#### Transcript Ch. 21.5-21.9x

```Hall Effect for Opposite Charges
Fm = qv ´ B
E
v
E
B
B
V>0
>0
-
+
v
V<0
????
-
+
Magnetic Force on a Current-carrying Wire
Current: many charges are moving
Superposition: add up forces on individual charges
Fm = qv ´ B
Number of moving charges in short wire:
Total force:
I
Fm = ( nADl ) qv ´ B = ( nqAv ) Dl ´ B
Force of a short wire: Fm = IDl ´ B
In metals: charges q are negative. Will this equation still work?
Exercise
Fm = IDl ´ B
What is the total force on this wire?
What is the magnitude of the force
on part L if:
I = 10 A
B=1T
L = 10 cm
Fm = ILB = 1 N
Experiment
F
I
B
Fm = IDl ´ B
What will happen if current
is reversed?
What will happen if magnet
is reversed?
Forces Between Parallel Wires
Definition of 1 Ampere:
Ampere is defined as a current at which two very long
parallel wires 1 m apart create a force on each other of
2.10-7 N per meter length.
From this also follows that 0/(4) = 10-7 T.m/A
Forces Between Parallel Wires
For long wire:
m0 2 I1
B1 =
4p d
Magnetic force on lower wire:
Fm = IDl ´ B
F21 = I 2 LB1 sin 90 o
m0 2I1
F21 = I 2 L
4p d
Magnetic force on upper wire:
m0 2 I 2
B2 =
4p d
F12 = I1LB2 sin 90 o
m0 2I 2
F12 = I1 L
4p d
What if current runs in opposite directions?
Electric forces: “likes repel, unlikes attract”
Magnetic forces: “likes attract, unlikes repel”
Why does the Wire Move?
Magnetic force acts only on moving charges
Mobile electrons experience magnetic force
Atomic cores are not moving – no magnetic force!
Atomic cores feel an unbalanced force. Wire moves
down!
The motion of the wire is an
electric side-effect of the magnetic
force on the moving electrons.
Ee^ = v B
Currents Due to Magnetic Forces
Metal bar
F = qE + qv ´ B
Fm = ( -e) v ´ B
polarization
Static equilibrium:
F =0
Fm
(-e) E = -(-e)v ´ B
E = vB
Metal is in static equilibrium: but E  0 inside!
What is V ?
DV = EL = vBL
Currents Due to Magnetic Forces
Metal bar
Static equilibrium:
E = vB
DV = EL = vBL
Fm
How much force do we
need to apply to keep
the bar moving at constant
speed?
Does this polarized bar remind you
Non-Coulomb Work
Non-Coulomb force
drives e against FE
Fm
Non-Coulomb work:
FNC L = evBL
What is emf of this ‘battery’?
FNC L
= vBL ‘motional emf’
Non-Coulomb work per unit charge:
e
DV = emf = vBL
Bar may have some resistance rint: DV = emf - rint I
Round-trip Potential
What electric fields are
produced by the surface
charges?
Is round trip V zero?
E
E
DV = - ò E × dl
Round trip due to these Coulomb electric fields has to be zero.
Moving Bar and Energy Conservation
P=IV=I(emf)
Are we getting something for nothing?
Bar – current I:
FI
F
Fm
FI = IDl ´ B = -F
FI = ILB
Work:
emf = vBL
x
W = FDx = ILBDx
W
Dx
= ILB
Power: P =
Dt
Dt
Main principle of electric generators:
Mechanical power is converted to electric power
P = ILBv
P = I (emf )
Magnetic Torque on a Magnetic Dipole Moment
A current carrying loop has a tendency to twist in magnetic field
Compass needle: collection of atomic current loops
Fm = IDl ´ B
m = IA = Ihw
Magnetic Torque: Quantitative Analysis
m = Ihw
Torque () = distance from the axle
(lever arm) times perpendicular
component of the force.
Fm = IDl ´ B
Fm = IwB
F^ = IwB sin q
æh
ö
t = 2ç IwB sin q ÷ = IwhB sin q
è2
ø
t = mB sin q
t = m´B
Works with loops of any shape!
Magnetic Dipole Moment: Potential Energy
m = Ihw
Calculate amount of work
needed to rotate from angle I
to f:
é æ h öù
dE = 2 ê F^ ç dq ÷ ú
ë è 2 øû
qf
æh ö
W = DU m = ò 2IwBsin q ç dq ÷
è2 ø
q
i
qf
DU m = IwhB ò sinq dq = IwhB [ - cosq ]
qi
DUm = - m B éë cosq f - cosqi ùû
U m = - mB cosq
qf
qi
Potential energy for a
magnetic dipole moment
Um = -m × B
Magnetic Dipole Moment: Potential Energy
Potential energy for a
magnetic dipole moment
Um = -m × B
U= min
-µB
0
max
µB
0
What is the energy difference between the highest and the lowest state?
Picture of the U and µ in magnetic field – important in atomic
and nuclear physics.
Reference Frame
m0 qv ´ rˆ
B=
=0
2
4p r
Any magnetic field?
m0 qv ´ rˆ
B=
¹0
2
4p r
charged tape
Magnetic Forces in Moving Reference Frames
Two protons
Electric force:
F21,e
+e
1
r
v
F21,m
2 B1
+e v
E1
F21,e
1 e2
= q2 E1 =
rˆ
2
4pe 0 r
Magnetic field:
m0 q1v1 ´ rˆ
B1 =
4p r 2
Magnetic force:
F21,m = q2v2 ´ B1
F21,m
m0 e 2 v 2
= q2vB1 =
4p r 2
Magnetic Forces in Moving Reference Frames
Electric force:
+e
1
r
Magnetic force:
Ratio:
F21,e
m0e 0
F21,m
m0 e 2 v 2
=
4p r 2
v
F21,m
2 B1
+e v
E1
1
F21,e
1 e2
=
4pe 0 r 2
(
)
8 2
= 3 ´ 10 (m/s)2
=c2
it is not accidental!
F21,m æ m0 e2v 2 ö æ 1 e 2 ö
÷
÷ /ç
= çç
2 ÷ ç
2 ÷
F21,e è 4p r ø è 4pe 0 r ø
F21,m
= ( m0e 0 ) v 2
F21,e
F21,m v 2
= 2
F21,e c
Magnetic Forces in Moving Reference Frames
+e
1
r
F21,m v 2
= 2
F21,e c
v
F21,m
2 B1
+e v
E1
F21,e
Full Lorentz force:
For v<<c the magnetic force is much
smaller than electric force
How can we detect the magnetic force on
a current carrying wire?
F = F21,e - F21,m
downward
1 e2 æ v 2 ö
ç1 - 2 ÷÷
=
2 ç
4pe 0 r è c ø
Magnetic Forces in Moving Reference Frames
1 e2 æ v 2 ö
20 ns F = 4pe r 2 çç1 - c 2 ÷÷
0
è
ø
+e
1
2
1
e
15 ns F =
4pe 0 r 2
Who will see protons hit
floor and ceiling first?
Time must run slower in moving frame.
r
v
F21,m
2 B1
+e v
E1
F21,e
Einstein 1905:
“On the electrodynamics of moving bodies”
Moving Through a Uniform Magnetic Field
Blab = 0,0,- B
E
Fm = qv ´ B
Fe = qE
v
E = Ex
'
x
E =
'
y
E =
'
y
(E
y
- vBz )
1- v / c
2
2
- vBz
1- v / c
2
2
E =
'
z
(E
z
+ vB y )
1 - v 2 / c2
» vB if v << c
The Principle of Relativity
There may be different mechanisms for different observers
in different reference frames, but all observers can
correctly predict what will happen in their own frames,
using the same relativistically correct physical laws.
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