PHYS 2421 - Fields and Waves
Idea: We have seen: currents can produce B fields
We will now see: B fields can produce currentsFacts: Current is produced in closed loops when the magnetic flux changes
Notice: As the magnet moves in and out, the B-flux
changes creating a current Direction of current changes if B field
increases or decreases Magnitude of current proportional to velocity
of magnet, i.e. to rate of change of B-flux No change of B-flux no current
A similar case
As the coil moves in and out, the B-flux changes creating a current
Direction of current changes if B field increases or decreases
Magnitude of current proportional to velocity of coil, i.e. to rate of change of B-flux
No change of B-flux no current
And another one
No motion of coils As the current in inner
coil sets in, the B-flux changes creating a current
With steady current in inner coil there is no change in B-flux no current
Direction of current changes if current in inner coil changes
More cases
Constant B-flux no current
Decreasing B-flux + current
Increasing B-flux -- current
And yet one more case
Summary of Section 29.1
Bddt
A changing flux produces a current, i.e. it can be taken as a EMF
EMF
Changing flux
Sign to be explained later . . . A curious
application:
Bddt
A changing flux produces a current, i.e. it can be taken as a EMF
cosB B A BA
Consider a B field traversing an area
The magnetic flux is
BddtIf the field (or the area)
is changing, its rate of change is
and the EMF it produces is
Faraday’s law of induction
Michael Faraday
1791-1867
Hmwk: Probls. 29.2, 5, 9 (11th Ed.), 2, 4, 9 (12th or 13th Eds.)
Soln 29.2: a) 1.44x10-5 T, 0; b) 3.6x10-4 V
0.020 0.012 0.242 T/s m mVBd dB Adt dt
B d BAd dB Adt dt dt
B B A BA
Check units:
2 2
/T N N N Jm m m m Vs Ams C s s C C
Summary of Section 29.2
Bddt
Faraday’s law of
induction
Hmwk Sect/ 29.2: Probls. 29.2, 5 and 9 (11th Ed.) or
29.2, 4 and 9 (12th or 13th Eds.)
The induced B field will have the direction needed
to reduce the change in flux
Lenz’s law
Book’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
A changing fluxInduces a current in
wirewhich produces a second B
fieldThe induced B field will have the direction needed to reduce the
change in fluxLenz’s
lawMore examples
There are two possible directions for the induced current, which is the correct one?
Homework : Problems 29.16 and 17 (11th Ed.) or
15 and 17 (12th or 13th Eds.)
Which current direction produces this induced B?
Increasing B induced B should oppose original B
Which current direction produces this induced B?
Decreasing B induced B should align original B
Which current direction produces this induced B?
Increasing B induced B should oppose original B
Summary of Section 29.3
The induced B field will have the direction needed
to reduce the change in flux
Lenz’s law
Homework : Problems 29.16 and 17 (11th Ed.) or
15 and 17 (12th and 13th Eds.)
Consider the following example
Moving conducting
barIncreasing
area increasing flux
Induced current on
circuitMagnitude and direction of
EMF?
B d BA d Ad Bdt dt dt
d Lx dxB BL BLvdt dt
Direction? The induced B field will reduce the change in fluxIncreasing area increasing flux
induced B should oppose original B
Counterclockwise direction
Magnitude and direction of electric field induced?
abV vBL
Homework: Problem 29.20 (11th or 12th Eds.), or 28 (13th Ed.)
(Answer: a) 5.6 V; b) counterclockwise; c) 0.22 A)
Magnitude and direction of electric
field induced?
Take field inside wire as uniform
abV EL E vB
v B dl
In general, for any moving
loopBd
dt
Equivalent to
2.5 0.6 0.10
0.15
m/s T m V
vBL
0.15 5.0 V A0.03 I
R
Extra: E field inside rod:
EMF:
Induced current
5 0.10 0.6 0.3 A m T NF ILB Force on rod
2.5 0.6 1.5 1.5 m/s T Tm/s N/CE vB
Homework: Problems 29.24 (11th Ed.), or 25 (12th Ed.)
or 27 (13th Ed.)
Summary of Section 29.4
BLv
E vB
EMF:
Electric field
Homework: Problems 29.20 and 24 (11th Ed.) or
20 and 25 (12th Ed.), or 28 and 27 (13th Ed.)
Idea:A changing
flux
Which induces a current on the circuitWhat is the electric field
induced?
Bddt
Induces an EMF
(voltage)Which induces an E-
field
Remember connections between EMFs and E-fields
Parallel plates:
/V Ed E V d a
a bb
V V E dl dV E dl
Potential gradient (Sect 23.5)
, , x y zV V VE E Ex y z
In general:
Bd E dldt
From which the E-field can be obtained
For a loop, eg.
122
B Bd dE dl Edl E r Edt r dt
The direction is determined
using the right-hand rule
Bd E dldt
Let us see an example . . .
7 4
0
6
4 10 500 100 4 10
25 10
2Wb turns A m
Am m s
V
Bd dB dIA n Adt dt dt
0
6 4
1 12 2
1 25 10 2 102 0.02
V V/m m
Bd dIE n Ar dt r dt
Homework sect. 29.5: Probl. 29.27 & 31 (11th Ed.),
or 28 & 30 (12th Ed.), or 36 & 40 (13th Ed.)
Summary of Section 29.5A changing flux induces an
EMF which induces an E-field
12
B Bd dE dl Edt r dt
For a loop:
PHYS 2421 - Fields and Waves