Sierzega: Electromagnetic Induction 2
Faraday’s Law of Electromagnetic Induction
2.1 Represent and reason The rectangular loop with a resistor is pulled at constant velocity through a
uniform external magnetic field that points into the paper in the regions shown with the crosses (x).
Complete the table that follows to determine qualitatively the shape of the induced current-versus-time
graph.
Flux 
Draw a
qualitative fluxversus-time
graph for the
t1
t2
process
(positive in and
negative out).
Draw a
Induced magnetic field Bind
qualitative
induced
magnetic fieldversus-time
t1
t2
graph for the
process.
Draw a
Induced current I
qualitative
induced currentversus-time
t1
t2
graph for the
process.
t3
t4
Time t
t3
t4
Time t
t3
t4
Time t
Sierzega: Electromagnetic Induction 2
2.2 Observe and explain In the table below, the results of four experiments are shown in
which a changing magnetic field produced by an electromagnet passes through a loop. This
changing B field causes a changing flux  through the loop and an induced current i ind
around the loop of resistance R. The product iindR is also plotted as a function of time.
(a) Draw another graph in the table cells that shows the induced emf  ind in the loop.
Coil resistance is 1.0 .
Coil resistance is 3.0 .
 (T m2)
 (T m2)
0.6
0.6
0
1
2
3
t (s)
0
1
2
3
1
2
3
1
2
3
1
2
3
t (s)
iind (A)
iind (A)
1
2
3
t (s)
t (s)
0.10
0.30
iind R (A )
iind R (A )
1
2
3
t (s)
0.30
0.30
 (V)
 (V)
1
2
3
t (s)
Coil resistance
is 2.0 .
2
t (s)
Coil resistance
is 6.0 .
2
 (T m )
 (T m )
0.6
0.6
t (s)
1
2
3
0
1
iin R (A )
2
3
2
3
t (s)
0
1
2
0
1
2
3
(b) Devise a relationship between /t and  ind. Do not forget the sign!
3
0
iin (A)
t (s)
1
2
3
0
iin (A)
t (s)
0.30
0
1
iin R (A )
2
3
t (s)
t (s)
0.60
0.60
0
 (V)
0.10
1
2
3
t (s)
0
 (V)
1
t (s)
t (s)
i
Sierzega: Electromagnetic Induction 2
2.3 Observe and explain The analysis of the following experiment provides an expression for the
potential difference produced in a loop moving into, through, and out of a magnetic field. This is
called motional emf.
Experiment
Analysis
Analysis
A metal bar of length L moves
at constant velocity through a
magnetic field that points into
the paper (the crosses).
(a) Indicate on the bar how the
magnetic force of the field on
charges in the bar redistributes
charge in the bar.
x
x
x
x
x
x
x
x
(b) This charge redistribution, which
occurs quickly, produces an electric
field inside the bar that prevents
further charge redistribution. Draw
the E-field lines.
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
(c) Apply Newton’s second law for a charge in the
(d) Use the expression that relates electric field Ey
middle of the bar—now in an E field and a B field.
and the potential difference over a distance ∆V/L
with the previous results to determine an expression
for the potential difference (emf) across the ends of
the bar.
(e) Below, you see a rectangular metal conductor entering the magnetic field, completely in the magnetic
field, and leaving the magnetic field described above. Use the results of (a) and (b) to draw charge
distributions on the front and back vertical parts of the rectangle due to the force of the magnetic field on
charge in the metal. Note: The magnetic field only exerts forces on charge in the metal parts that are in the
field, not on those outside the fields. Will current flow? If so, indicate the direction.
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
(f) Use the results of (d) to write
an expression for the potential
difference induced around the
rectangular metal conductor
while entering the field.
x
x
x
x
(g) Use the results of (d) to write
an expression for the potential
difference induced around the
rectangular metal conductor when
completely in the field. Note that
there are now charge distributions
that cancel.
x
x
(h) Use the results of (d) to write
an expression for the potential
difference induced around the
rectangular metal conductor
while leaving the field.
Sierzega: Electromagnetic Induction 2
2.4 Observe and explain We repeat the previous activity, only this time using ideas of flux and induced
emf. When you are finished, check to see if the results are consistent with those in Activity 2.3.
The same rectangular metal conductor as in Activity 2.3e is entering the magnetic field, completely in the
magnetic field, and leaving a magnetic field that points into the paper.
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
(a) Draw a graph showing the magnetic flux through the opening of the metal conductor as a function of
time while the rectangle is entering the magnetic field, completely in the magnetic field, and leaving the
magnetic field. Then use the flux graph to make a graph of the induced emf for the same time interval.

t

Entering
In field
Leaving
t
(b) Write an expression for the
changing flux as the rectangle
enters the field. Then use this
expression to determine the emf
while the rectangle is entering the
field.
(c) Write an expression for the
changing flux as the rectangle is
completely in the field. Then use
this expression to determine the
emf while the rectangle is in the
field.
(d) Compare the expressions in
(b) and (c) with the expressions
determined in 2.3 f and g. (We
skip the calculation for when the
rectangle is leaving the field—it’s
a little more messy.)
Did you know?
Before moving forward, you should know the following:
 magnetic flux:  = AB cos 
(units: [T*m2] = [Wb] Weber)


Faraday’s law of electromagnetic induction: The average magnitude of the induced emf ind in
a coil with N loops is the magnitude of the ratio of the magnetic flux change in the loop  to
the time interval ∆t during which that flux change occurred multiplied by the number N of loops:
|ind|  N |  t|
Lenz’s law: the direction of an induced current is such that its induced magnetic field opposes
the change in the external magnetic flux that caused the induced current.
Sierzega: Electromagnetic Induction 2
2.5 Reason The magnitude of the magnetic field in each situation described below is 0.50 T. For each
situation, in the table that follows write an expression for the magnetic flux through the loop.
Situation
Loop and B in the plane of the paper.
Write an expression for the flux

Loop perpendicular to the
paper and B in plane of the paper

Loop in the plane of the paper
Square loop of side a in
the plane of the paper.

B into the paper.
x
x
x
x
x
x
x
x
37o x
Sierzega: Electromagnetic Induction 2
2.6 Represent and reason Four situations are shown in which the external flux through a loop is plotted
as a function of time. In the table that follows, draw another graph that shows the induced emf in the
loop as a function of time.
0.6 T•m2
0.4 T•m2
0s
1s
2s
0s
3s
 (V)
 (V)
+0.4
+0.6
0s
1s
2s
3s
time
0s
1s
1s
2s
2s
3s
3s
4s
4s
time
–0.6
–0.4
0.6 T•m2
0.4 T•m2
0.15
0s
 (V)
1s
2s
2.5 s
0s
+0.4
0s
–0.4
1s
2s
3s
4s
 (V)
1s
2s
3s
time
+0.6
0s
–0.6
1s
2s
3s
4s
time
Sierzega: Electromagnetic Induction 2
2.7 A horizontal bar is pulled at constant velocity through a downward-pointing magnetic field. The bar
slides on two horizontal, frictionless metal rails moving away from a resistor connected between
their ends. Derive an expression for the induced current through the resistor of resistance R in terms
of any or all quantities that you choose to include in a sketch of this system. Be sure to identify the
quantities in the sketch.
a. Draw top view sketches showing the rails and the bar location at an initial time and at a later
time. Include symbols for quantities involved in the problem.
b. Describe the assumptions you are you making.
c. Draw lines in the graphs below that represent the process. Flux through the area surrounded by
rails, the moving bar and the resistor at the end of rails is plotted as a function of time.
d. Draw a consistent emf-versus-clock reading graph
e. Represent flux and emf mathematically.
f. Combine the mathematical representation with Ohm’s law to get the desired expression for the
current.
2.8 Consider the situation above, but with the bar moving toward the resistor instead. An external force
does work as the movable bar moves, and this work is converted to electrical energy. Because the
“circuit” is in a magnetic field, the flux through it changes with time, inducing a current.
a. What is the direction of the induced current in the resistor?
b. If the bar is 20 cm long and is pulled at a steady speed of 10 cm/s, what is the induced current if
the resistor has a value of 5.0 Ω and the circuit is in a uniform magnetic field of 0.25 T?
Did you know?
Motional emf: An electric potential difference is produced between two points that depends on the
magnitude of the electric field, E, in the rod and the distance L between the two ends
motional emf = EL = vBL
For problems involving conducting objects moving in a magnetic field, we can use either Faraday’s law or
the motional emf expression to determine the emf produced. Both methods yield the same result.