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FI 2201 Electromagnetism Alexander A. Iskandar, Ph.D. Physics of Magnetism and Photonics Research Group Electrodynamics ELECTROMOTIVE FORCE AND FARADAY’S LAW 1 Ohm’s Law • To make a current flow, we have to push the charges. How fast they move depends on the nature of the material. r • For most substances, the current density is proportional J r to the force per unit charge, f : r r J =σ f the proportionality “constant” σ is a second rank tensor, as are the susceptibilities, but many common media are “linear” in the sense that the conductivity σ can be considered a scalar scalar. • A perfect conductor has σ → ∞, and by contrast, a resistor has small conductivity. • The reciprocal of conductivity is called resistivity, ρ = 1 σ , which is a characteristics of the material. Alexander A. Iskandar Electromagnetism 3 Ohm’s Law • In principle, the force could be anything (chemical etc), but we concentrate on the electromagnetic force ( r r r r F = q E+v×B ) → r r r r f = E+v×B • In real substance, the velocity is very small, thus the second term above is neglected. Hence, we have r r J =σ E • This relation is called Ohm’s Law. • There is no contradiction with the fact that inside a conductor d t th the electric l t i fi field ld iis zero, since i iin an electrical l ti l circuit, the wires are made of good conductors, thus r r J E = →0 σ • Example 7.2, Problem 7.3 Alexander A. Iskandar Electromagnetism 4 2 Ohm’s Law • Ohm’s law should strike you as strange, at first, because you also know that r r r J =ρv r r • Suppose r J is constant. Then v and E should be constant. r r But if E is constant, charges should accelerate at a = qE m , r rendering it impossible to have a constant v . So which relation is wrong? • What happens is that collisions between free charges in a current (like electrons) with fixed or slower moving charges (like the ions the electrons leave behind), and other free current carriers, keep the acceleration from going on for very long. Alexander A. Iskandar Electromagnetism 5 Ohm’s Law • In collisions with the ions, the kinetic energy gained by the free carriers from the field is largely transferred to the ions, and the electron starts over. • We therefore reconcile Ohm’s Law with the definition of current density by supposing that the collision process r results in a well-defined average velocity, v , also called the drift velocity which is a constant, and write r r J =ρv • In fact fact, we don’t don t need to suppose; we can show that that’s that s the way it is, in a crude classical model of what is mostly a quantum phenomenon. Alexander A. Iskandar Electromagnetism 6 3 Electromotive Force (emf) • Note that in a typical electric circuit (for example, a light bulb) connected to a battery, the current is the same all the way around the loop. • Why is it constant around the loop ? r • Recall that the only driving force on the charges, f s , is confined on the source (battery). • Suppose that there is an accumulation of charges on some part of the wire, such that the current is not constant This accumulation of charges will create an constant. electrostatic field that will smooth out the flow of charges. • Thus, we can write r r r f = f s + Eelectrostatic Alexander A. Iskandar Electromagnetism 7 Electromotive Force (emf) • To calculate the work done by this force in taking a charge around the loop, we line integrate around the closed loop to yield r r r r r r r ∫ f ⋅ dl = ∫ ( f C C s ) + E ⋅ dl = ∫ fs ⋅ dl ≡ E C • This non-zero result is called the Electromotive force (emf) of the circuit. Alexander A. Iskandar Electromagnetism 8 4 Motional emf • Moving a conducting wire in a magnetic field can also resulted in motion of charges in the wire. This is called motional emf. • Consider the experiment of moving a loop in a magnetic field. In this case, the force that push the charges in r motion is the magnetic force v ⊗ r B r r r fs = v × B h R x Alexander A. Iskandar 9 Electromagnetism Motional emf • The motional emf is calculated as before r r dx da dΦ B E = ∫ f s ⋅ d l = vBh = − Bh = − B =− dt dt dt C the minus sign accounts for the fact the rate of change of the area a is negative and the magnetic flux is dΦ B = B da . • It turns out that this last relation is valid much more generally – independent of the shape of the loop, r homogeneity of B. r • Suppose S i id the inside th wire i th the d drift ift velocity l it iis u , th then th the total magnetic force on a charge can be deduced to be r r f mag v r u vB r r r f mag = w × B r w uB Alexander A. Iskandar Electromagnetism 10 5 Motional emf • Consider a loop of wire moving, and perhaps even r changing shape, through a region with a static B , and follow the point A. • In time dt rit moves a distance vdt, and with the line element d l sweeps out an area r r r da = v dt × d l A r dl at time t θ r v dt Alexander A. Iskandar at time (t + dt) da 11 Electromagnetism Motional emf • The change in magnetic flux through the loop, that’s admitted by the border ribbon is ∫ B ⋅ da = dt ∫ B ⋅ (v × d l ) r dΦ B = Φ B (t + dt ) − Φ B (t ) = r r r r C ribbon • Now suppose a current runs in the loop. If the drift velocity r of the carriers (relative to the loop) is u , and their total r r r r r velocity w = v + u , then since u must be parallel to d l then r r r r v × dl = w× dl A r dl θ r v dt Alexander A. Iskandar at time t da Electromagnetism at time (t + dt) 12 6 Motional emf • Hence, ( ) ( ) ( ) r r r r r r r r r r r B ⋅ w × d l = d l ⋅ B × w = −d l ⋅ w × B = −d l ⋅ f mag • Thus, Thus ( or ) ( E =− dΦ B dt ) r r r r r r r r dΦ B = − ∫ B ⋅ w × d l = − ∫ w × B ⋅ d l = − ∫ f mag ⋅ d l dt C C C Alexander A. Iskandar Electromagnetism 13 Electromagnetic Induction What if the field in the region varies, with the loop stationary? • Relativity: as long as the relative motion is the same, the same emf must be obtained as before. (We see this experimentally too.) • In this case, though, it’s no longer clear what exerts the r force that moves the charges, since v = 0 . • Faraday gave an ingenious explanation to this. He postulate an induced, non-electrostatic, electric field can be obtained from changing of magnetic field : r r dΦ B E = ∫ E ⋅ dl = − dt C • With Stokes theorem r r r r r r d r r dB ∫ E ⋅ d l = S∫ ∇ × E ⋅ da = − dt S∫ B ⋅ da → ∇ × E = − dt C ( Alexander A. Iskandar ) Electromagnetism 14 7 Electromagnetic Induction r r dB ∇× E = − dt Faraday’s Law of Induction • This means that a non-static electric field can be induced by a nonstatic magnetic field. • That is, a current canr be induced in a loop of conductor by changing the flux of B through it, no matterr how the flux changes: motion of the loop, or change in B . • The minus sign in Faraday’s law indicates that a changing magnetic flux will induce an electric field and current such that the magnetic field induces by the current leads to a flux change in the opposite direction. This is called Lenz’s Law. Alexander A. Iskandar 15 Electromagnetism Induced Electric Field r E with Faraday’s Law • Calculations of induced electric field r proceed just like calculations of B from steady currents using Ampère’s Law. • Note the following relations: r r dB ∇× E = − dt r r ∇ × B = µ0 J or in integral form r r dΦ E ∫C ⋅ d l = − dt B r r ∫ B ⋅ d l = µ0 I enc C • Also r ∇⋅B = 0 r ∇ ⋅ E = 0 ← ρ = 0, only currents changes • Apparently they’re the same, rwith the interchange of r r E⇔B Alexander A. Iskandar r µ0 J ⇔ − dB dt Electromagnetism µ0 I enc ⇔ − dΦ B dt 16 8 Induced Electric Field • We can use this similarity to calculate induced electric fields using the machinery of magnetostatics (namely Ampere’s Law). • Example 7.7, 7.8 Alexander A. Iskandar Electromagnetism 17 9