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 E K K E v d A L B K K Recall that voltage difference was defined as V E dr For the case of E fields created A by charges the voltage difference when we go around a closed loop is zero since voltage K K depends only on position not on path V E dr 0 A A Definition magnetic flux through some surface S B S K K B dA if B const and A flat K K B A BA cos B B cos A area A Units T m2 weber Wb Last update 10 30 2009 Dubson Phys1120 Notes University of Colorado F 2 Faraday s Law in words An induced emf E is created by changing magnetic flux Faraday s Law in symbols E 1 loop loop of wire d M dt If B constant emf E 0 If B is changing with time V B in uniform If have several loops E voltmeter E N loops N d 0 dt d dt We can change the magnetic flux in several ways V N 2 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 dB increasing at a rate 0 1 T s What is the magnitude of the dt B 10 cm V 10 cm emf E induced in the loop Answer E d d BA dB A 0 01m 2 0 1T s 10 3 V 1mV dt dt dt What is the emf if N 1000 loops E N d BA 1000 10 3 V 1V dt Last update 10 30 2009 Dubson Phys1120 Notes University of Colorado F 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 induced B B increasing Bind OR induced I Iind Answer Binduced downward opposes the increase in original B Bind B decreasing Here induced B is upward to oppose the decrease in the original B Iind 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 wire loop v B in here through the wire CW or CCW Answer CCW The flux is increasing as the loop enters the field In B 0 here 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 I Does the magnetic field exert a net force on the loop as it enters F 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 Last update 10 30 2009 Dubson Phys1120 Notes University of Colorado F 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 conventional current I electron current downward B in uniform parallel to the wire the direction of the motion of the loop to the right or some combination of net force on wire motion of wire and of fixed positive ions in wire these directions The conduction electrons in the right half of our wire are actually moving both downward and to the right But electron current motion of conduction electron in wire 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 Last update 10 30 2009 Dubson Phys1120 Notes University of Colorado F 5 Electrical Generators Convert mechanical energy KE into electrical energy just the crank 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 I B light bulb Eddy Currents If 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 K K K current I and the B field are always such as to cause a magnetic force F I L B which slows the motion of the metal Again …
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