Chapter 30 Electromagnetic Induction
Chapter 29 – Electromagnetic Induction
I.Intro - we know that electric currents will produce magnetic fields - is it possible that magnetic fields produce currents?
A.Demonstrate the existence of currents due to the motion of magnets or due to changing magnetic fields – note the direction of the induced currents.
1.wire moved through a magnetic field
2.loops of wire moved through a magnetic field
3.bar magnet moved through a loop
4.bar magnet moved through many loops
5.current turned on and off in one set of loops magnetically connected to a second set of loops
B.The currents that are produced are called "induced currents" that are produced by "induced emf's."
C.In all the above cases, there is either relative motion between charges and the magnetic field, or there is a change in the magnetic field.
Below is an example of a "induced emf," where a magnetic field is present, and there is movement (“motion”) through the field.
Take a metal rod and move it with a velocity v through a magnetic field. Examine the forces on the charges inside the rod as the rod is moved to the right (). Note that the force moves positive charges to the top of the rod and the negative charges to the bottom of the rod. This separation is like the separation of charges produced by the chemical reaction inside a battery. Thus, the resulting potential difference is referred to as an “emf.” In this case, an “induced emf.”
D.Both Michael Faraday and Joseph Henry analyzed situations where induced emf’s were produced and found that the "induced emf" depended on the rate at which the magnetic flux changed.
Digression: Remember that the magnetic flux is a measure of the number of lines of force of the magnetic field that pass through an area. Mathematically, it is written as:
for flux through an area ,
for flux through the total area,
for flux through the area A when B is constant.
The induced emf can be written as:
for induced currents in N loops:
Notice that the emf is produced by changes in : B, A, the angle between B and A, or any combination.
E.The negative sign is merely an indication of the direction in which the "induced current" flows. Lenz's Law gives the flow direction as follows:
The induced current is in such a direction that it produces a magnetic field that opposes the change that is occurring in the original magnetic field.
Examples:
1.Find the direction of the induced current in a loop of wire due to changes in the magnetic field shown.
a.
b.
2.A bar magnet is moved in and out of a solenoid.
3.A loop of wire surrounds a solenoid where the current in the solenoid is increasing or decreasing.
4.The planes of two loops are parallel and switch S is closed or opened.
II.Motional emf's are produced when a conductor moves in a magnetic field.
A.Find the induced emf (the voltage) in a rod of length L moving through a uniform magnetic field, B = constant.
What if = 90o?
B.Generalize for a small segment of a conductor, . This is your starting point if the magnetic field varies and/or the conductor is not straight.
C.Rotational motion changing an area.
DExamples:
1.A rod of length L slides without friction on two parallel conducting rails. Find the emf, the current through the resistor, the force needed to keep the rod moving at a constant velocity, the mechanical power required to move the rod at the velocity v, and the electric power dissipated in the resistor.
2.Find the emf in a rod of length L rotating around one end with an angular velocity .
3.Find the emf in an AC (alternating current) generator. Assume the magnetic field is uniform between the poles of the magnet, the area of the rotating loop is A, and the wires of the loop have resistance R.
4.A circular disk rotates with one part of it moving through a magnetic field - eddy currents.
5.Demo - drop magnets through a tube. What happens? Why?
III.Induced emf’s due to electric fields produced by a changing magnetic field strength.
A.A circular loop of wire is placed in a uniform magnetic field that is changing with time.
Suppose , find the induced emf in the loop. Take the radius of the loop to be R.
What’s causing the charges to flow (move) in the wire? Could it be an E-field? Could a changing magnetic flux produce an E-filed?
Where is the induced emf in the above loop? It is not localized but is distributed uniformly in the wire, that is, each segment of the wire L will have an emf . The total emf around the entire loop is or = V We know
,
and around a complete loop
= V = .
Also, from Faraday's Law = - . So finally
,
or,
This says that a changing B-field (or magnetic flux) produces an E-field, and it is this E-field that drives the charges around the loop. (The electric field exists whether or not a loop of wire is present, and is generally referred to as a nonelectrostatic or nonconservative E-field.)
B.Example: A magnetic field is confined to a circular region of radius b = 0.25 m. A circular loop of wire of radius a = 0.1 m is centered in the magnetic field, and the resistance of the wire loop is 3 ohms. The magnetic field is increasing at the rate of 0.2 T/s.
a)Draw the direction of the electric field at various places both inside and outside the region where the magnetic field exists.
b)Find the magnitude of the electric field, E, at radii r inside and outside the region where the magnetic field exists.
c)Find the emf in the loop of wire.
d)Find the current in the loop of wire.
e)Find the potential difference between two points on the loop.
f)If the loop of wire is cut so that there is a slight gap present, then what is the potential difference across the gap?
g)Move the ring out of the magnetic field. Is the emf still there even though the ring is not there?
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