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Physics 6B
Magnetic Induction
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Before we can talk about induction we need to understand magnetic flux.
You can think of flux as the number of field lines passing through an area.
Here is the formula:
flux  B  B  A  cos( )
Note that there are 3 parts to this formula: magnetic field strength, Area, and angle.
We will learn that magnetic induction occurs if any of these change.
In fact, the rate of change of flux is the induced voltage:
Here is the formula:
Vinduced 
B
t
The negative sign helps to remind us of what induction does –
induction always opposes the change in the flux. We call this Lenz’s Law.
Next we will look at a few specific examples of a changing magnetic flux:
Prepared by Vince Zaccone
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1) A loop of wire entering a uniform magnetic field.
X
X
X
X

B
X
X
X
X
X
X
X
X
v
X
X
X
X
X
X
X
X
X
X
X
X
Question: which direction is the induced current in the loop?
Prepared by Vince Zaccone
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1) A loop of wire entering a uniform magnetic field.
X
X
X
X

B
X
X
X
X
X
X
X
X
v
I induced
B induced
X
X
X
X
X
X
X
X
X
X
X
X
As the loop enters the B-field, the flux increases into the page, so there will be a current induced in the loop to
oppose this increasing flux. The direction of the induced current is Counter-Clockwise, because that direction will
create an induced B-field through the loop, out of the page (opposing the increase into the page).
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1) A loop of wire entering a uniform magnetic field.
X
X
X
X

B
X
X
X
X
X
X
X
X
v
I induced
B induced
X
X
X
X
X
X
X
X
X
X
X
X
You might also notice that there will be a net force on the loop to the right, opposing its motion (try the right hand
rule on each side of the loop). Induction opposes the change by trying to push the loop back out of the B-field.
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2) A loop of wire rotating in a uniform magnetic field.
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X

B
Question: which direction is the induced current in the loop?
Prepared by Vince Zaccone
For Campus Learning
Assistance Services at UCSB
2) A loop of wire rotating in a uniform magnetic field.
X
X
X
X

B
X
X
X
X
X
X
X
X
X B induced
X
X
X
X
X
X
X
X
I induced
X
X
X
X
As shown, the loop is intercepting the maximum flux. As it rotates, the flux will be decreasing, so there will be an
induced current to oppose this change. The direction of the current will be clockwise, which creates an induced
magnetic field through the loop, into the page.
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3) A stationary loop of wire in a magnetic field that is increasing in strength.
X
X

B(increase)
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Question: which direction is the induced current in the loop?
Prepared by Vince Zaccone
For Campus Learning
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3) A stationary loop of wire in a magnetic field that is increasing in strength.
X
X
X

B(increase)
X
X
I induced
X
X
X
X
X
X
X
B induced
X
X
X
X
X
X
X
X
X
X
X
X
The increasing B-field causes the flux to increase, inducing current in the loop. The direction of the current is
Counter-Clockwise, creating an induced B-field out of the page through the loop, opposing the change in the flux.
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4) A bar magnet approaching a stationary loop of wire.
N
S
Question: which direction is the induced current in the loop, as viewed from the right?
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4) A bar magnet approaching a stationary loop of wire.
Iinduced
N
S

Bmagnet

Binduced
View from the right
X
X

Bmagnet
X
X

Binduced
X
X
I induced
When viewed from the right, this looks like a circular loop with an increasing (because the magnet is approaching)
B-field directed through the loop, to the left. The induced current will oppose this increasing field by creating some
flux in the opposite direction (to the right, or out of the page in the view from the right). This means the induced
current must be Counter-Clockwise.
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5) A rotating loop of wire near a stationary bar magnet.
N
S
Question: which direction is the induced current in the loop, as viewed from the right?
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5) A rotating loop of wire near a stationary bar magnet.
View from the right
N
S

Bmagnet

Binduced
X
X
Iinduced

Bmagnet
X
X X
Binduced
X
X
Iinduced
As shown, the loop is intercepting maximum flux, so when it rotates, the flux decreases. Induction will create
some flux through the loop to the right, to oppose the change. This means the induced current is Clockwise.
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6) A loop of wire approaching a straight wire with a steady current.
Current = I (steady)
Question: which direction is the induced current in the loop?
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6) A loop of wire approaching a straight wire with a steady current.
Iinduced
X
B wire
X
X
Binduced
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Current = I (steady)
In the vicinity of the loop, the wire creates a magnetic field directed into the page. As the loop approaches, the
field gets stronger, so the flux is increasing. To oppose this, the induced current must be Counter-Clockwise to
create some induced flux out of the page.
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7) A stationary loop of wire near a straight wire with decreasing current.
Current = I (decreasing)
Question: which direction is the induced current in the loop?
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7) A stationary loop of wire near a straight wire with decreasing current.
Iinduced
X
B wire
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X

B
X induced
Current = I (decreasing)
In the vicinity of the loop, the wire creates a magnetic field directed into the page. As the current decreases, the
field gets weaker, so the flux is decreasing. To oppose this, the induced current must be Clockwise to create
some induced flux into the page.
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Here’s one where we can calculate the effect of the induction:
Suppose there is a uniform B-Field pointing into the page. A circuit is placed in the field so that it intercepts the
maximum magnetic flux. One side of the rectangular circuit is free to slide back and forth along the rails. Let’s
first try to figure out what happens qualitatively:
R
V
Prepared by Vince Zaccone
For Campus Learning
Assistance Services at UCSB
Here’s one where we can calculate the effect of the induction:
Suppose there is a uniform B-Field pointing into the page. A circuit is placed in the field so that it intercepts the
maximum magnetic flux. One side of the rectangular circuit is free to slide back and forth along the rails. Let’s
first try to figure out what happens qualitatively:
Iinitial
R
V
First we can find the direction of the current in the given loop. Current will flow clockwise. In the diagram it is
labeled Iinitial.
Prepared by Vince Zaccone
For Campus Learning
Assistance Services at UCSB
Here’s one where we can calculate the effect of the induction:
Suppose there is a uniform B-Field pointing into the page. A circuit is placed in the field so that it intercepts the
maximum magnetic flux. One side of the rectangular circuit is free to slide back and forth along the rails. Let’s
first try to figure out what happens qualitatively:
Iinitial
R
Fmag
V
First we can find the direction of the current in the given loop. Current will flow clockwise. In the diagram it is
labeled Iinitial.
The force on the sliding wire can also be found from the right hand rule for magnetic forces. This force points to
Prepared by Vince Zaccone
the right. So the wire will slide to the right.
For Campus Learning
Assistance Services at UCSB
Here’s one where we can calculate the effect of the induction:
Suppose there is a uniform B-Field pointing into the page. A circuit is placed in the field so that it intercepts the
maximum magnetic flux. One side of the rectangular circuit is free to slide back and forth along the rails. Let’s
first try to figure out what happens qualitatively:
Iinitial
R
Fmag
V
Now that the wire is moving, we can talk about the induction effect (remember, induction only happens when the
flux is changing). The magnetic flux through the loop is increasing into the page, and there will be an induced
current in the wire that opposes this change.
Prepared by Vince Zaccone
For Campus Learning
Assistance Services at UCSB
Here’s one where we can calculate the effect of the induction:
Suppose there is a uniform B-Field pointing into the page. A circuit is placed in the field so that it intercepts the
maximum magnetic flux. One side of the rectangular circuit is free to slide back and forth along the rails. Let’s
first try to figure out what happens qualitatively:
Iinitial
R
V
Fmag
Iinduced
Now that the wire is moving, we can talk about the induction effect (remember, induction only happens when the
flux is changing). The magnetic flux through the loop is increasing into the page, and there will be an induced
current in the wire that opposes this change.
This induced current will flow counter-clockwise, creating some induced flux to oppose the increasing flux.
Prepared by Vince Zaccone
For Campus Learning
Assistance Services at UCSB
Here’s one where we can calculate the effect of the induction:
Suppose there is a uniform B-Field pointing into the page. A circuit is placed in the field so that it intercepts the
maximum magnetic flux. One side of the rectangular circuit is free to slide back and forth along the rails. Let’s
first try to figure out what happens qualitatively:
Iinitial
R
V
Finduced
Fmag
Iinduced
Now that the wire is moving, we can talk about the induction effect (remember, induction only happens when the
flux is changing). The magnetic flux through the loop is increasing into the page, and there will be an induced
current in the wire that opposes this change.
This induced current will flow counter-clockwise, creating some induced flux to oppose the increasing flux.
Notice – this also creates an induced force on the wire to the left.
Prepared by Vince Zaccone
For Campus Learning
Assistance Services at UCSB
Here’s one where we can calculate the effect of the induction:
Suppose there is a uniform B-Field pointing into the page. A circuit is placed in the field so that it intercepts the
maximum magnetic flux. One side of the rectangular circuit is free to slide back and forth along the rails. Let’s
first try to figure out what happens qualitatively:
Iinitial
R
V
Finduced
Fmag
Iinduced
So the wire initially accelerates to the right, but the faster it moves, the stronger the induction, and the closer it
gets to some maximum speed (where the induced force will match up with the initial Fmag. We can try to calculate
this maximum speed.
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x
L
Iinitial
R
V
Finduced
Fmag
Iinduced
To get the max speed, we will have to use our formula for the induced voltage. As soon as the induced voltage
matches up with the battery voltage we will be at equilibrium, and the wire will be moving at maximum speed.
Vinduced 
B (B  A  cos )

t
t
 x 
Vinduced  B  L     B  L  v
 t 
B  L  vmax  Vbattery
vmax 
Vbattery
B L
In this formula, the only thing changing is the horizontal distance
which is labeled x in the diagram above. If we replace the Area
with L*x and realize that we can ignore the angle (90 degrees), we
get a simpler formula, involving the velocity of the wire.
Setting this induced voltage equal to the battery voltage, we will get
the maximum speed. Another option would be to set Finduced = Fmag.
Should get the same result, after using Ohm’s Law.
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