ring

!  Thus, this generator produces an induced potential difference that varies from positive to negative and back

!  A generator that produces alternating voltages and the resulting alternating current is also called an alternator



!  Faraday’s Law of Induction states that a changing magnetic flux produces an induced potential difference

!  Consider a test charge q moving in a circular path with radius r in an electric field

!  The work done is equal to the integral of the scalar product of the force and the displacement

!  For now we assume that the electric field is constant and that is has field lines that are circular

!  Considering one revolution of the test charge we obtain the work done on the test charge to be

!  The work done by a constant electric field is qΔVind so

Induced Electric Field

W =Fids∫ = q

Eids∫ = qE( ) 2πr( )

qΔVind = 2πrqE ⇒ ΔVind = 2πrE


Induced Electric Field !  We can generalize this result by considering the work

on a test particle with charge q moving along an arbitrary closed path as

!  Again substituting qΔVind for the work we obtain

!  Since the induced potential difference is the change in flux

!  A changing magnetic field induces an electric field !  This applies to any closed path drawn in a changing

magnetic field

W =Fids∫ = q

Eids∫

ΔVind =Eids∫

Eids∫ = − dΦB

dt


Inductance of a Solenoid !  Consider a long solenoid with N turns carrying a current i !  This current creates a magnetic field in the center of the

solenoid resulting in a magnetic flux of ΦB !  The same magnetic flux goes through each of the N

windings of the solenoid !  It is customary to define the flux linkage as the product of

the number of windings and the magnetic flux, NΦB !  Inside the solenoid, the magnetic flux is

!  Remembering that for a solenoid ΦB =

B•dA∫∫ = BA

B = µ0ni = µ0

N

i


Inductance of a Solenoid !  We can then write the flux linkage as

!  The flux linkage is a constant times the current !  We define this constant to be the inductance L !  Thus the inductance is a measure of the flux linkage

produced by the solenoid per unit current !  For an solenoid to behave in this manner, it must not have

any magnetic materials in its core !  The unit of inductance is the henry (H), named after

American physicist Joseph Henry (1797 – 1878)

[L] = [ΦB]

[i]⇒1 H = 1 T m2

1 A

NΦB = NBA = N µ0

N

i⎛⎝⎜

⎞⎠⎟ A = µ0

N 2

A⎛

⎝⎜⎞⎠⎟

i = Li


Inductance of a Solenoid

!  We can define the magnetic permeability of free space as !  The inductance of a solenoid is then

!  The inductance of a solenoid depends only on its geometry !  When a solenoid is used in an electric circuit, it is called an

inductor

µ0 = 4π ⋅10−7 H/m

Lsolenoid = µ0N 2

A

take n = N

Lsolenoid = µ0n( )2

A= µ0n2A


Self Inductance and Mutual Induction

!  Consider the situation in which two coils, or inductors, are close to each other

!  A current in the first coil produces magnetic flux in the second coil

!  Changing the current in the first coil will induce a potential difference in the second coil

!  However, the changing current in the first coil also induces a potential difference in itself

!  This phenomenon is called self-induction !  The resulting potential difference is termed the self-induced

potential difference !  Changing the current in the first coil also induces a potential

difference in the second coil !  This phenomenon is called mutual induction


Self Induction !  According to Faraday’s Law of Induction, the self-induced

potential difference for any inductor is given by

!  Thus, in any inductor, a self-induced potential difference appears when the current changes with time

!  This self-induced potential difference depends on the time rate of change of the current and the inductance of the device

!  Lenz’s Law provides the direction of the self-induced potential difference

!  The negative sign provides the clue that the induced potential difference always opposes any change in current

ΔVind ,L = −

d NΦB( )dt

= −d Li( )

dt= −L di

dt


Self Induction !  Consider a circuit with current flowing through

an inductor that increases with time !  The self-induced potential difference will oppose

the increase in current !  Consider a circuit with current flowing through

an inductor that is decreasing with time !  A self-induced potential difference will oppose

the decrease in current !  We have assumed that these inductors are ideal

inductors; that is, they have no resistance !  Induced potential differences manifest

themselves across the connections of the inductor

Documents

Electromagnetic Induction - Michigan State University · 2014. 3. 12. · • Covers Chapters 27, 28, 29 in the book • Magnetism, Magnetic Fields, Electromagnetic Induction •