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Magnetic flux

Magnetic flux is …

Magnetic flux

Magnetic flux is a measure of the number of magnetic field lines passing through something, such as a loop. If we define the area of the loop as a vector, with its direction perpendicular to the plane of the loop, the magnetic flux is given by:

where θ is the angle between the magnetic field and the area vector.

The unit of magnetic flux is the weber. 1 Wb = 1 T m2.

cosB BA

Magnetic flux

The more field lines pass through an area, the larger the flux.

Lots of flux. No flux.

Some flux.

cosB BA

Magnetic flux from a bar magnet

A bar magnet passes from right to left through a loop of wire. Flux is positive when the field lines pass one way through the loop, and negative when they go in the opposite direction. What, if anything, is wrong with the graph of magnetic flux for this situation?

1.There's nothing wrong

with it.

2. There is something

wrong with it (you

specify what it is)

Observations with a magnet, coil, and galvanometer

Let’s make a few observations, using a magnet near a coil connected to a galvanometer (a sensitive current meter). The magnet’s north pole points at the coil.

Situation Predict what the galvanometer needle does.

What the needle actually does.

The magnet is moved toward the coil.

No basis for prediction.

The magnet is moved away from the coil.

The magnet is moved toward the coil again, but more slowly.

The magnet is held at rest near the coil.

The magnet is held at rest and the coil is moved toward the magnet.






No basis for prediction. Needle deflects to the left.












Opposite to the above.











Opposite to the above. Needle deflects to the right.













Same as the first case, but needle deflects less.













Needle deflects a little bit to the left.















??














?? Nothing!














?? Nothing!


Same as the first case.













?? Nothing!


Same as the first case. Needle deflects to the left.


Let’s make a few more observations.



The magnet is reversed, so the south pole points at the coil. The magnet is moved toward the coil.


The magnet is taken to the other end of the coil, and the north pole is closer to the coil. The magnet is moved toward the coil.


The magnet is rotated back and forth near the end of the coil






Opposite to when the north pole was pointed at the coil.











The needle deflects to the right.













Opposite to the above.












Opposite to the above. The needle goes left.














Same as the south pole at the other end?
















Opposite to the previous.















Opposite to the previous. The needle goes left.

















The needle goes back and forth, too.
















The needle goes back and forth, too.

What we said.

Faraday’s LawFaraday's Law ties all of our observations into one neat package. It says that the voltage induced in a coil of N turns is given by the rate of change of the magnetic flux in the coil:

Faraday's Law:

This has an incredible number of practical applications, particularly in the generation and distribution of electricity.

We call the voltage induced by a changing magnetic flux an induced emf. This is because changing magnetic flux acts like a battery in a coil or loop, which is why there is a current when there's a complete circuit.

Nt

Graphs of flux and induced emfIn this situation, a bar magnet is moved at constant speed v toward a coil until the left end of the magnet is at the center of the coil. The magnet is held stationary for 3 seconds, and then moved to the right away from the coil with a speed v.

What does the graph of magnetic flux as a function of time look like? Assume positive flux corresponds to field lines passing through the coil to the left.

What does the graph of induced emf as a function of time look like?

Lenz’s LawExposing a coil or loop to a changing magnetic flux will generate a current if the circuit is complete. The direction of the current is given by Lenz's Law:

Lenz's Law: A changing magnetic flux induces an emf that produces a current which sets up a magnetic field that tends to oppose whatever produced the change.

Coils and loops don't like to change, and they will try to counteract any changes in magnetic flux imposed on them. They are not successful - the change can be made, but the coil or loop tries to oppose the change while the change is taking place. This tendency to oppose is why there is a minus sign in Faraday's Law.

A pictorial approach to Lenz’s LawAn easy way to approach Lenz’s Law situations, to figure out the direction of an induced current, it to draw a set of three pictures.

First, some review. In which direction is the magnetic field inside a loop if the loop has a counter-clockwise current? What if the current is clockwise?

A pictorial approach to Lenz’s LawExample: A wire loop in the plane of the page is in a uniform

magnetic field directed into the page. Over some time interval, the field is doubled. What direction is the induced current in the loop while the field is changing?

Step 1: Draw a “Before” picture, showing the field passing through the loop before the change takes place.

Step 2: Draw an “After” picture, showing the field passing through the loop after the change.

Step 3: Draw a “To Oppose” picture, showing the direction of the field the loop creates to oppose the change.

Step 4: Use the right-hand rule to determine which way the induced current goes in the loop to create that field.



Step 1: Draw a “Before” picture, showing the field passing through the loop before the change takes place.



Step 2: Draw an “After” picture, showing the field passing through the loop after the change.



Step 3: Draw a “To Oppose” picture, showing the direction of the field the loop creates to oppose the change.

Step 4: Use the right-hand rule to determine which way the induced current goes in the loop to create that field.

One field line is enough for the To Oppose picture - that's enough to determine the direction of the induced current.

Lenz’s Law example 1

A wire loop is located near a long straight current-carrying wire. The current in the wire is directed to the right.

With the current held constant in the long straight wire, the loop is moved up, away from the wire. In what direction is the induced current in the loop?

1. The induced current is clockwise.

2. The induced current is counter-clockwise.

3. There is no induced current.


The flux through the loop decreases, so the loop tries to add field lines that are directed out of the page to oppose the change. The induced current must go counter-clockwise to produced the required field.


With the current held constant in the long straight wire, the loop is moved parallel to the wire. In what direction is the induced current in the loop?





The flux through the loop is constant, so there is no change to oppose, and no induced current.


The loop is now placed directly on the wire with the wire bisecting the loop. If the current in the wire is increasing, in what direction is the induced current in

the loop?





The net flux through the loop is always zero, so there is no change to oppose, and no induced current.


A loop of wire, in the plane of the page, has an area of 0.5 m2 and a resistance of R = 0.1 Ω. There is a uniform magnetic field of B = 1.0 T passing through the loop into the page. In what direction is the induced current in the loop?





Nothing is changing, so there is no induced current.

Lenz’s Law example 4, continued

A loop of wire, in the plane of the page, has an area of 0.5 m2 and a resistance of R = 0.1 Ω. There is a uniform magnetic field of B = 1.0 T passing through the loop into the page. Now the magnetic field is reduced steadily from 1.0 T to 0 over a 10 second period. In what direction is the induced current in the loop?




Lenz’s Law example 4, continuedThe pictorial method tells us that the field from the loop must

be into the page, requiring a clockwise current.

Lenz’s Law example 4, continuedA loop of wire, in the plane of the page, has an area of 0.5 m2 and a resistance of R = 0.1 Ω. There is a uniform magnetic field of B = 1.0 T passing through the loop into the page. Now the magnetic field is reduced steadily from 1.0 T to 0 over a 10 second period. In what direction is the induced current in the loop?

What is the magnitude of the induced current in the loop?

Lenz’s Law example 4, continuedA = 0.5 m2 R = 0.1 Ω Bi = 1.0 T

Now the magnetic field is reduced steadily from 1.0 T to 0 over a 10 second period.

First, apply Faraday’s law to find the induced emf (voltage):

1.0 T0.1 T/s

10 s

B

t

2

( cos )cos

1 (0.5 m ) 1 ( 0.1 T/s) 0.05 V

BA BN N NA

t t t

Lenz’s Law example 4, continuedA = 0.5 m2 R = 0.1 Ω Bi = 1.0 T

Second, apply Ohm’s law to find the current:

0.05 V

0.05 V0.5 A

0.1 I

R

Lenz’s Law example 4, continuedIf we'd kept the magnetic field constant, what other ways could we have induced the same current (magnitude and direction) in the loop?

Lenz’s Law example 4, continuedIf we'd kept the magnetic field constant, what other ways could we have induced the same current (magnitude and direction) in the loop?

By changing the field we reduced the flux to zero over a 10 second period. Two other ways to do that are:

reduce the area to zero over a 10 second interval

rotate the loop by 90° over a 10 second interval

Lenz’s Law example 4, continuedWhat if, instead of reducing the field to zero in 10 seconds, we reduced it to zero in 2 seconds? Would anything change? Would anything stay the same?

Lenz’s Law example 4, continuedWhat if, instead of reducing the field to zero in 10 seconds, we reduced it to zero in 2 seconds? Would anything change? Would anything stay the same?

The current would still be clockwise, but it would be 5 times as large. So, we'd get five times the current for 1/5 of the time. The product of current and time would be the same, but that represents the total charge. The same total charge flows every time.

Graphs of flux and induced currentA square conducting loop moves at a constant velocity of 1 m/s to the right. At t = 0, its right side is at x = -6 m. A region of uniform magnetic field directed into the page is located between x = -4 m and x = 0. A second region of uniform magnetic field directed out of the page is located between x = 0 and x = 4 m. The field in the two regions has the same magnitude and the field is zero everywhere else. Assume the velocity of the loop is constant throughout its motion. Simulation

Graphs of flux and induced currentA square conducting loop moves at a constant velocity of 1 m/s to the right. At t = 0, its right side is at x = -6 m. A region of uniform magnetic field directed into the page is located between x = -4 m and x = 0. A second region of uniform magnetic field directed out of the page is located between x = 0 and x = 4 m. Plot graphs of the magnetic flux through the loop, and the induced current in the loop, as a function of time. Define into the page as positive for flux, and clockwise to be positive for current.

The flux graphRemember that flux is a measure of the number of field lines passing through the loop, with positive for, in this case, field lines into the page.

The graph of induced currentWhat is the connection between the graphs?

The graph of induced currentWhat is the connection between the graphs? The current graph is proportional to the negative slope of the flux graph.

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