Handout 9: Electromagnetic Induction - Motional EMF, Inductance, and Energy Storage, Summaries of Law

Download Handout 9: Electromagnetic Induction - Motional EMF, Inductance, and Energy Storage and more Summaries Law in PDF only on Docsity! EMF Handout 9: Electromagnetic Induction 1 ELECTROMAGNETIC INDUCTION (Y&F Chapters 30, 31; Ohanian Chapter 32) This handout covers: • Motional emf and the electric generator • Electromagnetic Induction and Faraday’s Law • Lenz’s Law • Induced electric field • Inductance • Magnetic energy The Electric and magnetic fields are inter-related The electric and magnetic fields are not independent. In fact: • A changing magnetic field induces an electric field • The reverse is also true: a changing electric field induces a magnetic field To see how this occurs, we first consider MOTIONAL emf Motional emf Consider a wire moving with velocity v in a magnetic field . Assume for simplicity that and are perpendicular The electrons in the wire experience a force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B (out of page) v EMF Handout 9: Electromagnetic Induction 2 This produces a separation of charge with an excess of negative charge at one end and positive charge at the other. There is therefore an INDUCED emf between the two ends. Induced emf ℰ = work done to move a unit positive charge from a to b against the magnetic force F = QvB = vB Work done = (Force)(Distance) ⇒ ℰ = vBL MOTIONAL emf The electric generator Assume that the ends of the wire are connected up through some external circuit (which we represent here by a simple resistor). Current I flows and dissipates electrical power in R. Where does this power come from? The current is due to electrons moving in the wire with some drift velocity velectron. The electrons therefore experience a magnetic force This force OPPOSES the motion of the wire through the field. v x B v B (out) F = -e(v x B) - - - + + + a b L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B (out of page) v R I velectron v B (out) F = -e(velectron x B) EMF Handout 9: Electromagnetic Induction 5 Induced electric field Consider again a wire moving in a magnetic field. Look at the situation from the point of view of someone who moves along with the wire: v = 0 ⇒ there is no magnetic force Therefore, this observer interprets the force acting on the electrons in the wire as being due to an INDUCED ELECTRIC FIELD. ⇒ around the closed path shown ≠ 0. ⇒ The induced electric field is not conservative. Faraday's Law is usually written in this form: FARADAY'S LAW MAXWELL'S 3rd EQUATION In words: The line integral of the electric field around a closed path is equal to minus the rate of change of magnetic flux through the path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B (out of page) v L E EMF Handout 9: Electromagnetic Induction 6 Mutual inductance Faraday's Law ⇒ a changing → induced emf Consider two nearby circuits: Changing current in Circuit 1 Changing magnetic field through Circuit 2 Induced emf in Circuit 2 Current flows in Circuit 2 Let I1(t) create magnetic field B1(t). Let ψ21 be the flux through circuit 2 due to I1 Clearly, B1 ∝ I1, so ψ21 ∝ I1 DEFINITION: The constant of proportionality between ψ21 and I1 is called the MUTUAL INDUCTANCE: From Faraday’s Law, the induced emf is given by ℰ21 = -M21 Circuit 1 Circuit 2 B1 I1 EMF Handout 9: Electromagnetic Induction 7 Note: • M21 depends on the shape, size, numbers of turns and relative positions of the two circuits • M21 = M12 so we need only use M as the symbol for mutual inductance • Ohanian uses L for mutual inductance • In the SI system, inductance is measured in Henrys (H) 1 H = Inductance that produces an emf of 1 Volt for a rate of change of current of 1 A s-1 1 H ≡ 1 V s A-1 • The usual unit for µo is H m-1 Self inductance Even a single circuit produces a magnetic field that passes through the circuit itself. So, if I changes → changes → Ψ changes → induced emf Definition: SELF INDUCTANCE Inductance and Lenz’s law ℰ21 = -M or -L Recall: The negative sign represents Lenz’s Law: the emf causes current to flow so as to oppose the change in flux that produces it.