Two questions: (1) How to find the force, F on the electric charge, Q excreted by the field E and/or B? (2) How fields E and/or B can be created? Maxwell’s equations Gauss’s law for electric field Electric charges create electric field: Gauss’s law for magnetic field Magnetic charges do not exist: For one not moving (v<<c) charge: Amperes law Electric current creates magnetic field: Faraday’s law A changing magnetic flux induces an EMF (As we will see later, this law should be extended) A changing magnetic field induces an electric field !

8. Electromagnetic induction (Faraday’s law) 2) EMF (review) 1) Flux (review) Units: (weber) [ΦB] = 1 Wb = 1 T·m2 3) Faraday’s law A changing magnetic flux induces an EMF A changing magnetic field induces an electric field ! This electric field is not a potential field. The field lines form a closed loops. - flux through a closed loop - EMF in the closed loop

4) Lenz’s law The direction of any magnetic induction effect is such as to oppose the cause of the effect For instance: a current produced by an induced emf moves in a direction so that its magnetic field opposes the original change in flux S S N N N N S S I I I I Example: If a North pole moves toward the loop in the plane of the page, in what direction is the induced current? Since the magnet is moving parallel to the loop, there is no magnetic flux through the loop. Thus the induced current is zero.

1) tilt the loop 2) change the loop area use thicker wires Example: In order to change the magnetic flux through the loop, what would you have to do? 1) drop the magnet move the magnet upwards move the magnet sideways all of the above only (1) and (2) Moving the magnet in any direction would change the magnetic field through the loop and thus the magnetic flux. 1) tilt the loop 2) change the loop area use thicker wires all of the above only (1) and (2) Since, changing the area or tilting the loop (which varies the projected area) would change the magnetic flux through the loop.

Example 1: A wire loop is being pulled away from a current-carrying wire. What is the direction of the induced current in the loop? 1) Clockwise 2) Counterclockwise 3) No induced current On the right side of the wire the magnetic flux is into the page and decreasing due to the fact that the loop is being pulled away. By Lenz’s Law, the induced B field will oppose this decrease. Thus, the new B field points into the page, which requires an induced clockwise current to produce such a B field. I Example 2: What is the induced current if the wire loop moves down? 1) Clockwise 2) Counterclockwise 3) No induced current The magnetic flux through the loop is not changing as it moves parallel to the wire. Therefore, there is no induced current. I

Example1: A wire loop is being pulled through a uniform magnetic field. What is the direction of the induced current? 1) Clockwise; 2) Counterclockwise; 3) No induced current x x x x x x x x x Since the magnetic field is uniform, the magnetic flux through the loop is not changing. Thus no current is induced. Example2: What is the direction of the induced current if the B field suddenly increases while the loop is in the region? 1) Clockwise 2) Counterclockwise 3) No induced current The increasing B field into the page must be countered by an induced flux out of the page. This can be accomplished by induced current in the counterclockwise direction in the wire loop. Example 3: A wire loop is being pulled through a uniform magnetic field that suddenly ends. What is the direction of the induced current? 1) Clockwise 2) Counterclockwise 3) No induced current x x x x x The B field into the page is disappearing in the loop, so it must be compensated by an induced flux also into the page. This can be accomplished by an induced current in the clockwise direction in the wire loop.

Example: If a coil is shrinking in a magnetic field pointing into the page, in what direction is the induced current? 1) Clockwise 2) Counterclockwise 3) No induced current The magnetic flux through the loop is decreasing, so the induced B field must try to reinforce it and therefore points in the same direction — into the page. According to the right-hand rule, an induced clockwise current will generate a magnetic field into the page. Example: If a coil is rotated as shown, in a magnetic field pointing to the left, in what direction is the induced current? 1) Clockwise 2) Counterclockwise 3) No induced current As the coil is rotated into the B field, the magnetic flux through it increases. According to Lenz’s Law, the induced B field has to oppose this increase, thus the new B field points to the right. An induced counterclockwise current produces just such a B field.

Example: Wire #1 (length L) forms a one-turn loop, & a bar magnet is dropped through. Wire #2 (length 2L) forms a two-turn loop, and the same magnet is dropped through. Compare the magnitude of the induced currents in these two cases. 1) I1 > I2 2) I1 < I2 3) I1 = I2  0 4) I1 = I2 = 0 N S 1 2 Induced emf is twice as large in the wire with 2 loops. The current is given by Ohm’s law: I = V/R. Since wire #2 is twice as long as wire #1, it has twice the resistance, so the current in both wires is the same. Example: A bar magnet is held above the floor and dropped. In 1, there is nothing between the magnet and the floor. In 2, the magnet falls through a copper loop. How will the magnet in case 2 fall in comparison to case 1? N S 2 1 1) it will fall slower; 2) it will fall faster; 3) it will fall the same When the magnet is falling from above the loop in 2, the induced current will produce a North pole on top of the loop, which repels the magnet. When the magnet is below the loop, the induced current will produce a North pole on the bottom of the loop, which attracts the South pole of the magnet.

Example 1: A coil of 600 turns with area 100 cm2 is placed in a uniform magnetic field. The angle between the direction of the field and the perpendicular to the loop is 60°. The field changes at the rate of 0.010 T/s. What is the magnitude of induced emf in the coil? Example 2:

Example 3: A 12.0-cm-diameter wire coil is initially oriented perpendicular to a 1.5 T magnetic field. The loop is rotated so that its plane is parallel to the field direction in 0.20 s. What is the average induced emf in the loop?

8a. Applications of Faraday’s law 1) Rotating loop: Example:

Example: A generator rotates at 60 Hz in a magnetic field of 0. 03 T Example: A generator rotates at 60 Hz in a magnetic field of 0.03 T. It has 1000 turns and produces voltage that is 120 V at a pick. What is the area of each turn of the coil?

8b. EMF induced in a moving conductor What is polarity of EMF? What would be the direction of the induced current, if rod slides on a conducting track? The B field points out of the page. The flux is increasing since the area is increasing. The induced B field opposes this change and therefore points into the page. Thus, the induced current runs clockwise according to the right-hand rule. Another method: FE B FB

Example: A uniform magnetic field B is perpendicular to the area bounded by the U-shaped conductor and a movable metal rod of length l. The rod is moving along the conductor at a speed v. The total resistance of the loop is R. What is the induced emf, the current in the loop, the magnetic force on the moving rod, and power needed to move the rod?

8c. Transformers

1 2 Example: What is the voltage across the lightbulb? The first transformer has a 2:1 ratio of turns, so the voltage doubles. But the second transformer has a 1:2 ratio, so the voltage is halved again. Therefore, the end result is the same as the original voltage. Example: Given that the intermediate current is 1 A, what is the current through the lightbulb? Power in = Power out 240 V  1 A = 120 V  ??? The unknown current is 2 A. Example: A 6 V battery is connected to one side of a transformer. Compared to the voltage drop across coil 1, the voltage across coil 2 is: Batteries provide DC current. Only a changing magnetic flux induces an EMF. Therefore, the voltage across coil 2 is zero. 1 2