Pîrî Reis University
Spring 2021
Physics 121
Chapter 29
Programme
Week Chapters Subjects
1-2 Ch.21 Electric Charge & Field E-charge, insulators, conductors, Coulomb’s law, E-field
3 Ch.23 Electric Potential E-potential energy, potential difference, charge distribution
4 C.24 Capacitance, Dielectrics Capacitors, capacitance, E-energy storage, dielectrics
5 C.25 Electric Currents & Res. Batteries, E-current, Ohm’s law, power, current density
EMF, resistors in series/parallel, Kirchhoff’s rules, RC
6 Ch.26 DC Circuits
circ’s
7 Review of Chs. 21-26 .
8 Midterm .
9 Ch.27 Magnetism Magnets, magnetic fields
10 Force on an E-current in a M-field, M-field due to a wire
Ch.28 Sources of
Magnetic Field
11 Ampère’s law, Biot-Savart law
12 C.29 EM Ind. & Faraday’s law Induced EMF, Faraday’s law, moving conductors
C.30 Inductance & AC
13 Inductance, magnetic energy, EM oscillations, AC circuits
Circuits
14 Review of Chs. 27-30 .
Chapter 29
Electromagnetic Induction
and Faraday’s Law
I. Induced EMF
II. Magnetic Flux
III. Faraday’s Law of Induction; Lenz’s Law
IV. EMF Induced in a Moving Conductor
In the last Lectures, we saw that electric currents create magnetic fields and
magnetic fields exert force on electric currents (moving charges). In this
Lecture, we will see that a magnetic field (actually, a changing magnetic
flux) can produce an electric current.
I. Induced EMF
Using the setup shown in the right figure, Michael
Faraday found that a changing magnetic field
induces an emf. When the switch is closed, coil-X
produces a magnetic field. The iron ring carries
this magnetic field into coil-Y. However, it is not
this magnetic field itself, but it is the change in this magnetic field that induces an emf
in coil-Y. The galvanometer connected to coil-Y shows no signal while the switch is
closed or open. It shows a signal only during a very short time period after the switch
closes or after it opens. Such a current in coil-Y is called induced current, and this
phenomenon is called electromagnetic induction. It can also be observed in a simpler
setup shown in the bottom figures.
When a magnet is moved quickly
into a coil of wire, a current is
induced in the wire (Fig. a). If the
magnet is quickly removed from
the coil, a current in the opposite
direction is induced (Fig. b). But,
if the coil and the magnet are at
rest with respect to each other,
no current is induced (Fig. c).
II. Magnetic Flux
The volumetric flow rate (dV/dt) of fluid through the wire rectangle of area A is v A
when the area of the rectangle is perpendicular to the velocity vector v, see figure
(a); whereas it is v A cosθ when the rectangle is tilted at an angle θ, see figure (b).
We will next replace the fluid flow velocity field v with the magnetic field B to get to
the concept of magnetic flux ΦB .
II. Magnetic Flux
Faraday’s law involves the concept of magnetic flux,
which refers to the magnetic field passing through a
given area.
B
B For uniform magnetic field B passing through an area
A, magnetic flux ΦB is defined as ΦB = B A cosθ.
Here, θ is the angle between the direction of B and
the line drawn perpendicular to the area. Therefore,
B we can write magnetic flux for a uniform field as
B
ΦB = B┴ A = B A┴ = B A cosθ
Here, B┴ = B cosθ is the component of B along the
perpendicular to the area, and similarly, A┴ = A cosθ
B is the projection of the area A perpendicular to the
field B (see figure).
II. Magnetic Flux
The area A of a surface can be represented by a
vector A whose magnitude is A and whose direction
is perpendicular to the surface.
B
B So the magnetic flux can be written as
B Magnetic flux has a simple intuitive interpretation in
terms of field lines. In previous lecture, we mentioned
B that field lines can be drawn so that the number (N)
passing through unit area perpendicular to the field
(A┴) is proportional to the field (B): that is, B ~ N/A┴.
Hence
B
So the flux through an area is proportional to the
number of field lines passing through that area.
II. Magnetic Flux
Exercise: Which of the following would cause a change in the magnetic flux through
a circle lying in the xz plane where the magnetic field is 10 T ĵ ?
(a)Changing the magnitude of the magnetic field. z
(b)Changing the size of the circle.
(c)Tipping the circle so that it is lying in the xy plane. A
(d)All of the above. (e) None of the above. B
ΦB = B┴ A = B A┴ = B A cosθ A
y
θ = 0 and cosθ = 1 when the circle is in xz plane. x
θ = π/2 and cosθ = 0 when the circle is in xy plane. B
(a) ΦB changes when B changes.
(b) ΦB changes when A changes. Therefore, the answer is (d).
(c) ΦB changes when θ changes.
Exercise (determining flux): A square loop of wire encloses
area A1 as shown in the figure. A uniform magnetic field B
perpendicular to the loop extends over the area A2. What is
the magnetic flux through the wire loop of area A1?
Answer: ΦB = B A2
II. Magnetic Flux
In the general case, if the magnetic field is not uniform and the surface is not flat, we
divide the surface into n small surface elements whose areas are ΔA1, ΔA2, … , ΔAn,
so that each ΔAi is small enough that (1) it can be considered flat, and (2) the
magnetic field varies so little over this small area that it can be considered uniform.
Then the magnetic flux through the entire surface is approximately
where B i is the field passing through ΔA i . B
In the limit, as we let ΔA i → 0, the sum becomes
an integral over the entire surface and the relation
becomes mathematically exact:
B
B
So, SI unit of magnetic flux is the T·m2, which is called a weber (Wb): 1 Wb = 1 T·m2.
III. Faraday’s Law of Induction; Lenz’s Law
Faraday found that the induced emf in a circuit loop is related to the rate of change of
the magnetic flux passing through the surface enclosed by that loop.
With the definition of flux, Faraday’s law of induction can be written as E = –dΦB/dt.
So, the emf induced in a circuit is equal to the negative of the rate of change of
magnetic flux through the circuit. If the circuit contains N loops, then
The minus sign in Faraday’s law defines the
direction of induced emf: a current produced by
an induced emf moves in a direction so that the
magnetic field created by that current opposes
the original change in flux. This is known as
Lenz’s law.
III. Faraday’s Law of Induction; Lenz’s Law
Example (induction stove): In an induction stove, an AC current
exists in a coil that is the “burner” (a burner that never gets hot,
see Fig.). Why will it heat a metal pan but not a glass container?
The AC current sets up a changing magnetic field that passes through the pan
bottom. This changing magnetic field induces a current in the metal pan bottom, and
since the pan has a resistance, this electric energy is transformed into thermal
energy, which heats the metal pan. A glass container has such high resistance that
little energy is transferred (P = V 2/R).
Exercise (practice with Lenz’s law): In which direction is the current induced in the
circular loop for each situation in the figure below?
Answer: (a)(c)(e) Counterclockwise. (b) Clockwise. (d) No current.
IV. EMF Induced in a Moving Conductor
A simple way to induce an emf is shown in the
figure. A uniform magnetic field B is perpendicular
to the area bounded by the U-shaped conductor
and the movable rod resting on it. If the rod is
made to move at a speed v, it travels a distance dx
= v dt in a time dt. So, the area of the loop
increases by an amount dA = l dx = l v dt in a time
dt. By Faraday’s law, magnitude of the induced
emf is given by
Ex. (electromagnetic blood-flow measurement): The rate of blood
flow in our body’s vessels can be measured using the apparatus
shown in the figure, since blood contains charged ions. Suppose
that the blood vessel is l = 2.0 mm in diameter, the magnetic field
is B = 0.080 T, and the measured emf is E = 0.10 mV. What is
the flow velocity v of the blood?
This is the situation shown in above figure. From the above equation, we can obtain
v = E /(Bl ) = (0.10 × 10–3 V)/[(0.080 T) (2.0 × 10–3 m)] = 0.63 m/s
Reference
Physics
for
Scientists & Engineers
with Modern Physics
4th edition
Giancoli
