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CU-Boulder PHYS 1120 - Faraday

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F-1 Faraday's Law Faraday'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 Ed= ⋅∫KKAvLE Recall that voltage difference was defined as BAVEdr∆=− ⋅∫KK. 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: AAVEdr0∆=− ⋅ =∫KK Definition: magnetic flux through some surface S, BSif constand A flatBdA BA BAcosΦ = ⋅ = ⋅ = θ∫KKKKB Units [Φ] = T⋅m2 = weber (Wb) B cos θ B θ A area A Last update: 10/30/2009 Dubson Phys1120 Notes, ©University of ColoradoF-2 Faraday's Law (in words): An induced emf (E) is created by changing magnetic flux. Faraday's Law (in symbols): M(1 loop)ddtΦ= −E If B = constant ⇒ emf = E = 0 If B is changing with time ⇒ d0dtΦ= ≠E. If have several loops, (N loops)dNdtΦ= −E loop of wire V We can change the magnetic flux Φ 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 θ 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 dB0.1T /sdt=+. What is the magnitude of the emf E induced in the loop? Answer: 23dd(BA) dBA (0.01m )(0.1T /s) 10 V 1mVdt dt dt−Φ== = = = =E What is the emf if N = 1000 loops? 3d(BA)N 1000 10 V 1Vdt−==×=E voltmeter B(in) uniform V N = 2 B 10 cm 10 cm V Last update: 10/30/2009 Dubson Phys1120 Notes, ©University of ColoradoF-3 Lenz's Law The 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 so Answer: 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 wire moving to the right enters a region where there is a uniform B-field (in). What is the direction of the current through the wire: CW or CCW? Answer: CCW The flux is increasing as the loop enters the field. In order to fight the increase, the induced B-field must be out-of-the-page. An induced CCW current will produce a B-field pointing out. Does the magnetic field exert a net force on the loop as it enters the field? Answer: Yes. The upward current on the right side of the loop will feel a force to the left (from Fwire = ILB and B increasinginduced BBindinduced IOR?? ⇒ IindB decreasingBind⇒ Iindwire loop B = 0 hereB (in) herev I F Last update: 10/30/2009 Dubson Phys1120 Notes, ©University of ColoradoF-4 R.H.R.). Notice that the direction of the force 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 we apply the force law F = q v × B. Should we pick the direction of the electron current (downward, parallel to the wire), the direction of the motion of the loop (to the right), or some combination of these directions? The conduction electrons in the right half of our wire are actually moving both downward and to the right. But only the downward motion matters, 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 to their 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. B(in) uniform conventional current I net force on wire motion of wire and of fixed positive ions in wire electron current motion of conduction electron in wireLast update: 10/30/2009 Dubson Phys1120 Notes, ©University of ColoradoF-5 Electrical Generators Convert mechanical energy (KE) into electrical energy (just the opposite of motors). A wire loop in a constant B-field (produced by a magnet) is turned by a crank. The changing magnetic flux in the loop produced an emf which drives a current. Eddy Currents If a piece of metal and a B-field are in relative motion in such a way as to cause a changing Φ through some loop within the metal, then the changing Φ 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 ( ) which FIL=×KKBKslows the motion of the metal . Again, if metal moving in a B-field makes a changing Φ and the direction of the force N(Faraday)on metaleddy currentIF(IL⇒ =⇒B)always slows


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CU-Boulder PHYS 1120 - Faraday

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