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Last time: how charged particles move in a
magnetic field.
Consider a uniform magnetic field into the
board, with conducting rod moving through it:
+ side
𝑣
Charges will move
- side
If part of a circuit, can
generate a current!
Which direction is the current through the resistor?
1) Up
2) Down
Note: as rod moves, there is an increasing magnetic flux
through the loop.
Magnetic Flux
Just like for electric fields, can define magnetic
flux through surface:
Φ𝐵 =
𝐵 ⋅ 𝑑𝐴
For uniform 𝐵 and a flat surface 𝐴:
Φ𝐵 = 𝐵𝐴 cos 𝜃
Faraday’s Law
Changing magnetic field can
induce a ℇ and 𝐼:
𝑑Φ𝐵
ℇ=−
𝑑𝑡
where Φ𝐵 is the magnetic flux.
- Solenoid with alternating
/ direct current.
- Vary number of windings.
What are ways that we can increase the amount
of current through the loop?
More windings?
Angle of loop relative to solenoid?
Shape of loop?
Top View
Solenoids
B
B
Wire Loops
Magnetic field ~inside of solenoid only 
area is the same for both  flux is the
same for both.
In figure (a), a solenoid produces a magnetic field whose strength
increases into the plane of the page. An induced emf is established
in a conducting loop surrounding the solenoid, and this emf lights
bulbs A and B. In figure (b), points P and Q are shorted. After the
short is inserted,
1. bulb A goes out; bulb B gets brighter.
2. bulb B goes out; bulb A gets brighter.
3. bulb A goes out; bulb B gets dimmer.
4. bulb B goes out; bulb A gets dimmer.
5. both bulbs go out.
6. none of the above
Lenz’s Law
The induced current creates a magnetic field
that opposes the change in magnetic flux
through the area enclosed by the loop.
𝑑Φ𝐵
ℇ=−
𝑑𝑡
Way to interpret the minus
sign in Faraday’s Law
The current through the wire is decreasing and
the loop is fixed relative to the wire. In which
direction is the induced current in the loop?
1. Clockwise
2. Counter-clockwise
3. No current is induced
The loop shown below is traveling towards the
wire with the velocity shown. The current, 𝐼, is
constant. In which direction is the induced
current in the loop?
1. Clockwise
2. Counter-clockwise
3. No current is induced
The loop shown below is traveling towards the
right at a constant distance from the bottom
wire. The current, 𝐼 , is constant. In which
direction is the induced current in the loop?
𝑣
1. Clockwise
2. Counter-clockwise
3. No current is induced
Cool Application: Generators and Motors
A uniform magnetic field can produce an EMF in
a conducting loop if:
1) The field changes in magnitude.
2) The loop changes position within the field.
3) The loop rotates within the field.
4) The loop is removed from the field.
5) The loop changes size.
6) All of the above
7) All except 2)
8) All except 3)
9) All except 4)
Relation to Electric Fields
Electric fields cause charges to move  induced
EMF can be related to 𝐸:
𝑑Φ𝐵
𝐸 ⋅ 𝑑𝑠 = −
𝑑𝑡
where integral is a closed path.
Note: Not necessary for a loop to be there (i.e.,
test charges) for an 𝐸 field to be created.
Eddy Currents
F
Plate moving between
poles of magnet:
F
N
S
S
Force tends to damp
motion of the plate.
N
Adding slots  fewer paths for current 
higher resistance  less force.
Question:
Let's say you take an ordinary wire coathanger
and straighten out the hook shaped part that
normally hangs over the coat rack. Now, you can
spin the (roughly) triangular part around by
twisting the straightened part between your
fingers. Estimate the EMF that you can generate
by spinning the hanger in the Earth's magnetic
field (about 5.3 x 10⁻⁵ T).
Lab:
Magnetic Field of a Slinky:
- Magnetic field sensor w/ LoggerPro
- Do theoretical predictions about
magnetic field hold?
Power supply
Interface
V
Ammeter
mV
Switch
Quantitative Question (31.12):
A coil of 15 turns and radius 10.0 cm surrounds a long
solenoid of radius 2.00 cm and 1000 turns/meter. The
current in the solenoid changes as 𝐼 = 5 sin 120𝑡 𝐴,
where 𝐼 is in amperes and t is in seconds. Find the
induced emf in the 15-turn coil as a function of time.