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. 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 (in words): An induced emf (E) is created by changing magnetic field. Definition: emf , E = a voltage difference (∆V = E d ) capable of doing useful work, generating power. 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 ) Definition: magnetic flux through a loop of area A = BBAcos B A⊥Φ = θ =B⊥ is component of B perpendicular to the area A. If B-field is perpendicular to the area A, then Φ = B A. Units [Φ] = T⋅m2 = weber (Wb) Faraday's Law (in symbols): (1 loop)t∆Φ= −∆E If B = constant ⇒ voltage = E = 0 If B is changing with time ⇒ 0t∆Φ= ≠∆E. If have several loops, (N loop)Nt∆Φ= −∆E B⊥ = B cos θ B θ area A loop of wire V voltmeter B(in) uniform V N = 2 Last update: 3/14/2007 Dubson Phys2020 Notes, ©University of ColoradoF-2 We can change the magnetic flux Φ in several ways: 1) change B (turn the magnetic field up or down) 2) change A (by altering shape of the loop) 3) change the angle θ between the B-field and the plane of the loop (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 B0.1T /st∆=+∆. What is the magnitude of the emf E induced in the loop? Answer: 23(BA) BA (0.01m )(0.1T/s) 10 V 1mVtt t−∆Φ ∆ ∆== = = = =∆∆ ∆E What is the emf if N = 1000 loops? 3(BA)N 1000 10 V 1Vt−∆==×=∆E 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. 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. B 10 cm 10 cm V I B crank light bulbLast update: 3/14/2007 Dubson Phys2020 Notes, ©University of ColoradoF-3 Example: a loop of wire in an external B-field which is increasing like so induced BB increasing Bind 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. A 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 R.H.R.). Notice that the direction of the force on the wire loop will slow its motion. 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 induced IOR?? ⇒ Iind BindB decreasing ⇒ Iindwire loop B = 0 hereB (in) herev I F Last update: 3/14/2007 Dubson Phys2020 Notes, ©University of ColoradoF-4 direction of the motion of the entire loop. Now, it is really the negative conduction electrons thare 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 cross B". Should we pick the direction of theelectron 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 ourwire are actually moving to bothat e es n n rd ddy Currents nd a B-field are in relative motion in such a way as to cause a changing Φ downward and to the right. But only the downward motion matters, because the motion to the right is effectively canceled by thmotion of the positive chargwithin the wire. Remember that thewire is electrically neutral; there are as many fixed positive ions ithe wire as there are mobile negative electrons. The force othe electrons due to their rightwamotion 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. EIf a piece of metal athrough 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 eddycurrent I and the B-field are always such as to cause a magnetic force (F = I L B sinθ ) whichslows the motion of the metal (as in the example above). Again, if metal moving in B-field produces a changing Φ N(Faraday)on metaleddy currentIF(IL⇒ =⇒ B)and the direction of the force always slows the motion. 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: 3/14/2007 Dubson Phys2020 Notes, ©University of ColoradoF-5 If the eddy current force did not slow the motion, but instead aided the motion, then we would ransformers l power distribution system in the civilized world depends on a simple device rimaryhave runaway motion ⇒ free energy ⇒ violation of energy conservation. TThe entire electricacalled a transformer. A transformer is a device for transforming AC voltage from one value (say 120 VAC) to another value (like 10 VAC or 2000 VAC). A transformer is made of 2 coils of wire, usually wrapped around an iron core. It is
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