Download Lecture 13 - UConn Physics

Physics 1202: Lecture 13
Today’s Agenda
• Announcements:
– Lectures posted on:
www.phys.uconn.edu/~rcote/
– HW assignments, solutions etc.
• Homework #4:
– Not this week ! (time to prepare midterm)
• Midterm 1:
– Friday Oct. 2
– Chaps. 15, 16 & 17.
Faraday's Law
n
B
B
N
q
S
v
S
B
N
v
FB = B· A = BAcosq
DF B
e =Dt
B
Faraday's Law
• Define the flux of the magnetic field B through a surface
A=An from:
n
FB = B· A = BAcosq
B
q
B
• Faraday's Law:
The emf induced around a closed circuit is determined by
the time rate of change of the magnetic flux through that
circuit.
DF B
e =Dt
The minus sign indicates direction of induced current
(given by Lenz's Law).
Faraday’’s law for many loops
• Circuit consists of N loops:
all same area
FB magn. flux through one loop
loops in “series”
emfs add!
DF B
e = -N
Dt
• Lenz's Law:
Lenz's Law
The induced current will appear in such a direction that it
opposes the change in flux that produced it.
B
B
S
N
v
N
S
v
• Conservation of energy considerations:
Claim: Direction of induced current must be so as to
oppose the change; otherwise conservation of energy
would be violated.
» Why???
• If current reinforced the change, then the change
would get bigger and that would in turn induce a
larger current which would increase the change,
etc..
Motional EMF
B
+
xxxxxxxxxx
+
Charges in the conductor experience
the force (electron = negative)
xxxxxxxxxx
FB = qv ´ B
-
v
FB
l
xxxxxxxxxx
xxxxxxxxxx
-
xxxxxxxxxx
-
The charges will be accumulated on
both ends of the conductor producing
the electric field E.
The accumulation of charges will stop when the magnetic force qvB is
balanced by electric force qE. Condition of equilibrium requires that
qE = qvB Þ E = vB
The electric field produced in the conductor is related to the potential
difference across the ends of the conductor
DV = El = lvB
Calculation
• Suppose we pull with velocity v a
coil of resistance R through a
region of constant magnetic field
B.
– What will be the induced current?
» What direction?
• Lenz’ Law  clockwise!!
– What is the magnitude?
» Magnetic Flux:
» Faraday’s Law:
\
xxxxxx
xxxxxx
xxxxxx
xxxxxx
x
DF B
e =Dt
DF B
Dx
= wB
= wBv
Dt
Dt

I
w
v
Calculation
• When pulling on the loop, power is
spent: a force must be applied to
compensate Fmag = IBw (to the left) x x x x x x
– Assuming constant v
xxxxxx
F
– F = -Fmag (i.e. to the right)
mag
xxxxxx
P = F v (from 1201)
–
so P=(IBw) v = B2w2v2 / R
xxxxxx
x
• But P = RI2

\
P = R (wBv / R)2 = B2w2v2 / R
Power is the same (as it should)
I
F
w
v
DB  E
• Faraday's law  a changing B induces
an emf which can produce a current in
a loop.
• In order for charges to move (i.e., the
current) there must be an electric field.
 \ we can state Faraday's law more
generally in terms of the E field which
is produced by a changing B field.
x x xEx x x x x x x
E
xxxxxxxxxx
r
xxxxxxxxxx
B
xxxxxxxxxx
E
x x x x x x x xEx x
• Suppose B is increasing into the screen as shown above. An E
field is induced in the direction shown. To move a charge q
around the circle would require an amount of work =
Example
An instrument based on induced emf has been used to measure projectile
speeds up to 6 km/s. A small magnet is imbedded in the projectile, as
shown in Figure below. The projectile passes through two coils separated
by a distance d. As the projectile passes through each coil a pulse of emf
is induced in the coil. The time interval between pulses can be measured
accurately with an oscilloscope, and thus the speed can be determined.
(a) Sketch a graph of DV versus t for the arrangement shown. Consider a
current that flows counterclockwise as viewed from the starting point of
the projectile as positive. On your graph, indicate which pulse is from coil
1 and which is from coil 2.
(b) If the pulse separation is 2.40 ms and d = 1.50 m, what is the projectile
speed ?
A Loop Moving Through a Magnetic Field
Schematic Diagram of an AC Generator
D
d (cos( wt))
dt
= - w sin( wt)
D (cos( wt))
DF B
= - NAB w sin( wt )
NAB
=
e= -N
Dt
Dt
(a) As the conducting plate enters the field (position 1), the eddy currents are counterclockwise. As the plate leaves
the field (position 2), the currents are clockwise. In either case, the force on the plate is opposite to the velocity, and
eventually the plate comes to rest. (b) When slots are cut in the conducting plate, the eddy currents are reduced and
the plate swings more freely through the magnetic field.
Transformers
• Device to change e (or the voltage)
• 2 coils wrapped around iron core
• Primary (P) and secondary (S)
• B-field inside core
• Time varying current in P (with NP)
DF B
=
eP - N P
Dt
• Time varying flux induces emf in secondary coil S (with NS)
DF B
=
eS - N S
Dt
• Same varying flux
• Faraday's Law:
\ DF B = - e P = - e S
Dt
NP
NS

NS
eS = eP
NP
Induction
Self-Inductance, RL Circuits
XXX
XXXX
XX
1
L/R
e1
V
f( x ) 0.5
L
0.0183156
0
0
1
2
3
4
t
Self-Inductance
• Consider the loop at the right.
• switch closed  current starts to flow in
the loop.
X XX X
X XX XX X
X XX X
• \ magnetic field produced in the area
enclosed by the loop.
• \ flux through loop changes
• \ emf induced in loop opposing initial emf
• Self-Induction: the act of a changing current through a
loop inducing an opposing current in that same loop.
Self-Inductance
• The magnetic field produced by the
current in the loop shown is
proportional to that current.
I
• The flux, therefore, is also proportional
to the current.
FB = B· A = BAcosq µI
• We define this constant of proportionality
between flux and current to be the
inductance L.
DF B
• We can also define the inductance L, e = Dt
using Faraday's Law, in terms of the
emf induced by a changing current.
Lº-
DI
=L
Dt
e
DI / D t
Self-Inductance
• The inductance of an inductor ( a set of coils in some
geometry, e.g., solenoid, toroid) then, like a capacitor, can be
calculated from its geometry alone if the device is constructed
from conductors and air.
• If extra material (e.g. iron core) is added, then we need to add
some knowledge of materials as we did for capacitors
(dielectrics) and resistors (resistivity)