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Magnetic flux and Faraday’s Law Other way to show: So far: ÖMagnetic fields arises by moving charges A coil experiences a current when magnetic passing through it varies Now: Changing the magnetic field arise electric field (Michael Faraday conducted an experiment to prove this) At the moment one closes the switch ammeter deflect and returns immediately to zero. Again deflection at the moment opening the switch The change in the magnetic field is described with the change in magnetic flux There is no contact between the coils. We called this phenomenon a induce current. If we induce a current we have to have a potential difference: induced emf (electromotive force) Electric Flux )el = E A cosT Magnetic Flux )mag = B A cosT [Tm2]=[Wb, Weber] In detail: The magnitude of the induced current/ induced emf is proportional to the change of the magnetic field. ÖThe more rapidly the change in the magnetic field the more the higher the induced current/emf Faraday’s Law of Induction: Induced emf is proportional to the rate of flux change emf N ') 't Magnitude: emf ) final ) initial I final I initial ') N 't N Example: Lenz Law (expresses the meaning of the minus) An induced current always flow in a direction that opposes the change that caused it. Motional emf: Qualitative Field is constant but the magnetic flux changes because rod accelerates due to gravity. Öinduce emf because of motion Lenz’s Law: A decrease in the surface causes a decrease in magnetic flux. Therefore the magnetic field from the induced current has to strengthen the field. The current flows counterclockwise. ÖCurrent through rod causes a force against gravity. If Fmag = Fg then rod moves with a constant velocity Motional emf: Quantitative Power F = I l B sinT = Bvl/R lB = B2 l2 v/R Pmech= F v = (B2 l2 v/R) v = B2 l2 v2/R Pelectric= I2 R = (Bvl/R)2 R = B2 l2 v2 /R Application: Generator No Fg in this case. We move the rod to the right side with a constant external force Fext. ǻ)mag = B ǻA with A = v ǻt l ǻ)mag= B v ǻt l Faradays Law to calculate emf emf N ') 't N Bvl't 't N=1 (1 loop) |emf| = Bvl Current through light bulb I = |emf|/R = Bvl/R Relationship between E-field and B-field V= El emf = Bvl El = Bvl or E = B v Definition of inductance L ') 'I ') emf N L and L= N [H, Henry = Vs/A] 't 't 'I Inductance Coil – coil induction is referred to as mutual induction. Example: Inductance of a solenoid A coil with a changing current/magnetic field can induce a current in itself: self inductance Example: Coil in electric circuit (or inductor in short resist changes in its electric current) By closing the switch it is not possible for current to rise immediately to final value. The inductor slows the process. (Remember capacitor) Capacitor IJ = RC Inductor IJ = L/R I Decreasing current: Inductor try to keep current constant. ĭ ~ I it must be that N RL Circuit Switch open: Break in the circuit. Switch closed: Initially current is zero. Current steadily increase with time. ÖMagnetic flux increase with time ÖInduce an emf (Faraday) in the opposite direction to the current (Lenz) ÖResistance against I ÖWork is done!! Self-induced emf ') because emf N 't ') BA 0 (P ( N / l ) I ) A N N 0 'I I 0 I or N / l n L P 0 A n 2l L ') 'I ~ 't 't emf (1 e t / W ) R emf (1 e tR / L ) R P0 A N 2 / l Work is required to establish a current through a inductor. This is a conservative process and the work/energy is stored in the magnetic field, similar to capacitor. An application: Transformers Changes voltage from one value to the other. In principle: Magnetic flux in a primary coil induces a voltage in a secondary coil. Pav= ½ I emf = ½ I (LI/t) t: time of increasing current Pav= ½ L I2 /t Energy stored: U = Pavt UB = ½ L I2 Energy density uB = magnetic energy / volume Example: Solenoid L P 0 A n 2l B P 0 nI U 1 2P0 U 1 P 0 A n 2l I 2 2 B 2 Al energy density u B 1 2 B 2P0 ') 't ') emf s N s 't if the number of coils N is different, voltage is different. Rearrange for ĭ give emf p N p Transformer equation emf p 't Np emf p Np emf s Ns emf s 't Ns or emf p emf s Np Ns P= V I = emf I Energy conservation: The higher the voltage the lower the current emf p emf s Np Ns Is Ip