1. Our goal for today
To go over the pictorial approach to Lenz’s law.

2. Lenz’s Law
Exposing 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.

3. drop a magnet through a copper tube
Lenz's Law is relevant when we drop a magnet through a copper tube.
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.
Different parts of the tube are exposed to a changing magnetic flux, because of the motion of the magnet.
Induced currents swirl around the tube, producing magnetic fields.
These magnetic fields oppose whatever produced the change in flux - they dramatically slow down the motion of the magnet.

4. A pictorial approach to Lenz’s Law
A nice 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?

5. A pictorial approach to Lenz’s Law
Example: 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.

6. A pictorial approach to Lenz’s Law
Example: 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.

7. A pictorial approach to Lenz’s Law
Example: 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 2: Draw an “After” picture, showing the field passing through the loop after the change.

8. A pictorial approach to Lenz’s Law
Example: 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 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.

9. 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.

10. Lenz’s Law example 1
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.

11. Lenz’s Law example 2
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?
1. The induced current is clockwise.
2. The induced current is counter-clockwise.
3. There is no induced current.

12. Lenz’s Law example 2
The flux through the loop is constant, so there is no change to oppose, and no induced current.

13. Lenz’s Law example 3
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?
1. The induced current is clockwise.
2. The induced current is counter-clockwise.
3. There is no induced current.

14. Lenz’s Law example 3
The net flux through the loop is always zero, so there is no change to oppose, and no induced current.

15. Lenz’s Law example 4
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?
1. The induced current is clockwise.
2. The induced current is counter-clockwise.
3. There is no induced current.

16. Lenz’s Law example 4
Nothing is changing, so there is no induced current.

17. 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?
1. The induced current is clockwise.
2. The induced current is counter-clockwise.
3. There is no induced current.

18. Lenz’s Law example 4, continued
The pictorial method tells us that the field from the loop must be into the page, requiring a clockwise current.

19. 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?
What is the magnitude of the induced current in the loop?

20. Lenz’s Law example 4, continued
A = 0.5 m 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):

21. Lenz’s Law example 4, continued
A = 0.5 m R = 0.1 Ω Bi = 1.0 T
Second, apply Ohm’s law to find the current:

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

23. Lenz’s Law example 4, continued
If 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

24. Lenz’s Law example 4, continued
What 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?

25. Lenz’s Law example 4, continued
What 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.