2 Expectations After this chapter, students will: Calculate the EMF resulting from the motion of conductors in a magnetic fieldUnderstand the concept of magnetic flux, and calculate the value of a magnetic fluxUnderstand and apply Faraday’s Law of electromagnetic inductionUnderstand and apply Lenz’s Law
3 Expectations After this chapter, students will: Apply Faraday’s and Lenz’s Laws to some particular devices:Electric generatorsElectrical transformersCalculate the mutual inductance of two conducting coilsCalculate the self-inductance of a conducting coil
4 Motional EMFA wire passes through a uniform magnetic field. The length of the wire, the magnetic field, and the velocity of the wire are all perpendicular to one another:
5 Motional EMFA positive charge in the wire experiences a magnetic force, directed upward:
6 Motional EMFA negative charge in the wire experiences the same magnetic force, but directed downward:These forces tend to separate the charges.
7 Motional EMFThe separation of the charges produces an electric field, E. It exerts an attractive force on the charges:E
8 Motional EMFIn the steady state (at equilibrium), the magnitudes of the magnetic force – separating the charges – and the Coulomb force – attracting them – are equal.E
9 Motional EMF Rewrite the electric field as a potential gradient: Substitute this result back into our earlier equation:E
10 Motional EMFSubstitute this result back into our earlier equation:E
11 Motional EMFThis is called motional EMF. It results from the constant velocity of the wire through the magnetic field, B.E
12 Motional EMFNow, our moving wire slides over two other wires, forming a circuit. A current will flow, and power is dissipated in the resistive load:
13 Motional EMF Where does this power come from? Consider the magnetic force acting on thecurrent in the slidingwire:
14 Motional EMFRight-hand rule #1 shows that this force opposes the motion of the wire. To move the wire at constant velocity requires an equal and opposite force.That force does work:The power:
15 Motional EMF The force’s magnitude was calculated as: Substituting: which is the same as thepower dissipated electrically.
16 Motional EMFSuppose that, instead of being perpendicular to the plane of the sliding-wire circuit, the magnetic field had made an angle f with the perpendicular to that plane.The perpendicularcomponent of B: B cos f
17 Motional EMFThe motional EMF:Rewrite the velocity:Substitute:
18 Motional EMFL Dx is simply the change in the loop area.
19 Motional EMF Define a quantity F : Then: F is called magnetic flux. SI units: T·m2 = Wb (Weber)
20 Magnetic Flux Wilhelm Eduard Weber 1804 – 1891 German physicist and mathematician
21 Faraday’s LawIn our previous result, we said that the induced EMF was equal to the time rate of change of magnetic flux through a conducting loop. This, rewritten slightly, is called Faraday’s Law:Why the minus sign?
22 Faraday’s Law Michael Faraday 1791 – 1867 English physicist and mathematician
23 Faraday’s LawTo make Faraday’s Law complete, consider adding N conducting loops (a coil):What can change the magnetic flux?B could change, in magnitude or directionA could changef could change (the coil could rotate)
24 Lenz’s Law Here is where we get the minus sign in Faraday’s Law: Lenz’s Law says that the direction of the induced current is always such as to oppose the change in magnetic flux that produced it.The minus sign in Faraday’s Law reminds us of that.
25 Lenz’s LawHeinrich Friedrich Emil Lenz1804 – 1865Russian physicist
26 Lenz’s LawLenz’s Law says that the direction of the induced current is always such as to oppose the change in magnetic flux that produced it.What does that mean?How can an induced current “oppose” a change in magnetic flux?
27 Lenz’s LawHow can an induced current “oppose” a change in magnetic flux?A changing magnetic flux induces a current.The induced current produces a magnetic field.The direction of the induced current determines the direction of the magnetic field it produces.The current will flow in the direction (remember right-hand rule #2) that produces a magnetic field that works against the original change in magnetic flux.
28 Faraday’s Law: the Generator A coil rotates with a constant angular speed in a magnetic field.but f changeswith time:
29 Faraday’s Law: the Generator So the flux also changes with time:Get the time rate of change (a calculus problem):Substitute into Faraday’s Law:
30 Faraday’s Law: the Generator The maximum voltage occurs when :What makes the voltage larger?more turns in the coila larger coil areaa stronger magnetic fielda larger angular speed
31 Back EMF in Electric Motors An electric motor also contains a coil rotating in a magnetic field.In accordance with Lenz’s Law, it generates a voltage, called the back EMF, that acts to oppose its motion.
32 Back EMF in Electric Motors Apply Kirchhoff’s loop rule:
33 Mutual Inductance A current in a coil produces a magnetic field. If the current changes, the magnetic field changes.Suppose another coil is nearby. Part of the magnetic field produced by the first coil occupies the inside of the second coil.
34 Mutual InductanceFaraday’s Law says that the changing magnetic flux in the second coil produces a voltage in that coil.The net flux in thesecondary:
35 Mutual InductanceConvert to an equation, using a constant of proportionality:
36 Mutual InductanceThe constant of proportionality is called the mutual inductance:
37 Mutual Inductance Substitute this into Faraday’s Law: SI units of mutual inductance: V·s / A = henry (H)
39 Self-InductanceChanging current in a primary coil induces a voltage in a secondary coil.Changing current in a coil also induces a voltage in that same coil.This is called self-inductance.
40 Self-InductanceThe self-induced voltage is calculated from Faraday’s Law, just as we did the mutual inductance.The result:The self-inductance, L, of a coil is also measured in henries. It is usually just called the inductance.
41 Mutual Inductance: Transformers A transformer is two coils wound around a common iron core.
42 Mutual Inductance: Transformers The self-induced voltage in the primary is:Through mutual induction, and EMF appears in the secondary:Their ratio:
43 Mutual Inductance: Transformers This transformer equation is normally written:The principle of energy conservation requires that the power in both coils be equal (neglecting heating losses in the core).
44 Inductors and Stored Energy When current flows in an inductor, work has been done to create the magnetic field in the coil. As long as the current flows, energy is stored in that field, according to
45 Inductors and Stored Energy In general, a volume in which a magnetic field exists has an energy density (energy per unit volume) stored in the field: