PHYS 2421 - Fields and Waves
Idea: We have seen: currents can produce fields We will now see: fields can produce currents Facts: Current is produced in closed loops when the magnetic flux changes Notice: As the magnet moves in and out, the -flux changes creating a current Direction of current changes if field increases or decreases Magnitude of current proportional to velocity of magnet, i.e. to rate of change of -flux No change of -flux no current
A similar case And another one As the coil moves in and out, the -flux changes creating a current Direction of current changes if field increases or decreases Magnitude of current proportional to velocity of coil, i.e. to rate of change of -flux No change of -flux no current No motion of coils As the current in inner coil sets in, the -flux changes creating a current With steady current in inner coil there is no change in -flux no current Direction of current changes if current in inner coil changes
More cases And yet one more case Constant -flux no current Decreasing -flux + current Increasing -flux -- current
Summary of Section 29.1 A changing flux produces a current, i.e. it can be taken as a EMF d dt Changing flux Sign to be explained later... EMF A curious application:
A changing flux produces a current, i.e. it can be taken as a EMF Consider a field traversing an area The magnetic flux is AAcos If the field (or the area) is changing, its rate of change is d dt and the EMF it produces is d dt Faraday s law of induction Michael Faraday 1791-1867
A A d d A d A dt dt dt d d A 0.020 2 T/s 0.012 m 0.24mV dt dt Check units: T N N N J s Ams C / s s C C 2 2 m m m m V Hmwk: Probls. 29.2, 5, 9 (11 th Ed.) or 2, 4, 9 (12 th Ed.) Soln 29.2: a) 1.44x10-5 T, 0; b) 3.6x10-4 V
Summary of Section 29.2 Faraday s law of induction d dt Hmwk Sect/ 29.2: Probls. 29.2, 5 and 9 (11 th Ed.) or 29.2, 4 and 9 (12 th Ed.) or
The induced field will have the Lenz s law direction needed to reduce the change in flux ook s version: The direction of any magnetic induction effect is such as to oppose the cause of the effect Complicated? Let us see some examples
Direction of induced EMF There are two possible directions for the induced current, which is the correct one? A changing flux Induces a current in wire which produces a second field The induced field will have the direction needed to reduce the change in flux More examples Lenz s law Homework : Problems 29.16 and 17 (11 th Ed.) or 15 and 17 (12 th Ed.)
Increasing induced should oppose original Which current direction produces this induced? Decreasing induced should align original Which current direction produces this induced? Increasing induced should oppose original Which current direction produces this induced?
Summary of Section 29.3 Lenz s law The induced field will have the direction needed to reduce the change in flux Homework : Problems 29.16 and 17 (11 th Ed.) or 15 and 17 (12 th Ed.)
Consider the following example Moving conducting bar Increasing area increasing flux Induced current on circuit Magnitude and direction of EMF?
d d A d A dt dt dt d Lx dx L Lv dt dt Direction? The induced field will reduce the change in flux Increasing area increasing flux induced should oppose original Counterclockwise direction Magnitude and direction of electric field induced?
Magnitude and direction of electric field induced? V ab vl Take field inside wire as uniform Vab EL E v In general, for any moving loop v dl Equivalent to d dt Homework: Problem 29.20 (11 th or 12 th Eds.) (Answer: a) 5.6 V; b) counterclockwise; c) 0.22 A)
EMF: vl 2.5 m/s 0.6 T 0.10 m 0.15 V Induced current Force on rod F Extra: E field inside rod: E I 0.15 V 5.0 R 0.03 IL 5 A 0.10 m 0.6 T 0.3 N A v 2.5 m/s 0.6 T 1.5 Tm/s 1.5 N/C Homework: Problems 29.24 and 25 (11 th Ed.) or 25 and 26 (12 th Ed.)
Summary of Section 29.4 EMF: Lv Electric field E v Homework: Problems 29.20, 24 and 25 (11 th Ed.) or 20, 25 and 26 (12 th Ed.)
Idea: A changing flux Induces an EMF (voltage) Which induces an E-field d dt Which induces a current on the circuit What is the electric field induced?
Remember connections between EMFs and E-fields Parallel plates: V Ed E V / d Potential gradient (Sect 23.5) a a b b V V E dl dv E dl V x, V E Ey, E V z x y z In general: d dt E dl From which the E-field can be obtained
d dt E dl For a loop, eg. d 1 d E dl Edl E2 r E dt 2 r dt The direction is determined using the right-hand rule Let us see an example...
d d di A 0n A dt dt dt Wb turns A 4 10 500 100 4 10 m Am m s 2510 7 4 6 V 1 d 1 di E 0n A 2r dt 2r dt 1 6 4 2510 V 210 V/m 2 0.02 m 2
Summary of Section 29.5 For a loop: A changing flux induces an EMF which induces an E-field d 1 d E dl E dt 2 r dt Homework sect. 29.5: Probl. 29.31 (11 th Ed.) or 30 (12 th Ed.)
PHYS 2421 - Fields and Waves