F-1Faraday's LawFaraday's Law is one of 4 basic equations of the theory of electromagnetism, called Maxwell's Equations. We have said before that - charges makes electric fields. (Gauss's Law)- currents make magnetic fields. (Ampere's Law)This is the truth, but not the whole truth. Michael Faraday (British physicist, c.1850) showed that there is a second way to make an electric field: - a changing magnetic field makes an electric field. (Faraday's Law)Around 1860, James Maxwell(Scottish physicist) showed that there is a second way to make a magnetic field:- a changing electric field makes an magnetic field. (modification of Ampere's Law)Before stating Faraday's Law, we must define some new terms:Definition: emf , E , is (roughly speaking) a voltage difference (-V = E d ) capable of doing continuous useful work.. Think of emf as a battery voltage. Batteries have an emf, but resistors do not, even though a resistor R can have a voltage difference across it (-V = I R )Technically, the emf around a closed loop L is defined as E d= ��vvl�LE Recall that voltage difference was defined as BAV E drD =- ��vv. For the case of E-fields created by charges, the voltage difference when we go around a closed loop is zero, since voltage depends only on position, not on path: AAV E dr 0D =- � =�vvDefinition: magnetic flux through some surface S,BSif constand A flatB dA B A BA cosF = � = � = q�v vv vB Last update: 1/14/2019 Dubson Phys1120 Notes, -University of Colorado BB cos --area AAF-2Units [-] = T-m2 = weber (Wb) Faraday's Law (in words): An induced emf (E) is created by changing magnetic flux.Faraday's Law (in symbols): M(1 loop)dd tF= -EIf B = constant - emf = E = 0If B is changing with time - d0d tF= �E.If have severalloops, (N loops)dNd tF= -E We can change the magnetic flux - in several ways:1) change B (increase or decrease magnitude of magnetic field)2) change A (by altering shape of the loop)3) change the angle - between B and the area vector A (by rotating the loop, say)Example of Faraday's Law: We have a square wire loop of area A =10 cm - 10 cm, perpendicular to a magnetic field B which is increasing at a rate d B0.1 T / sd t= +. What is the magnitude of theemf E induced in the loop?Answer: 2 3d d(BA) d BA (0.01m )(0.1T / s) 10 V 1mVd t d t d t-F= = = = = =ELast update: 1/14/2019 Dubson Phys1120 Notes, -University of Colorado voltmeterVloop of wireB(in)uniformN = 2VB10 cm10 cmVF-3What is the emf if N = 1000 loops? 3d(BA)N 1000 10 V 1Vd t-= = � =ELast update: 1/14/2019 Dubson Phys1120 Notes, -University of ColoradoF-4Lenz's LawThe minus sign in Faraday's law is a reminder of .. Lenz's Law: the induced emf E induces a current that flows in the direction which creates an induced B-field that opposes the change in flux.Example: a loop of wire in an external B-field which is increasing like soAnswer: Binduced downward opposes the increase in original B.Here, induced B is upward to oppose the decrease in the original B.Lenz's Law says "Change is bad! Fight the change! Maintain the status quo."Example of use of Lenz's Law A square loop of wiremoving to the right enters a region where there is auniform B-field (in). What is the direction of the currentthrough the wire: CW or CCW? Answer: CCWThe flux is increasing as the loop enters the field. Inorder to fight the increase, the induced B-field must beout-of-the-page. An induced CCW current will produce a B-fieldpointing out. Does the magnetic field exert a net force on the loop as it entersthe field? Answer: Yes. The upward current on the right side ofLast update: 1/14/2019 Dubson Phys1120 Notes, -University of Colorado B increasinginduced Binduced IOR??-BindIindB decreasingBindIind-wire loopB = 0hereB (in)herevIFF-5the loop will feel a force to the left (from Fwire = ILB and R.H.R.). Notice that the direction of theforce on the wire loop will slow its motion. Aside: There is a subtlety in this problem that we have glossed over. To get the direction of the force on the right-hand side of the wire, we assumed that the direction of the (imaginary positive)moving charges in the wire is upward, along the direction of the current, and not to the right, along the direction of the motion of the entire loop. Now, it is really the negative conduction electrons that are moving within the wire, but we still have the problem of understanding which velocity v we should pick when weapply the force law F = q v - B.Should we pick the direction of theelectron current (downward,parallel to the wire), the directionof the motion of the loop (to theright), or some combination ofthese directions? The conductionelectrons in the right half of ourwire are actually moving bothdownward and to the right. Butonly the downward motionmatters, because the motion to the right is effectively canceled by the motion of the positive charges within the wire. Remember that the wire is electrically neutral; there are as many fixed positive ions in the wire as there are mobile negative electrons. The force on the electrons due totheir rightward motion is exactly canceled by the force on the positive charges, which have exactly the same rightward motion. But the force on the conduction electrons due to their downward motion is not canceled out, and this is the cause of the net force on the wire. Last update: 1/14/2019 Dubson Phys1120 Notes, -University of Colorado B(in) uniformconventional current Ielectron current motion of wire and of fixed positive ions in wiremotion of conduction electron in wirenet force on wireF-6Electrical GeneratorsConvert mechanical energy (KE) into electrical energy (just theopposite of motors). A wire loop in a constant B-field (producedby a magnet) is turned by a crank. The changing magnetic flux inthe loop produced an emf which drives a current. Eddy CurrentsIf a piece of metal and a B-field are in relative motion in such a way as to cause a changing - through some loop within the metal, then the changing - creates an emf E which drives a current I. This induced current is called an eddy current. The relative direction of this eddy current I and the B-field are always such as to cause a magnetic force ( F I L B= �v v v ) which slows the motion of the metal . Again, if metal moving in a B-field makes a changing - {(Faraday)on metaleddy currentI F ( I L B)� =� and the direction of the force always slows the motion.If the eddy current force did not slow the motion, but
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