Physics 169. Luis anchordoqui. Kitt Peak National Observatory. Monday, March 27, 17

Transcription

1 Physics 169 Kitt Peak National Observatory Luis anchordoqui 1

2 Question teady electric current can give steady magnetic field Because of symmetry between electricity and magnetism we can ask: teady magnetic field can give steady electric current OR Changing magnetic field can give steady electric current When a magnet is moved toward a loop of wire sensitive ammeter deflects indicating that current is induced in the loop Ammeter N Ammeter (a) When magnet is held stationary there is no induced current in the loop N even when the magnet is inside the loop (b) When magnet is moved away from loop ammeter deflects in opposite direction indicating that induced current is in opposite direction Ammeter N 2

3 Answer teady electric current can give steady magnetic field Because of symmetry between electricity and magnetism we can ask: teady magnetic field can give steady electric current OR Changing magnetic field can give steady electric current When a magnet is moved toward a loop of wire sensitive ammeter deflects indicating that current is induced in the loop Ammeter N Ammeter (a) When magnet is held stationary there is no induced current in the loop N even when the magnet is inside the loop (b) When magnet is moved away from loop ammeter deflects in opposite direction indicating that induced current is in opposite direction Ammeter N 3

4 9.1 Magnetic Flux Magnetic flux through surface Unit of m Graphical m = Z Weber (Wb) 1Wb = 1Tm 2 ~B d ~ A m number of magnetic field lines passing through surface Flux through plane is zero when magnetic field is parallel to plane surface d A d A B B Flux through plane is maximum when magnetic field is perpendicular to plane 4

5 9.2 Faraday s Law Ammeter witch + Battery Primary coil econdary coil Active Figure 31.2 Faraday s experiment. When the switch in the primary circuit is closed, Faraday s the ammeter experiment in the secondary circuit deflects momentarily. The emf induced in the secondary circuit is caused by the changing magnetic field through the secondary coil. ron At the Active Figu at can open and close th and observe the curre ammeter. When switch in primary circuit is closed ammeter in secondary circuit deflects momentarily emf induced in secondary circuit is caused by changing magnetic field through secondary coil Faraday s law of induction nduced emf E = N d m dt number of coils in circuit 5

6 B = Constant B = Constant ˆB = Constant B = Constant A = Constant  = Constant db/dt = 0 A = Constant da/dt = 0 A = Constant dâ/dt = 0 E =0 E > 0 E > 0 E > 0 Note nduced emf drives a current throughout circuit similar to function of a battery Difference here is that induced emf is distributed throughout circuit consequence we cannot define a potential difference between any two points in circuit! 6

7 uppose there is an induced current in loop can we define V AB? Recall V AB = V A V B = ir > 0 ) V A >V B Going anti-clockwise (same as ) A f we start from going to then we get B f we start from going to then we get B A i V A >V B V B >V A A battery A loop ) We cannot define V AB!! This situation is like when we study interior of a battery o provides energy needed to drive charge carriers around circuit by n chemical reactions changing magnetic flux sources of emf non-electric means 7

8 9.3 Lenz s Law Flux of magnetic field due to induced current opposes change in flux that causes induced current nduced current is in such a direction as to oppose changes that produces it 3 ncorporating Lenz s law into Faraday s Law E = N d m dt f d m dt > 0, m " ) E appears ) nduced current appears ) ~B -field due to ) change in m ) m # induced current so that 8

9 Lenz s Law is consequence from principle of conservation of energy uppose bar is given slight push to right B in This motion sets up a counterclockwise current in the loop BUT R F B v What happens if we assume that current is clockwise such that direction of magnetic force exerted on bar is to the right? This force would accelerate the rod and increase its velocity This (in turn) would cause area enclosed by loop to increase more rapidly this would result in increase in induced current which would cause increase in force which would produce increase in current... and so on... R ystem would acquire energy with no input of energy v F This is clearly inconsistent with all experience and violates law of energy conservation B We are forced to conclude that current must be counterclockwise 9

10 F B v R (b) Likewise if bar is push to the left 10

11 produces v opposes the change F B in the external magnetic flux. etic flux due to an external field directed into the page is increascurrent, if it is to oppose this change, must produce a field e page. Hence, the induced current must be directed countere bar moves to the right. (Use the right-hand rule to verify this When magnet is moved toward stationary conducting loop bar is moving to (b) the left, as in Figure current 31.13b, is induced the external in the direction magthe area enclosed by the loop decreases with time. Because the shown into the page, the direction of the induced current must v be produce a field that also is directed into the page. n either case, nt tends to maintain the original flux through the area enclosed N. this situation using energy considerations. uppose that the bar is to the right. n the preceding analysis, we found that this motion lockwise current Magnetic in field the loop. lines What shown happens are those if due we assume to bar magnet that the This induced current produces its own magnetic field directed to the left (a) that counteracts the increasing external flux N Example (b) v N N Example v N (a) Magnetic field lines shown are those due to induced current in ring (b) (c) 11 (d)

12 When magnet is moved away from stationary conducting loop N Example current is induced in direction shown v N (b) Magnetic field lines shown are those due to (c) bar magnet This induced current produces Figure magnetic field directed (a) When to the the magnet right is moved toward the stationar loop, a current and is induced so counteracts in the direction decreasing shown. external The magnetic flux fie are those due to the bar magnet. (b) This induced current produce magnetic field directed to the left that counteracts the increasing e magnetic field lines shown are those due to the induced current in (c) When the magnet is moved away from the stationary conductin is induced in the direction shown. The magnetic field lines shown a N the bar magnet. (d) This induced current produces a magnetic fiel right and so counteracts the decreasing external flux. The magneti shown are those due to the induced current in the ring. Field lines shown are those due to induced current in ring (d) 12 (d)

13 Question 13

14 Answer 14

15 9.4 Motional EMF traight conductor of length is moving through uniform ~B -field directed into the page L Assume conductor is moving with constant under influence of some external agent ~v? ~ B Electrons in conductor experience force FB ~ = q~v B ~ L ~v B ~ directed along the length perpendicular to both and Under influence of this force electrons move to lower end of conductor and accumulate there leaving net positive charge at upper end Because of this charge separation electric field ~E is produced inside conductor Charges accumulate at both ends until downward magnetic force qvb on charges remaining in conductor is balanced by the upward electric force qe 15

16 At this point electrons move only with random thermal motion Equilibrium requires that ~F E + ~ F B =0 ) q ~ E + q~v ~ B =0 ) ~ E = ~v ~ B Voltage across ends of conductor V = Z L 0 ~E d~s V = EL ) Voltage V = vbl Potential difference is maintained between ends of conductor as long as the conductor continues to move through the uniform magnetic field 16

17 uppose moving wire slides without friction on stationary Motional emf can drive electric current in -shape conductor i U U -shape conductor ) Power is dissipated in circuit ) P out = Vi Joule s heating What is source of this power? Look at the forces acting on conducting rod: Magnetic force ~F m = i ~ L ~ B F m = ilb (pointing left) For wire to continue to move at constant velocity we need to apply an external force ~F ext = F ~ m = ilb (pointing right) 17

18 ) Power required to keep rod moving P in = ~ F ext ~v = iblv = ibl dx dt = ib d(xl) dt = i d(ba) dt xl = A BA = ince energy is not being stored in system ) P in + P out =0 iv + i d m dt We recover Faraday s Law ( area enclosed by circuit) ( magnetic flux) =0 m ) V = d m dt 18

19 Generators and Motors Assume circuit loop is rotating at constant angular velocity (ource of rotation steam produced by burner or water falling from dam)! Magnetic flux through loop B = N Z loop changes with time! =!t number of coils ~B d ~ A = NBAcos nduced emf E = nduced current d B dt = NBA d dt i = E R = NBA! R sin!t (cos!t) = NBA!sin!t 19

20 Alternating current (AC) voltage generator ε ε max t Power has to be provided by source of rotation to overcome torque z} { agram of an AC generator. An emf is induced ~ = Ni A ~ B ~ in (b) The alternating emf induced in the loop ) = NiAB sin acting on a current loop in a magnetic field (b) ~µ At the Act at can adjust the rotation and t field to see th emf generated Net effect of torque is to oppose rotation of coil 20

21 Electric motor is a generator operating in reverse Replace load resistance R with a battery of emf E ) With battery there is a current in coil and it experiencestorque in B-field E ) Rotation of coil leads to an induced emf E ind in direction opposite that of battery Lenz s law ) i = E E ind R ) As motor speeds up E ind ", ) i # P electric = i 2 R + P mechanical Electric power input Mechanical power delivered 21

22 9.5 nduced Electric Field We have seen that a changing magnetic flux induces an emf and a current in a conducting loop n the same way we can relate induced current in conducting loop to an electric field by claiming that electric field is created in conductor as a result of the changing magnetic flux 22

23 nduced electric field is nonconservative unlike electrostatic field produced by stationary charges We can illustrate this point by considering conducting loop of radius r situated in uniform magnetic field that is perpendicular to plane of loop f magnetic field changes with time according to Faraday s law emf E = d B /dt induced in loop E E r nduction of current in loop implies presence of induced electric field which must be tangent to the loop E E B in Because electric force acting on charge work done by electric field in moving charge once around loop because this is direction in which charges in the wire move Figure A conducting loop in response to electric force ~E Work done by -field in moving test charge once around loop These two expressions for work done must be equal ~E q = q ~ E = qe = qe2 r 23

24 ) we see that qe = qe2 r E = E E2 r 1 = 2 r = r db 2 dt d B n general emf for any closed path can be expressed as line integral E = ~E d~s ~E d~s = d dt dt nduced electric field is a nonconservative field that is generated by a changing magnetic field 24

25 ) Faraday s Law becomes C ~E ind d~s = d dt Z ~B d ~ A Direction of d ~ A L.H. = ntegral around a closed loop C R.H. = ntegral over a surface bounded by C determined by direction of line integration C (Right-Hand Rule) 25

26 UMMARY Regular -field nduced ~E -field ~E created by charges created by changing B-field ~E -field lines start from E ~ -field lines form closed loops +q and end on q charge can define electric potential so that we can discuss potential difference between two points Conservative force field Electric potential cannot be defined (or, potential has no meaning) Non-conservative force field Classification of electric and magnetic effects depend on frame of reference of observer!!! e.g. For motional emf observer in reference frame of moving loop ~E will NOT see an induced -field but just a regular -field To be continued next semester in pecial Relativity same bat-time, same bat-channel 26 ~E

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