resistance R
“induction”
Changing magnetic field “induces” a current in a wire loop
magnitude? clockwise/counterclockwise?
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 
B(r ) , surface S
magnetic flux
^n
For planar surface drawn on page, choose n out of page
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 
B(r ) , surface S
magnetic flux

B ^
n
dA

Φ = ∫ ( B(x, y)⋅ n̂)dA
Flat surface, uniform B:



Φ = ∫ ( B ⋅ n̂)dA =( B ⋅ n̂) ∫ dA =( B ⋅ n̂)A
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B(r), surface S
Magnetic flux through a
surface

B1
A1
^
n

B2
^n
A2
Flat surface, B uniform on each part:


Φ = ( B1 ⋅ n̂)A1 + ( B2 ⋅ n̂)A2
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“Magnetic flux through a loop” =
Magnetic flux through a surface for
which the loop is a boundary
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clicker
In the green square, B = (3.0 T) khat.
Otherwise B = 0.
What is the magnetic flux through the
black square loop?
1.0 m
2.0 m
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(a)  zero
(b)  +3.0 Wb
(c)  -3.0 Wb
(d)  +12 Wb
(e)  -12 Wb
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Current flowing in a loop produces a magnetic field
and a magnetic flux through the loop
i
Magnetic field at the center
due to counterclockwise i is
out of the page
Magnetic flux is positive
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Current flowing in a loop produces a magnetic field
and a magnetic flux through the loop
i
Magnetic field at the center
due to clockwise i is
into the page
Magnetic flux is negative
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resistance R
“induction”
Changing magnetic field “induces” a current in a wire loop
magnitude? clockwise/counterclockwise?
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Changing flux Φ(t)
Current flows in loop as
if a battery were included
with emf = - dΦ/dt
i = |dΦ/dt|/R
What is the minus sign?
clockwise/counterclockwise
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Changing flux Φ(t)
i = |dΦ/dt|/R
Lenz’s law:
The induced current
produces a magnetic flux
that OPPOSES THE
CHANGE
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Changing flux Φ(t)
Current flows in loop as
if a battery were included
with emf = - dΦ/dt
i = (dΦ/dt)/R
What is the minus sign?
clockwise/counterclockwise
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Simplest way to change the flux
Change the magnetic field
example 1: loop inside a solenoid with changing I
Uniform field inside solenoid
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clicker
i
wire of resistance R
made into a loop
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If B is out of the page and
the current flows clockwise,
what is the direction of the
force on a piece of the wire?
(a)  towards center
(b)  away from the center
(c)  out of the screen
(d) into the screen
(e) The force is zero
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Answer with cool demo:
What happens to the aluminum can?
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clicker
i
wire of resistance R
made into a loop
4/9/14
If B is out of the page and
the current flows clockwise,
what is the direction of the
force on a piece of the wire?
(a)  towards center
(b)  away from the center
(c)  out of the screen
(d) into the screen
(e) The force is zero
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OTHER WAYS TO CHANGE THE MAGNETIC FLUX
THROUGH A LOOP
Rotate the loop in a uniform field:
angle between B and n changes, so Φ changes
Move the loop into or out of a region with B field
Change the size of the loop in a uniform field
Examples:
pull the loop closed
a moving metal rod on U-shaped wire
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Move the loop into or out of a region with B field
Magnetic force on
induced current in loop
opposite to direction of
motion
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EDDY CURRENTS AND FORCES
Piece of metal moves in B field
Currents induced are called “eddy currents”
Force on induced currents opposite to the direction of motion
Lower resistivity means larger currents, larger forces
DEMOS
•  ROLLING DISKS
•  Magnet rolling down the ramp
•  Magnet falling down the copper tube
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