magnetism magnetic force. magnetic force outline lorentz force charged particles in a crossed field...
TRANSCRIPT
Magnetism
Magnetic Force
Magnetic Force Outline
• Lorentz Force
• Charged particles in a crossed field
• Hall Effect
• Circulating charged particles
• Motors
• Bio-Savart Law
Class Objectives
• Define the Lorentz Force equation.
• Show it can be used to find the magnitude and direction of the force.
• Quickly review field lines.
• Define cross fields.
• Hall effect produced by a crossed field.
• Derive the equation for the Hall voltage.
Magnetic Force
• The magnetic field is defined from the Lorentz Force Law, BvqEqF
Magnetic Force
• The magnetic field is defined from the Lorentz Force Law,
• Specifically, for a particle with charge q moving through a field B with a velocity v,
• That is q times the cross product of v and B.
BvqEqF
BvqF
Magnetic Force
• The cross product may be rewritten so that,
• The angle is measured from the direction of the velocity to the magnetic field .
• NB: the smallest angle between the vectors!
sinvBqF
v B
v x BB
v
Magnetic Force
Magnetic Force
• The diagrams show the direction of the force acting on a positive charge.
• The force acting on a negative charge is in the opposite direction.
+
-
v
F
F
B
Bv
Magnetic Force
• The direction of the force F acting on a charged particle moving with velocity v through a magnetic field B is always perpendicular to v and B.
Magnetic Force
• The SI unit for B is the tesla (T) newton per coulomb-meter per second and follows from the before mentioned equation .
• 1 tesla = 1 N/(Cm/s)
Bvq
F
sin
Magnetic Field Lines
Review
Magnetic Field Lines
• Magnetic field lines are used to represent the magnetic field, similar to electric field lines to represent the electric field.
• The magnetic field for various magnets are shown on the next slide.
Magnetic Field Lines
Crossed Fields
Crossed Fields
• Both an electric field E and a magnetic field B can act on a charged particle. When they act perpendicular to each other they are said to be ‘crossed fields’.
Crossed Fields
• Examples of crossed fields are: cathode ray tube, velocity selector, mass spectrometer.
Crossed Fields
Hall Effect
Hall Effect
• An interesting property of a conductor in a crossed field is the Hall effect.
Hall Effect
• An interesting property of a conductor in a crossed field is the Hall effect.
• Consider a conductor of width d carrying a current i in a magnetic field B as shown.
i i
d
xx x x
x xxx
xxxx
xxxx
B
Dimensions:
Cross sectional area: ALength: x
Hall Effect
• Electrons drift with a drift velocity vd as shown.
• When the magnetic field is turned on the electrons are deflected upwards.
i i
d
xx x x
x xxx
xxxx
xxxx
B
-vd
FB
FB
Hall Effect
• As time goes on electrons build up making on side –ve and the other +ve.
i i
d
xx x x
x xxx
xxxx
xxxx
B
-vd
- - - - -
+ + + + +High
Low
FB
Hall Effect
• As time goes on electrons build up making on side –ve and the other +ve.
• This creates an electric field from +ve to –ve.
i i
xx x x
x xxx
xxxx
xxxx
B
-vd
- - - - -
+ + + + +High
Low
FB
EFE
Hall Effect
• The electric field pushed the electrons downwards.
• The continues until equilibrium where the electric force just cancels the magnetic force.
i i
xx x x
x xxx
xxxx
xxxx
B
-vd
- - - - -
+ + + + +High
Low
FB
EFE
Hall Effect
• At this point the electrons move along the conductor with no further collection at the top of the conductor and increase in E.
i i
xx x x
x xxx
xxxx
xxxx
B
-vd
- - - - -
+ + + + +High
Low
FB
EFE
Hall Effect
• The hall potential V is given by, V=Ed
Hall Effect
• When in balance,EB FF
BeveE d
neA
ivwhere d
Hall Effect
• When in balance,
• Recall,
EB FF BeveE d
neA
ivwhere d
dt
dqi
dxneAdq
dx
A
A wire dt
dxneAi dvneA
Hall Effect
• Substituting for E, vd into we get,BeveE d
Vle
Bin
d
Alwhere
A circulating charged particle
Magnetic Force
• A charged particle moving in a plane perpendicular to a magnetic field will move in a circular orbit.
• The magnetic force acts as a centripetal force.
• Its direction is given by the right hand rule.
Magnetic Force
Magnetic Force
• Recall: for a charged particle moving in a circle of radius R,
• As so we can show that,
R
mvFB
2
R
mvqvB
2
qB
mvR
qB
mT
2
m
qBf
2,
m
qB,
Magnetic Force on a current carrying wire
Magnetic Force
• Consider a wire of length L, in a magnetic field, through which a current I passes.
x x
x
x
xx x
xI
B
Magnetic Force
• Consider a wire of length L, in a magnetic field, through which a current I passes.
• The force acting on an element of the wire dl is given by,
x x
x
x
xx x
xI
B
BLIdFd B
Magnetic Force
• Thus we can write the force acting on the wire, BIdLdFB
L
B dLBIF0
BILFB
Magnetic Force
• Thus we can write the force acting on the wire,
• In general,
BIdLdFB
L
B dLBIF0
BILFB
sinBILFB
Magnetic Force
• The force on a wire can be extended to that on a current loop.
Magnetic Force
• The force on a wire can be extended to that on a current loop.
• An example of which is a motor.
Interlude
Next….
The Biot-Savart Law
Biot-Savart Law
Objective
• Investigate the magnetic field due to a current carrying conductor.
• Define the Biot-Savart Law
• Use the law of Biot-Savart to find the magnetic field due to a wire.
Biot-Savart Law
• So far we have only considered a wire in an external field B. Using Biot-Savart law we find the field at a point due to the wire.
Biot-Savart Law
• We will illustrate the Biot-Savart Law.
Biot-Savart Law
• Biot-Savart law:
rldr
IBd ˆ
4 20
20
4
sin
r
IdldB
Biot-Savart Law
• Where is the permeability of free space.
• And is the vector from dl to the point P.
0ATm /104 7
0
r̂
Biot-Savart Law
• Example: Find B at a point P from a long straight wire.
l
Biot-Savart Law
• Sol: rldr
IBd ˆ
4 20
20
4
sin
r
IdldB
l
Biot-Savart Law
• We rewrite the equation in terms of the angle the line extrapolated from makes with x-axis at the point P.
• Why?
• Because it’s more useful. l
r̂
Biot-Savart Law
• Sol:
• From the diagram,
• And hence
rldr
IBd ˆ
4 20
20
4
sin
r
IdldB
180
90 l
Biot-Savart Law
• Sol:
• From the diagram,
• And hence
rldr
IBd ˆ
4 20
20
4
sin
r
IdldB
180
90 90sinsin cos
l
ABBABA cossincossinsin
Biot-Savart Law
• Hence,
• As well,
• Therefore,
20
4
cos
r
IdldB
x
ltan
r
xcos 22 lxr
x
dIdB
4
cos0
l
Biot-Savart Law
• For the case where B is due to a length AB,
A
B
d
x
idBB
B
cos4
0
0
sinsin
40 x
i
Biot-Savart Law
• For the case where B is due to a length AB,
• If AB is taken to infinity,
A
B
d
x
idBB
B
cos4
0
0
sinsin
40 x
i
x
iB
2
0