Transcript Magnetic field lines and flux

Magnetic field lines and magnetic flux
The magnetic field is a vector field
We can introduce magnetic field lines in the same we introduced electric
field lines
Let’s recall: Properties of field lines:
-Imaginary curve such that tangent at any point is along the B-field in this point
-density of field lines in a given region allows to picture the magnitude of the B-field
At any particular point in space the B-field has a well defined direction
Only one field line can pass through each point
Field lines never cross
Anything new
Click Wastson
Magnetic field lines are not force lines !
So what do they visualize then:
Magnetic field lines have the direction that a compass needle would point at each location
Examples of magnetic field lines
Current through wire
Right hand rule gives
direction of field
Permanent magnet
Clicker question
What happens if you cut a permanent magnet in half?
1) You get a separate north pole and south pole similar to electric plus and
minus charges
2) There is no magnetic charge. Any permanent magnet has two poles,
if you cut a magnetic dipole in half you end up with two dipoles.
Magnetic flux
The magnetic field is a vector field
we can define a magnetic flux
Remember:
Flux : scalar quantity, ,
which results from a surface integration over a vector field.
B   B d A
A
vector
field
surface
magnetic flux
through surface
The SI unit of magnetic flux
B   [ B][ A]  T m 2  Wb
Wb read Weber in honor of Wilhelm Weber
Is there something like Gauss’ law for magnetic flux
Yes, and it surprisingly simple with deep fundamental meaning
Remember Gauss law for electric flux and let’s apply it by enclosing electric dipoles:
Since the total enclosed charge is zero we have
E 
 E dA
Qenclosed
0
0
has never been observed (and that is why physicists keep looking for it -> “I want my Nobel prize” )
Since there is no such thing as magnetic monopoles
B
dA
0
The magnetic flux through any closed surface is zero
(magnetic monopoles have never been observed,
magnetic field lines always close )
If you don’t feel sufficiently confused yet read also “Have physicists seen magnetic monopoles?”
Magnetic force on a current carrying conductor
Let’s consider a conducting wire carrying a current I=jA
in a B-Field  to the current density vector j
Lorentz force on an individual charge q with drift velocity
vd (remember Drude model) reads
F  qvd B
For N charges we have therefore a total force
F  Nqvd B
In the wire of volume V=l A we have N=n lA charges q
F  nlAqvd B using the transport expression for the current density
j
j  qnvd  vd 
qn
F  l Aj B
I
F  I l B If B makes an angle  with the wire
F  I l B sin 
F  I l B
Where l is a vector pointing in the direction
of the current and has the magnitude l
If the conductor is not straight
consider infinitesimal short segments contributing with d F
 I dl  B
F  dF
Example: Magnetic force on a curved conductor
Let’s find the total magnetic force on the conductor
Start with the straight segment:
l  le x
B  Be z
ex e y
l  B  l 0
0 0
F straight  IlBe y
ez
0  lBe y
B
Curved part:
r  R  cos ,sin  ,0
dr
d
d 
R  cos  ,sin  ,0  d
d
d
dl  R   sin  ,cos ,0 d
ex
ey

dl 
F curved   d F  I  d  R sin 
0
0
R cos 
0
ez
0
B

 I  d RB cos e x  RB sin  e y   IRBe x sin 
0
F total  IB  l  2R  e y

0
 IRBe y cos 
 2 IRBe y
The force a straight wire of length l+2R would experience.
That makes sense when considering the symmetry.
Luke, the Force runs strong in your family. Pass on what you have learned.
If you think you deserve a break see also http://www.youtube.com/watch?v=o7ENNyGlmQY

0
Force and torque on a current loop
r1  b / 2
Fb
T1 
sin 
2
r2 with r 2  b / 2
Fb
T2 
sin 
2
r1 with
T  Fb sin 
 IBab sin 
T  IA B sin 
Image from our textbook Young and Freedman
Magnitude of torque
on current loop
Let’s generalize into a full vector notation
T  IA B sin 
=: the absolute value of the magnetic moment of the loop
We are used to assign a vector to an area
It appears natural to define a vector
I A
A is normal to the loop area.
Its direction is determined by the right hand rule
Since  is the angle between

and B
T   B
Vector torque of on current loop
This equation is in wonderful analogy to the torque on an electric dipole in an E-field
The direct current motor
Click here for a java applet
http://www.magnet.fsu.edu/education/tutorials/java/dcmotor/index.html

T
B
From the right hand rule we get
direction of the magnetic moment
B-field points from N to S
Determines the direction of 