Today’s agendum: Induced emf.

Today’s agendum: • Induced emf. • You must understand how changing magnetic flux can induce an emf, and be able to determine the direction of the induced emf. • Faraday’s “Law.” • You must be able to use Faraday’s “Law” to calculate the emf induced in a circuit. • Lenz’s “Law.” • You must be able to use Lenz’s “Law” to determine the direction induced current, and therefore induced emf. • Generators (part 1). • You must understand how generators work, and use Faraday’s “Law” to calculate numerical values of parameters associated with generators.

Induced emf and Faraday’s “Law” Magnetic Induction We have found that an electric current can give rise to a magnetic field… I wonder if a magnetic field can somehow give rise to an electric current…

It is observed experimentally that changes in magnetic flux induce an emf in a conductor. B An electric current is induced if there is a closed circuit (e.g., loop of wire) in the changing magnetic flux. I B A constant magnetic flux does not induce an emf—it takes a changing magnetic flux.

Passing the coil through the magnet would induce an emf in the coil. They need to calibrate their meter!

N S v move magnet toward coil Note that “change” may or may not not require observable (to you) motion.  A magnet may move through a loop of wire, or a loop of wire may be moved through a magnetic field (as suggested in the previous slide). These involve observable motion.  A magnet may move through a loop of wire I region of magnetic field this part of the loop is closest to your eyes change area of loop inside magnetic field N S rotate coil in magnetic field

changing I induced I changing B  A changing current in a loop of wire gives rise to a changing magnetic field (predicted by Ampere’s “Law”) which can induce a current in another nearby loop of wire. In this case, nothing observable (to your eye) is moving, although, of course microscopically, electrons are in motion. Induced emf is produced by a changing magnetic flux.

We can quantify the induced emf described qualitatively in the last few slides by using magnetic flux. Experimentally, if the flux through N loops of wire changes by dB in a time dt, the induced emf is Faraday’s “Law” of Magnetic Induction Faraday’s “Law” of Induction is one of the fundamental laws of electricity and magnetism. I wonder why the – sign… Your text, pages 997-998, shows how to determine the direction of the induced emf. Argh! Lenz’s Law, coming soon, is much easier.

Faraday’s “Law” of Magnetic Induction where the magnetic flux is This is another version of Faraday’s “Law”: We’ll use this version in a later lecture. In a future lecture, we’ll work with Web page with pictures of a whole bunch of applications:http://sol.sci.uop.edu/~jfalward/electromagneticinduction/electromagneticinduction.html

N S I v + - Example: move a magnet towards a coil of wire. N=5 turns A=0.002 m2

Homework hint: if B varies but loop  B. Ways to induce an emf:  change B  change the area of the loop in the field

=90 =45 =0 Ways to induce an emf (continued):  change the orientation of the loop in the field

.01 T =1 s Example: a uniform (but time-varying) magnetic field passes through a circular coil whose normal is parallel to the magnetic field. The coil’s area is 10-2 m2 and it has a resistance of 1 m. B varies with time as shown in the graph. Plot the current in the coil as a function of time. For 0 < t < 3: For 3 < t < 5:

Example: a uniform (but time-varying) magnetic field passes through a circular coil whose normal is parallel to the magnetic field. The coil’s area is 10-2 m2 and it has a resistance of 1 m. B varies with time as shown in the graph. Plot the current in the coil as a function of time. For 5 < t < 11: I(t) .01 T =1 s -.0333 A

N S I v + - Experimentally… Lenz’s law—An induced emf always gives rise to a current whose magnetic field opposes the change in flux.* If Lenz’s law were not true—if there were a + sign in Faraday’s law—then a changing magnetic field would produce a current, which would further increase the magnetic field, further increasing the current, making the magnetic field still bigger… *Think of the current resulting from the induced emf as “trying” to maintain the status quo—to prevent change.

…violating conservation of energy and ripping apart the very fabric of the universe…

Practice with Lenz’s Law. In which direction is the current induced in the coil for each situation shown? (counterclockwise) (no current)

(counterclockwise) (clockwise)

Rotating the coil about the vertical diameter by pulling the left side toward the reader and pushing the right side away from the reader in a magnetic field that points from right to left in the plane of the page. (counterclockwise) Remember Now that you are experts on the application of Lenz’s “Law”, remember this: You can use Faraday’s “Law” to calculate the magnitude of the emf (or whatever the problem wants. Then use Lenz’s “Law” to figure out the direction of the induced current (or the direction of whatever the problem wants).

Today’s agendum: • Induced emf. • You must understand how changing magnetic flux can induce an emf, and be able to determine the direction of the induced emf. • Faraday’s “Law.” • You must be able to use Faraday’s “Law” to calculate the emf induced in a circuit. • Lenz’s “Law.” • You must be able to use Lenz’s “Law” to determine the direction induced current, and therefore induced emf. • Generators. • You must understand how generators work, and use Faraday’s “Law” to calculate numerical values of parameters associated with generators.

B A  side view Motional emf: an overview An emf is induced in a conductor moving in a magnetic field. Your text introduces four ways of producing motional emf. We will cover the first two in this lecture. 1. Flux change through a conducting loop produces an emf:rotating loop. start with this derive these

B B                                                             2. Flux change through a conducting loop produces an emf:expanding loop. v start with these ℓ dA x=vdt derive these

v B B             –                         ℓ                         + Next time we will look at two more examples of motional emf… 3. Conductor moving in a magnetic field experiences an emf: magnetic force on charged particles. start with these derive this You could also solve this using Faraday’s” Law” by constructing a “virtual” circuit using “virtual” conductors.

                                                                                                    4. Flux change through a conducting loop produces an emf:moving loop. start with this derive these

B A  side view S N Generators and Motors: a basic introduction Take a loop of wire in a magnetic field and rotate it with an angular speed . Choose 0=0. Then Generators are an application of motional emf.

B A  side view If there are N loops in the coil The NBA equation! || is maximum when  = t = 90° or 270°; i.e., when B is zero. The rate at which the magnetic flux is changing is then maximum. On the other hand,  is zero when the magnetic flux is maximum.

emf, current and power from a generator

Example: the armature of a 60 Hz ac generator rotates in a 0.15 T magnetic field. If the area of the coil is 2x10-2 m2, how many loops must the coil contain if the peak output is to be max = 170 V?

B B                                                             Another Kind of Generator: A Slidewire Generator Recall that one of the ways to induce an emf is to change the area of the loop in the magnetic field. Let’s see how this works. v A U-shaped conductor and a moveable conducting rod are placed in a magnetic field, as shown. ℓ The rod moves to the right with a constant speed v for a time dt. dA vdt x The rod moves a distance vdt and the area of the loop inside the magnetic field increases by an amount dA = ℓ vdt .

v B B                         ℓ                         dA             vdt The loop is perpendicular to the magnetic field, so the magnetic flux through the loop is . The emf induced in the conductor can be calculated using Faraday’s law: x B and v are vector magnitudes, so they are always +. Wire length is always +. You use Lenz’s law to get the direction of the current.

B                               Direction of current? The induced emf causes current to flow in the loop. Magnetic flux inside the loop increases (more area). v System “wants” to make the flux stay the same, so the current gives rise to a field inside the loop into the plane of the paper (to counteract the “extra” flux). ℓ I dA vdt x Clockwise current!

As the bar moves through the magnetic field, it “feels” a force v B       A constant pulling force, equal in magnitude and opposite in direction, must be applied to keep the bar moving with a constant velocity. FM FP             ℓ             I x

Power and current. If the loop has resistance R, the current is v B       And the power is             ℓ             I (as expected). x Mechanical energy (from the pulling force) has been converted into electrical energy, and the electrical energy is then dissipated by the resistance of the wire.

Today’s agendum: • Motional emf. • You must be able to apply Faraday’s and Lenz’s “Laws” to calculate motional emf, as well as current and power in circuits powered by motional emf. • Motors and Generators (part 2). • In the previous lecture, we used Faraday’s “Law” to calculate numerical values of parameters associated with generators. You must also understand conceptually how motors and generators work. • Back emf. • You must be able to use Lenz’s “Law” to explain back emf.

B A  side view Motional emf An emf is induced in a conductor moving in a magnetic field. In the last lecture you learned about two examples of motional emf. 1. Flux change through a conducting loop produces an emf:rotating loop. start with this derive these

B B                                                             2. Flux change through a conducting loop produces an emf:expanding loop. v start with these ℓ dA x=vdt derive these

v B B             –                         ℓ                         + Today we’ll look at two more examples of motional emf. 3. Conductor moving in a magnetic field experiences an emf: magnetic force on charged particles. start with these derive this You could also solve this using Faraday’s “Law” by constructing a “virtual” circuit using “virtual” conductors.

                                                                                                    4. Flux change through a conducting loop produces an emf:moving loop. start with this derive these

Example 3 of motional emf: moving conductor in B field. Example 4 of motional emf: flux change through conducting loop. (Entire loop is moving.) Let’s work out these two examples now. Remember, it’s the flux change that produces the emf. Flux has no direction associated with it. However, the presence of flux is due to the presence of a magnetic field, which does have a direction, and allows us to use Lenz’s “Law” to determine the “direction” of current and emf.

Example 3 of motional emf: moving conductor in B field. Motional emf is the emf induced in a conductor moving in a magnetic field. “up” v B B If a conductor (purple bar) moves with speed v in a magnetic field, the electrons in the bar experience a force             –                         ℓ                         + The force on the electrons is “up,” so the “top” end of the bar acquires a net – charge and the “bottom” end of the bar acquires a net + charge.

The separated charges in the bar produce an electric field pointing “up” the bar. The emf across the length of the bar is “up” The electric field exerts a “downward” force on the electrons: v B B             –                         ℓ An equilibrium condition is reached, where the magnetic and electric forces are equal in magnitude and opposite in direction.                         +

Example 4 of motional emf: flux change through conducting loop. (Entire loop is moving.) I’ll include some numbers with this example.

                                                                                                    A square coil of side 5 cm contains 100 loops and is positioned perpendicular to a uniform 0.6 T magnetic field. It is quickly and uniformly pulled from the field (moving  to B) to a region where the field drops abruptly to zero. It takes 0.10 s to remove the coil, whose resistance is 100 . B = 0.6 T 5 cm

                                                                                                    (a) Find the change in flux through the coil. B = Bf - Bi = 0 - BA = -(0.6 T)(0.05 m)2 = -1.5x10-3 Wb.

                                                                                                    (b) Find the current and emf induced. Current will begin to flow when the coil starts to exit the magnetic field. Because of the resistance of the coil, the current will eventually stop flowing after the coil has left the magnetic field.  final initial The current must flow clockwise to induce an “inward” magnetic field (which replaces the “removed” magnetic field).

                                          v The induced emf is A x

“uniformly” pulled The induced current is

(c) How much energy is dissipated in the coil? Current flows “only*” during the time flux changes. W = P t = I2R t = (1.5x10-2 A)2 (100 ) (0.1 s) = 2.25x10-3 J (d) Discuss the forces involved in this example. The loop has to be “pulled” out of the magnetic field, so there is a pulling force, which does work. The “pulling” force is opposed by a magnetic force on the current flowing in the wire. If the loop is pulled “uniformly” out of the magnetic field (no acceleration) the pulling and magnetic forces are equal in magnitude and opposite in direction. *Remember: if there were no resistance in the loop, the current would flow indefinitely. However, the resistance quickly halts the flow of current once the magnetic flux stops changing.

Today’s agendum: Induced emf.