Download Lecture19

Announcements
Current Coil
•A single circular loop has an axial field
B( z ) 
0iR 2
•Use Biot-Savart law to derive
2( R 2  z 2 )3 / 2
B( z ) 
0iR 2
2( z 3 )
•Far away from the loop
Current Coil as a Dipole

0iR 2
B( z ) 
2( z 3 )


0iA
0 
B( z ) 

3
2 ( z ) 2 ( z 3 )
•Far away from the loop
Rewrite in different form
•So we can yet again consider a current-loop
as a magnetic dipole
Magnetic Flux
•Magnetic Flux is the amount of magnetic field flowing through
a surface
•It is the magnetic field times the area, when the field is
perpendicular to the surface
•It is zero if the magnetic field is parallel to the surface
•Normally denoted by symbol B.
•Units are T·m2, also known as a Weber (Wb)
Magnetic
Field
B = BA
Magnetic
Field
B = 0
Magnetic Flux
•When magnetic field is at an angle, only the part perpendicular
to the surface counts (does this ring a bell? Remember Gauss’s
B
law? )
•Multiply by cos 
Bn

•For a non-constant magnetic
field, or a curvy surface, you
have to integrate over the surface
B = BnA= BA cos 
 B   B  dA   B cos dA
Quiz
A sphere of radius R is placed near a very long straight wire
that carries a steady current I. The total magnetic flux
passing through the sphere is:
A) oI
B) oI/(4R2)
C) 4R2oI
D) zero
EMF from Magnets
•Wire sliding through a uniform magnetic field
F
v
q
•The wire contains charges q which feel a force due to
magnetic field
•The charges, if they are moving, undergo work
W  Fa  qvBa
E  W / q  vaB
a
EMF from Magnets II
v
b
db
dA
dB
B 
E  vaB   aB  
dt
dt
dt
•Force is caused by cutting across magnetic fields
dB
E 
dt
a
Faraday’s Law
10 
5T
10 m
As the bar moves a current is induced!
3 m/s
2m
dB
E 
dt
There are no batteries anywhere, so we say that a
current is induced, by an induced emf.
Hence, an electric current can be induced in a circuit by a changing
magnetic field, in the opposite direction to the change in flux.
Faraday’s Law
10 
5T
10 m
What is the current induced in this circuit?
A) 30A
B) 3A
C) 10A
D) 6A
3 m/s
2m
Changing the Flux
dB
E 
dt
1) Change the field within the coil
2) Changing the area of the coil (the explanation we did)
3) Changing the angle between the field and the coil (common method)
Lenz’s Law
•EMF causes current in opposite direction
compared to right-hand rule
•Current produces a magnetic field according to
the right-hand rule
•Current appears in the loop that tries to
maintain constant magnetic field
•Consider a conducting ring in a
magnetic field
•Suddenly, the magnetic field is reduced
•This causes current to start flowing
•Current recreates magnetic field
Arises from conservation of energy!
dB
E 
dt
Voltage, Current, and Power
10 
5T
3 m/s
10 m
E = vaB = 30 V
•What is voltage?
I = V/R = E/R = 3 A
•What is current?
•What is power
P = IV = I E = 90 W
consumed by resistor?
•What is force on wire
F = BIL = 30 N
due to the magnetic field?
P = Fv = 90 W
•What is power needed
to move wire?
2m
Faraday and Changing the Flux
The circuit shown is in a uniform magnetic field that is
into the page and is decreasing in magnitude at the rate 150 T/s.
The current in the circuit is:
dB
E 
dt
V=IR=> I=0.4A without considering the
field. The current flows counterclockwuse
Apply Faraday’s law: the changing flux is:
(0.12)(0.12)(-150), so the Emf is 2.2V, produces an
opposed current of 0.22A.
So the total current is 0.18A
Generality of Faraday’s Law
•Faraday’s law applies when moving a
wire in a magnetic field
dB
E 
dt
However, changing
magnetic fields also
produce an EMF.
What is the nature of the force?
It is not a magnetic force because the
charges are not necessarily moving
Changing magnetic fields must produce electric fields!
Faraday’s Law and Electric Fields
dB
E 
dt
q
E
 E  ds
dB
 E  ds   dt
Faraday’s Law and Electric Fields
dB
 E  ds   dt
. A cylindrical region of radius R = 3.0 cm contains a uniform magnetic field parallel to its
axis. The field is 0 outside the cylinder. If the field is changing at the rate 0.60 T/s,
the electric field induced at a point 2R from the cylinder axis is:
Using Faraday’s law: 2 (2R)E =-(R2) dB/dt,
so E= (-(R2) /4) dB/dt=0.0045 V/m
Inductance
•How much voltage does it take to change
the current through this solenoid?
N turns
Length l
dB
E 
N
dt
B   R B 
2
R
Current I
B
 0 NI
l
0 NR I
2
l
d  B 0 N 2 R 2 dI
V  E  N

dt
l
dt
Inductance
N turns
Length l
•Any loop of wire (especially a coil) will
have a certain reluctance to change its
current
•This reluctance is called the inductance
•Denoted by the letter L
dI
V  L
dt
V s
1 H 1
A
•Unit is Vs/A also called a henry (H)
•Circuit diagram looks like this:
R
Current I
Inductance of a Solenoid
N turns
Length l
d  B 0 N 2 R 2 dI
V  E  N

dt
l
dt
dI
V  L
dt
L
R
Current I
0 N A
2
l
 0 n 2 Al