1830s Michael Faraday Joseph Henry. Induced EMF

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2 1830s Michael Faraday Joseph Henry Induced EMF

3 There can be EMF produced in a number of ways: A time varying magnetic field An area whose size is varying A time varying angle between r and Any combination of the above ds r

4 Faraday s Law of Induction The induced EMF in a closed loop equals the negative of the time rate of change of magnetic flux through the loop EMF = dφ Caution: induction EMF should be not confused with electrostatics: the effect is dynamical r E r dr = dφ Caution: electric field is not potential anymore: electric field acquired circulation!

5 r ur dφ = d A = da = dacosφ

6 Ampere s law orientation of the contour! contour ur r dl = μ I 0 Caution: Sign of the currents enclosed by the contour are determined by the orientation of the contour. For this orientation of the contour (anticlockwise), currents I 1 and I 3 are positive while I 3 is negative.

7 r E r dr = dφ For this orientation of the area element the orientation of the contour (direction of the integral) is anticlockwise,

8 Sign of Induction (orientation of the area is UP) induced current has clock-wise direction Orientation of the oval -- as if it is lying on the floor

9 Sign of Induction (orientation of the area is DOWN); direction of the current does not depend on the choice of orientation! Orientation of the oval -- as if it is on the ceiling The same as in (a) on the previous page, but for opposite orientation; induced current has clock-wise direction

10 Lenz s principle (law): H.F.E. Lenz ( ) the direction of induction effect is such as to oppose the cause the effect Caution: the induced current opposes the change in flux through the circuit not the flux itself induced current has anti-clock-wise direction induced current has clock-wise direction d Φ = EMF = IR induced current changed its direction after the magnet passed through the circle

11 Lenz s principle (law): H.F.E. Lenz ( ) the direction of induction effect is such as to oppose the cause the effect Caution: the induced current opposes the change in flux through the circuit not the flux itself d Φ = EMF = IR

12 Check: Units of the magnetic field and EMF The force F r on a charge q moving with a velocity The magnitude of the force F = qvsinθ [ ] = Newtons /( Coulomb meter / v r sec) 1 Ttesla ( ) = 1 N/ Cms / = 1 N/ Am flux [ Φ ] = 1T m 2 T m = N m s/ C = 1J s = 1Wb 1V = 1 Wb/ s / C = 1V s dφ = EMF

13 Induction in a coil with N turns EMF = N dφ

14 Example 29.4 Generator (a simple alternator) Caution: don t confuse generator with a motor. Generator of electricity is rotated by an external source (water, wind, gasoline ) Φ () t =Φ cosωt 0 Φ () t EMF = =Φ0ω sinωt

15 Example 29.5 DC Generator Φ () t =Φ cosωt 0 EMF = dφ ()/ t =Φω sin ωt 0

16 Example 29.5 motor s back EMF The motor s back EMF is the emf induced by the changing magnetic flux in rotating coil of the motor Φ = Φ () t N 0 cos t ω EMF = NAω sin ωt sin ωt = 2/ π π EMF EMF ω = f = 2 NA 4NA 112V rev f = = (0.2 T)(0.1 m) sec

17 Example 29.5 motor s back EMF EMF = NAω sin ωt V = EMF + Ir input back 2 Pinput = I EMFback + I r P = P + P input work dissipation Comment: power distribution is similar to the examples of Chapter 26, but instead of a lamp now it is a motor

18 A series DC motor: What happens when a motor suddenly stops? Did it stop because it was burnt, or it was burnt because it had been stopped by a jam? Example 27.11, see also Example to understand what is going on here, and why the current has so different values? 120V = E+ 4A 2Ω E = 112V P = A= W P = A Ω= W input stalled ab heat (4 ) / 120 / 2 60 heat (60 ) I = V r = V Ω= A P = A Ω= W

19 Generator versus Motor: In what direction the current flows?? Generator: EMF = V + Ir P applied output = = P + output EMF I P dissip Φ () t =Φ cosωt input Motor: V = EMF + Ir back 2 Pinput = VinputI = I EMFback + I r P = P + P 0 EMF = dφ ()/ t =Φω sin ωt input work dissipation 0

20 Example 29.6 Slidewire generator The increase in the magnetic flux caused by the increase of the area induces the EMF and current. Caution: actually, there are TWO forces. One (not shown) is applied on the right, and it causes sliding with velocity v. The other force is due to the induced current. It acts in the direction opposite to the direction of sliding (Lenz s law).

21 x x x x x x x x Example 29.6 Slidewire generator power distribution in the presence of an output power x x x Motor x x x x x x x x EMF = dφ / = Lv EMF = Voutput + Ir ( Lv Voutput )/ r = I r ur ur F = IL applied P = vf = vli = EMF I applied x x x x x F x x x x x x x x x x x x x applied Caution: here, only the applied force that causes sliding is shown. The other force that is due to the induced current is not shown. It acts in the direction opposite to the applied force. P = vl( Lv V )/ r P = P + P applied output applied output dissip!!! P = V I P = Lv V r 2 output output dissip ( output) /

22 Example 29.6 Slidewire generator; motional EMF Attention: to generate EMF there is no need in a material frame, like in slidewire. It may be virtual. Then, it is called motional EMF. This EMF is not a mystery. It originates from a probe force acting on a probe charge moving together with a rod.

23 Rail-gun; the motor-counterpart of the slidewire generator V = EMF + Ir battery back EMF = L v back 2 Pinput = VbatteryI = I EMFback + I r I EMF = I( Lv) = v( IL) = vf back P = P + P input mechanical dissipation

24 Faraday disc dynamo: two ways to get the answer Lenz s law: the current directed down toward the sliding contact b creates a force directed to the left, thus, opposing disc s rotation F E = = qv sinθ v sinθ R R R 2 (1) EMF = E( r) dr = v( r) dr = ω r dr = ωr / dϕ R 2 (2) EMF = dφ ( t)/ = r dr =ωr /2 0

25 Induced electric fields caused by the varying magnetic flux. r r E dr = dφ How can it be that the magnetic field induces electric fields outside its location? Caution: Notice that it is a magnetic field changing in time.

26 Eddy currents Attention: current in the transmitting coil should be pulsing.

27 Symmetry: if there are induced electric fields caused by varying magnetic field, why not to look for magnetic fields caused by varying electric fields? r E r dr = dφ ur r ( dφ E dr = μ ) 0 Ic + ε

28 Symmetry (Clerk Maxwell, ): if there are induced electric fields caused by varying magnetic field, why not to look for magnetic fields caused by varying electric fields? r r E dr dφ = ur r ( dφ E dr = μ ) 0 Ic + ε The flux term is called displacement current I = ε dφ / j = εde/ displacement E displ

29 Comparison of the displacement current Idisplacement with the transport current I c for charging capacitor r r E dr dφ = ur r ( dφ E dr = μ ) 0 Ic + ε q= CV = ( εs/ d)( Ed) = εse = εφ I = dq/ = ε dφ / I = I c E c displacement displ displacement I = εdφ / j = εde / E E

30 Maxwell s equations in the empty space: induced electric fields caused by varying magnetic field, magnetic fields caused by varying electric fields r E r dr = dφ ur r dr = με 0 0 dφ E Two coupled equations are typical for waves. Theoretical discovery of the electromagnetic waves was made by Maxwell.

31 Maxwell s equations in the empty space: induced electric fields caused by varying magnetic field, magnetic fields caused by varying electric fields r E r dr = dφ ur r dr = με 0 0 dφ E Two coupled equations are typical for waves. Theoretical discovery of the electromagnetic waves was made by Maxwell.

32 Additional Material: superconductivity

33 Meissner effect and magnetic levitation Eddy currents do not let a magnetic field to penetrate inside a superconductor (the Meissner effect); The current in the superconductor in its turn creates a backward magnetic field; This backward magnetic field may cause the levitation of a body, which was the origin of the magnetic field which generated the Eddy current. (Manifestation of the Lenz law at work!)