Download B 1 - Purdue Physics

Last time
MAGNETIC FORCE
point charge
Result of Cross Product
is Perpendicular to both
Right-Hand Rule:
1)
2)
1
and
Today
• Magnet force on currents
• Hall effect
• Relativity effect
iClicker Question
Small metal ball has charge q = +0.05C and
mass, m = 0.025kg. Ball enters a region of
magnetic field B = 0.5 T that is perpendicular
to its velocity v = 200m/s. Centripetal
acceleration = v2/r. What is radius of the
curve ball will move thru in magnetic field?
A.
B.
C.
D.
E.
200m
100m
250m
120m
50m
ẑ
v  vyˆ
x̂
. .
. .
. .
. .
B  Bxˆ
3
ŷ
ẑ
v  vyˆ
x̂
. .
. .
. .
. .
B  Bxˆ
ŷ
a) F = qv1B = (0.05)(200)(0.5)
= 5N
b) (see diagram) Force in –z
direction
c) F = ma = qv1B = 5N
a = 5N/0.025kg = 200 m/s2
d) v2/r = 200m/s2 , r = v2/(200m/s2) =
(200m/s)2/200m/s2 = 200m
4
Biot-Savart Law
BIOT-SAVART LAW
point charge
We need to understand
how these are related
BIOT-SAVART LAW
current in a wire
= length of this
chunk of wire
5
GOAL: Show
(wire)
(point charge)
(POSITIVE)
POINT CHARGE
MANY POINT
CHARGES
N particles
particles per
volume ΔV
ΔV
Move the vector symbol
CHUNK
OF WIRE
= length of this
chunk of wire
6
Magnetic Force on a charge or wire
MAGNETIC FORCE
point charge
We just showed how
these are related
MAGNETIC FORCE
current in a wire
= length of this
chunk of wire
7
Magnetic Force on a Wire
Current-carrying wire in an applied magnetic field B
Bapplied
Bapplied
I
Which way will the wire move?
8
Very Close to the Wire
Very close to the wire: r << L
CLOSE TO
THE WIRE
B
I
9
http://physick.wikispaces.com/Electric+Current
iClicker question: Two long parallel wires carry
currents of 5 A and 10 A in the same direction
as shown. What is the direction of the magnetic
force acting on the 10-A wire?
a)
b)
c)
d)
e)
Perpendicular to the plane of the
page and into the page
Perpendicular to the plane of the
page and out of the page
Downward
Inward toward the other wire
Outward away from the other wire
10
Force Between Parallel Wires
Bapplied = Magnetic field applied by the red wire. Blue wire feels a force down.
Bapplied
I
Bapplied
Fon blue wire
I
What about reciprocity? (Equal and opposite forces)
11
iClicker question: Two long parallel wires carry
currents of 5 A and 10 A in opposite directions
as shown. What is the direction of the magnetic
force on the 10-A wire?
a)
b)
c)
d)
e)
Perpendicular to the plane of the
page and into the page
Perpendicular to the plane of the
page and out of the page
Upward
Downward
Outward away from the other wire
12
Force Between (Anti) Parallel Wires
Bapplied = Magnetic field applied by the red wire. Blue wire feels a force up.
x
x
Bapplied
Fon blue wire
I
Bapplied
x
Fon red wire
Bapplied
I
x
Bapplied
Bapplied = Magnetic field applied by the blue wire. Red wire feels a force down.
13
Hall Effect
By measuring the Hall effect for a particular material,
we can determine the sign of the moving particles
that make up the current
Why would it be anything other than electrons? (Negative charges)
Semiconductors: sometimes current is carried by electrons,
but sometimes it is carried by the "holes".
In semiconductors, "holes" (missing electrons)
in the electron sea behave like positive charges.
14
Hall Effect
MAGNETIC FORCE
point charge
A hole in the electron sea
behaves like a proton.
Inside a Material:
Proton
or "Hole"
x
Bapplied
+ + + + + + + + + + + Moving holes
get pushed
to the top
x
Bapplied
Electron
- - - - - - - - - - 15
Moving electrons
get pushed
to the bottom
Moving holes
get pushed
to the top
Moving electrons
get pushed
to the bottom
Hall Effect
MAGNETIC FORCE
point charge
A hole in the electron sea
behaves like a proton.
Inside a Material:
+ + + + + + + + + + +
ΔV = Hall Voltage
- - - - - - - - - - x
Bapplied
+ + + + + + + + + + +
How long does this go on?
 Until the Hall Voltage
is strong enough to balance
the magnetic force
ΔV = Hall Voltage
- - - - - - - - - - 16
Measuring the Hall Effect
http://www.machinerylubrication.com
1. Apply B-Field
2. Apply Current I
3. Measure "Hall Voltage"
17
Currents Due to Magnetic Forces
Metal bar
F = qE + qv ´ B
Fm = ( -e) v ´ B
polarization
Fm
How much force do we
need to apply to keep
the bar moving at constant
speed?
18
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 )
19
Reference Frame
m0 qv ´ rˆ
B=
=0
2
4p r
Any magnetic field?
m0 qv ´ rˆ
B=
¹0
2
4p r
charged tape
20
iClicker Question
Jessie
Jack
charged tape
In case Jack have two positively changed tapes side by side. For the interaction
between the two tapes :
A). Jack and Jessie: purely through E force.
B). Jack: only through E force. Jessie: only through B force.
C). Jack and Jessie: through a mixture of B and E forces.
D). Jack: though only E forces. Jessie: through a mixture of B and E forces.
21
Magnetic Forces in Moving Reference Frames
Two protons
+e
1
r
v
F21,m
2 B1
+e v
E1
F21,e
Electric force: (assume independent of frame?)
1 e2
F21,e = 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
22
Magnetic Forces in Moving Reference Frames
+e
1
r
Electric force:
F21,e
1 e2
=
4pe 0 r 2
F21,m
m0 e 2 v 2
=
4p r 2
v
F21,m
2 B1
+e v
E1
F21,e
Magnetic force:
Ratio:
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
23
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
1 e2 æ v 2 ö
ç1 - 2 ÷÷
=
2 ç
4pe 0 r è c ø
downward
24
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
v
r
F21,m
2 B1
+e v
E1
Einstein 1905:
“On the electrodynamics of moving bodies”
Twin paradox
F21,e
Who will see protons hit
floor and ceiling first?
Time must run slower in moving frame.
http://en.wikipedia.org/wiki/Twin_paradox
25
Relativistic Field Transformations
Our detailed derivations are not correct for relativistic speeds,
According to the theory of relativity:
(E
- vBz )
E = Ex
E =
Bx' = Bx
v
æ
ö
B
+
E
ç y
z÷
2
c
ø
By' = è
1 - v 2 / c2
'
x
'
y
y
1- v / c
2
2
E =
'
z
(E
z
+ vB y )
1 - v 2 / c2
v
æ
ö
B
E
ç z
y÷
2
c
ø
Bz' = è
1 - v 2 /26
c2
Magnetic Field of a Moving Particle
q
E=
B=0
2
4pe 0 r
v
æ
ö
v
B
E
ç z
÷
- 2 Ey
y
2
c
ø=
c
Moving: Bz' = è
1 - v2 / c2
1 - v 2 / c2
Still:
1
v 1 q
Slow case: v<<c  B = - 2
c 4pe 0 r 2
1
= c2
m0 qv Field transformation is consistent
'
m0e 0
Bz = with Biot-Savart law
4p r 2
'
z
Electric and magnetic fields are interrelated
Magnetic fields are relativistic consequence of electric fields
27
Electric Field of a Rapidly Moving Particle
E = Ex
'
x
E =
'
y
E =
'
y
(E
y
- vBz )
1- v / c
2
2
Ey
1- v / c
2
2
E =
'
z
E =
'
z
28
(E
z
+ vB y )
1 - v 2 / c2
Ez
1 - v 2 / c2
Interpretation depends on reference frame
E
Blab = 0,0,- B
Fm = qv ´ B
Fe = qE
v
E  Ex =0
'
x
E
E =
'
y
'
y
E


y
 vBz 
1 v / c
2
- vBz
1- v / c
2
2
2
E
'
z
E


z
 vBy 
1 v / c
2
» vB if v << c
29
2
0