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Lecture 14 — Chapter 29
Electromagnetic Induction
Lecture 14 — Chapter 29
Electromagnetic Induction
Basic Principles
A changing magnetic field creates an induced electric field
— Discovered by Faraday (experimentally)
A changing electric field creates an induced magnetic field
— Deduced by Maxwell (theoretically)
Equivalent statements:
Changing magnetic flux creates an induced emf
Changing electric flux creates a “displacement current”
Electrostatic fields:
produced by charges according to Coulomb’s Law
Induced electric fields:
produced by changing magnetic fields
For both: F = q E
Practical Applications
•Electrical generators
• Convert mechanical, thermal, chemical, gravitational or nuclear energy into electrical energy
• Electrical energy can be transmitted over long distances, and consumed far from the original
energy source
•Magnetic readers for data storage
•
•
•
Magnetic computer disks
Magnetic card readers
Magnetic tape
•Electric guitar pick-ups
•Metal detectors
•Induction stoves
•Wireless charging
Review: Magnetic Flux
Definition of the magnetic flux through a
surface element with area dA:
•
•
The flux is the product of the component of B
perpendicular to the surface with the area
The angle ϕ is the angle between the normal to
the surface and the direction of B
The flux: Φ B = BdAcosφ
Direction for dA is usually chosen to point
either outwards or inwards from the normal
of the surface. Either choice will give the
same answer, as we will see.
Review: Magnetic Flux
Faraday’s Law of Induction
The induced emf in a closed loop equals the negative of the time
rate of change of the magnetic flux through the loop.
dΦ B
ε=−
dt
The minus sign in Faraday’s Law is important!
It indicates the direction of the induced emf.
We’ll examine how in a moment.
Example: Induced emf and current in a loop
The uniform magnetic field shown is increasing at a rate of dB/dt = 0.020 T/s.
A loop with area 0.012 m2 normal to the field has resistance 5.0 Ω. Find the induced current in the loop.
Φ B = BA
Defined A in the direction of B
dΦ B
dB
ε=−
= −A
= −(0.012m2 )(0.02T / s) = −0.24mV
dt
dt
I=
ε −0.24mV
=
= −0.048mA
R
5Ω
Positive current would flow in a right-hand sense relative to A (i.e. CCW).
This negative current flows in the opposite direction.
dΦ B
ε=−
dt
direction of induced emf
Direction of induced emf: recap
•Define direction for A
•Find the direction of ΦB
•
If A & B are in the same direction ΦB >0
•
If A & B are in the opposite direction ΦB <0
•Determine if dΦB/dt is positive or negative
•
If ΦB > 0 & B increases -or- ΦB < 0 & B decreases: dΦB/dt > 0
•
If ΦB > 0 & B decreases -or- ΦB < 0 & B increases: dΦB/dt < 0
•The emf will be signed in the opposite direction of dΦB/dt
•Direction of emf, ξ
•
•
ξ > 0: in the right-hand direction around A
ξ < 0: opposite the right-hand direction around A
Quiz: Induced emf
A circular loop of wire is in a region of spatially uniform magnetic field. The magnetic field
is directed into the plane of the figure. If the magnetic field magnitude is constant,
A.the induced emf is clockwise.
B.the induced emf is counterclockwise.
C.the induced emf is zero.
D.The answer depends on the strength of the field.
dΦ B
ε=−
dt
Quiz: Induced emf
A circular loop of wire is in a region of spatially uniform magnetic field. The magnetic field
is directed into the plane of the figure. If the magnetic field magnitude is constant,
A.the induced emf is clockwise.
B.the induced emf is counterclockwise.
C.the induced emf is zero.
D.The answer depends on the strength of the field.
A changing magnetic field creates an induced electric field
ε=−
B is constant here.
dΦ B
dt
Quiz: Induced emf
A circular loop of wire is in a region of spatially uniform magnetic field. The magnetic field
is directed into the plane of the figure. If the magnetic field magnitude is decreasing,
A.the induced emf is clockwise.
B.the induced emf is counterclockwise.
C.the induced emf is zero.
D.The answer depends on the strength of the field.
ε=−
dΦ B
dt
Quiz: Induced emf
A circular loop of wire is in a region of spatially uniform magnetic field. The magnetic field
is directed into the plane of the figure. If the magnetic field magnitude is decreasing,
A.the induced emf is clockwise.
B.the induced emf is counterclockwise.
dΦ B
ε=−
dt
C.the induced emf is zero.
D.The answer depends on the strength of the field.
If the flux rate of change is decreasing (becoming more negative, or
less positive), the emf is positive, & in right-hand direction around A.
Let’s take A going out of the page (anti-parallel to B), thus ΦB < 0. Since B decreases (becomes less
negative), dΦB/dt > 0. => emf < 0 and in opposite direction given by RH direction around A, thus
clockwise. (Follow up: Take A going into the page, do you get the same answer ?)
Lenz’s law
Lenz’s law is an alternative means for determining the direction of current flow in
the previous example.
Lenz’s law states that…
The direction of any magnetic induction effect is such as to oppose
the cause of the effect.
In our case, “cause” = changing magnetic flux and “effect” = induced current.
Lenz’s law and induced currents
In this example, there is a uniform B field through the coil. The magnitude of the field is
increasing, so there is an induced emf.
According to Lenz’s Law, the induced emf will produce a current which opposes the change
that causes it.
The change that causes it is an increase in the magnitude of the original B field, so the
induced current will be such as to produce an induced B field opposite to the increase in B.
Changes in Flux
dΦ B
ε=−
dt
There are three basic ways the magnetic flux through a loop can change, resulting
in an induced emf due to Faraday’s Law:
1.The magnitude of the magnetic field can change (dB/dt)
2.The area (A) of the loop in the magnetic field can change
3.The angle of the normal to the loop with the field direction (ϕ) can change.
•
This could be due to the orientation of the loop changing, or the direction of the
magnetic field changing (or both)
Quiz: induce current I
B=
A circular loop of wire is placed next to a long straight wire.
The current I in the long straight wire is increasing. What current does this induce in the circular loop?
A.a clockwise current
B.a counterclockwise current
C.zero current
D.not enough information given to decide
ε=−
dΦ B
dt
µ0 I
2π r
ε
I=
R
Quiz: induce current I
B=
A circular loop of wire is placed next to a long straight wire.
The current I in the long straight wire is increasing. What current does this induce in the circular loop?
A.a clockwise current
µ0 I
2π r
I=
ε
R
dΦ B
ε=−
dt
B.a counterclockwise current
C.zero current
D.not enough information given to decide
B (from the straight wire) points out of the page at the circular loop. Since the current in the
straight wire increases, B increases.
Let’s take A in the same direction as B, so that ΦB > 0. Since |B| increases, dΦB/dt > 0. Thus, emf < 0 and in the opposite direction given by RH direction around A, thus clockwise.
Announcements
— Ch 28 HW assigned on Mastering Physics
— Review these notes including the clicker questions.