Today in Physics 217: EMF, induction, and Faraday s Law

Transcription

1 Today in Physics 217: EMF, induction, and Faraday s Law Electromotive force v Motional EMF The alternating-current generator Induction, Faraday s Law, Lenz s Law I R h I Faraday s Law and Ampère s Law s Simple example uses of Faraday s and Lenz s laws v ox o 20 November 2002 Physics 217, Fall

2 Electromotive force (EMF) A more general constitutive relation like Ohm s Law can be written in order to account for ways of moving charges other than electric fields: J = σ f. Here f has the same units as E, but has more to it than just an electrostatic field: f = Estatic + fs, where f S represents anything else that can move charge: chemical reactions, conveyor belts, trained bacteria, The line integral of f around a closed path is called the electromotive force, E: 0 E f d = E d + f d = f d static S S. 20 November 2002 Physics 217, Fall

3 Motional EMF The simplest case of EMF is provided by motion in a magnetic field: v fs =. c Consider a loop of wire, partly enclosed by a region of constant, moving at speed v. What s E? vh h ds E = v d = = c c c dt 1 da = A = area filled with c dt 1 dφ =. Motional EMF c dt 20 November 2002 Physics 217, Fall v R h v ox o s

4 Motional EMF (continued) It turns out that this last relation is valid much more generally -- independent of the shape of the loop, homogeneity of -- which we will show now. It s helpful to keep the simpler example in mind, though. Consider a loop of wire moving, and perhaps even changing shape, through a region with a static, and follow the point A. In dt it moves a distance vdt, and with the line element d it sweeps out an area da= vdt d. order Time: t t + dt 20 November 2002 Physics 217, Fall A vdt da A d

5 Motional EMF (continued) The change in magnetic flux through the loop, that s admitted by the border (shaded in cyan), is ( ) ( ) a ( v ) Φ = ( v d ). border dφ =Φ t + dt Φ t = d = dt d d dt Now suppose a current runs in the loop. If the drift velocity of the carriers (relative to the loop) is u, and their total velocity w = u + v, then since u must be parallel to d, v d = w d., 20 November 2002 Physics 217, Fall

6 Motional EMF (continued) Now, ( d ) d ( ) d ( ) w = w = w = cd mag 1 dφ = fmag d E c dt 1 dφ E =, c dt f, so, or Motional EMF same as in the simpler case. dφ In MKS: E =. dt 20 November 2002 Physics 217, Fall

7 Example: the AC generator Griffiths problem 7.10: A square loop (side b) is mounted on a vertical shaft and rotated at angular velocity ω. A uniform magnetic field is perpendicular to the axis. Find the EMF, and the current driven through a resistor R in series with the loop, for this alternating-current generator. b ω b R 20 November 2002 Physics 217, Fall

8 The AC generator (continued) 2 2 Φ = a= b cosθ = b cosωt 1 dφ ω 2 E = = b sinωt c dt c 2 E ωb I = = sinωt R cr b θ For MKS, leave off the factor of c. ω 20 November 2002 Physics 217, Fall

9 Induction and Faraday s Law What if the field region moves, with the loop staying still? Relativity: as long as the relative motion is the same, the same EMF must be obtained as before. (We see it, experimentally, to work this way, too.) In this case, though, it s no longer clear what exerts the force that moves the charges, since v = 0. Thus we have to postulate an induced, non-electrostatic, electric field: 1 dφ = E d Faraday s Law E =. c dt (integral form) ut E d = ( E) da 1 dφ 1 1 and d = da= da, c dt c dt c t 20 November 2002 Physics 217, Fall

10 Induction and Faraday s Law (continued) So 1 E = = c t t in MKS. Faraday s Law That is, a non-static electric field can be induced by a nonstatic magnetic field. That is, moreover, a current can be induced in a loop of conductor by changing the flux of through it, no matter how the flux changes: motion of the loop, or change in. The minus sign in Faraday s law indicates that a changing magnetic flux will induce an electric field and current such that the the current produces leads to a flux change in the opposite direction. This is called Lenz s Law. 20 November 2002 Physics 217, Fall

11 Faraday s Law and Ampère s Law Calculations of E and I with Faraday s Law proceed just like calculations of from steady currents using Ampère s Law. Note the following forms: 4π 1 = J E = c c t 4π 1 dφ d I = enclosed E d = c c dt E = 0 if ρ = 0 and = 0 only currents change Apparently they re the same, with 4 π 1,, 4 π 1 dφ E J I enclosed. c c t c c dt 20 November 2002 Physics 217, Fall

12 Example: polarity of motional EMF Which way does the current flow in the example of the loop being pulled out of the region of constant? v R As we saw earlier, vh vh E =, I =. c cr I h I Flux decreases as loop moves. If current were to flow clockwise, it would generate its own, and flux, that would oppose the decrease. v ox o s 20 November 2002 Physics 217, Fall

13 Example: a flux tube Consider a cylinder, radius a, of uniform magnetic field with timevariable amplitude (t). (Structures like this are seen on the surface of the Sun.) What is the electric field? Draw a circular loop, radius s: 1 dφ E d = c dt 1 2 d s d s d E2πs = πs E = φˆ = sˆ s< a c dt 2c dt 2c dt d a d = π a E = sˆ ( s> a) c dt 2cs dt 20 November 2002 Physics 217, Fall z (t) a ( )