Faraday s Law. d dt. d dt. BdS. B t dt ds B t dt ds B dt ds B t ds. B t dt ds B t dt ds B t ds

Transcription

1 Faraday s Law We now consider what happens in the non-static case. The fundamental result is Faraday s Law d Ed Bd where the LH is the line integral around any closed path and the RH is over the surface bounding the path with d directed as in tokes Theorem go around loop keeping it to your left. Curl the fingers of the right hand in that direction and your thumb will point in the direction of d. As before this is not a closed surface, but rather the top half of such a surface. To understand the physical significance of Faraday s Law we examine d surf Bd carefully. Clearly the integral changes due to two things. First the bounding surface can change, and second can change. We shall see that the first is nothing new. The second is. The change in the integral is B t d B t d B t d B t d B d B t d B B Bt d Bt d Btd t B B d Btd d Consider a length dl of the loop. If dl is moving with velocity it produces an increase in area given by

2 vd sin ince we need d to point up (in this orientation) we have Then vd B d t Hence it goes to zero in the derivative. Thus B d BdA d vd B t t d B B Bd d vd Bt d d vb t t Now consider an arbitrary loop fixed in space. Then surface B d E d E d t surface ince the surface is arbitrary B E t

3 Hence there are two effects here. The one just aove is clearly new since it describes a nonconservative electric field. The other E d d v B is not new. It is simply a result of the Lorentz force FqvB we have seen earlier F E d d q This effect is called a motional EMF. We change from voltage to EMF since V must be conservative whereas EMF is not. To see these effects in action, consider a simple generator, consisting of a square loop spinning in a magnetic field as shown v There will be a force on charges on side (1) directed into the board given by On side () it will be reversed (since is) r F qvbsint in 1 F qvbsin t out On sides () and () the force will be perpendicular to the wire. Hence

4 E d vbsint Bsint Thus there will be an EMF around the loop given by BAsin t Bsin t We have done this without using Faraday s Law at all. Clearly we could use Faraday s Law as follows d Ed BdA da ycos ˆ t x sin t B Byˆ BdA ByAycos ˆ ˆ txsint BAcost B just as above. If in addition we would have to add t B EMF d t The moral of the story is that there are two sources of EMF: motional EMF caused by changes in the position or shape of the loop, and EMF caused by time dependence of B. The former is familiar, while the latter is new.

5 We now have the modified Maxwell equations B E E t B B J Maxwell s insight was to notice that these are inconsistent with charge conservation J t In our case J B But the divergence of any curl is zero. Hence J This is the situation we noted earlier. To fix this we add F to J so that B JF B J F Then we need Thus we take E F E t t t B J E t In the presence of dielectrics this becomes

6 D B J t Hence the name displacement correct. The complete Maxwell equations in empty space become B E E t B B J In the presence of dielectric or magnetic materials these become B D E t D B H J t E t Energy in EM Fields Consider a volume V containing a charge density ρ and an electric field. The force on a charge q will be F qe The field will do work Fd Fv qe v in time. If the charges were moving along a circuit this would become Jd E If they are moving on a surface da E

7 Thus in a volume Hence the power provided by the field is J E dvol J Ed r But D D H J J H t t D JEd r' H Ed r' t Now EH H E EH EH HE EH D JEdr' HEEHE dr t B D H E d r' EHd t t B D JEH E d r' EHd t t We recognize

8 D E t as the rate at which the energy in the electric field is increasing. Then B H t must be the rate at which energy in the magnetic field is increasing. Thus the left hand side is the total rate at which energy is appearing in the volume. Thus the right hand side must be the rate at which energy is entering the volume. That means that E H is the energy/area/sec striking the surface bounding the volume. is called the Poynting vector. We now look at the electric and magnetic energy terms a little more closely. The energy involved in polarizing a molecule is dpe 1 PE F dr q E dr E q r i i i i i i i i t i But qr i i i p i the dipole moment of the molecule. If we now sum over all the molecules/volume we get dpe P E t imilarly we can find the rate at which energy is going into creating the magnetization of the material. The rate of energy input into a current loop is du M d d B t M I I I B da I da

9 Note that this step is possible because the loop is not changing we are simply increasing the current in it. We are also assuming does not change over the dimensions of the loop which is ok since they are infinitesimal loops. Then du M B m t where is the dipole moment of the loop. Then the rate of change of energy/volume is du M vol db M It is often useful to split the EM energy into three terms uem up um so that B D du Em P B H E E M t t t t du B D P B H E E M t t t t EM But B H M DEP u B B da B EM M E EP E M t t t 1 B E B E t t 1 uem B E

10 This is the energy in the fields themselves. To actively evaluate du P / and du M / we have to know the relationship between and, and between and or. If they are linear we get or du P P de x e de E xee t u P x e E For non-ferromagnetic materials we have du M x B m B x m B xmh B t t t or 1 x m um B B For ferromagnetic materials the situation is quite different. Now is not proportional to and in fact is a function of the path taken. Thus we can t define a magnetic potential energy for the magnetization of a ferromagnet. We can however, calculate the work done if we know the path the curve above. We can also calculate the field energy produced by the magnet as we have done above.