Download Ch. 21.1-21.4

A
B
C
D
Exercise: A Complicated Resistive Circuit
Find currents through resistors
I2
loop 1:
emf  r1I1  R1I1  R4 I 4  R7 I1  0
Loop 2
loop 2:
I1
 R2 I 2  r2 I 2  emf  R6 I 2  R3 I 3  0
I3
Loop 1
Loop 3
I4
Loop 4
loop 3:
I5
R4 I 4  R3 I 3  R5 I 5  0
nodes:
I1  I 2  I 3  I 4  0
I3  I 2  I5  0
I 4  I 5  I1  0
Five independent equations and five unknowns
Chapter 21
Magnetic Force
Magnetic Field of a Moving Charge
The Biot-Savart law for a
moving charge
 0 qv  rˆ
B
4 r 2
The Biot-Savart law for a
short piece of wire:

 0 Il  rˆ
B 
4 r 2
How does magnetic field affects other charges?
Magnetic Force on a Moving Charge
Direction of the magnetic force depends on:
the direction of B
the direction of v of the moving charge
the sign of the moving charge

 
Fmagnetic  qv  B
N
N
T

C  m/s A  m
q – charge of the particle
v – speed of the particle
B – magnetic field
Right Hand Rule for Magnetic Force

 
Fmagnetic  qv  B
Electron charge = -e:
The magnetic force on a moving electron is in opposite direction to
the direction of the cross product 𝑣 × 𝐵
Effect of B on the Speed of the Charge

 
Fmagnetic  qv  B
What is the effect on the magnitude of speed?
 
F  dl  0
Kinetic energy does not change
Magnetic field cannot change a particle’s energy!
Magnetic field cannot change a particle’s speed!
Magnetic force can only change the direction of velocity but
not its magnitude
Magnitude of the Magnetic Force

 
Fmagnetic  qv  B
Single electron in uniform B:

dp
 qvB sin 
dt

dv
q
 vB sin 
dt m
(v<<c)
e/me = 1.78.1011 C/kg
𝑓𝑜𝑟 𝑣 = 3 × 106 m/s ; 𝐵 = 1T
𝑎= (1.78.1011 C/kg) (3 × 106 m/s)(1T) = 5.3 × 106 m/s2
Same as acceleration due to an E= 3 × 106 V
𝑓𝑜𝑟 𝑣 = 0 ?
Motion in a Magnetic Field

 
Fmagnetic  qv  B
Confined area: deflection
What if we have large (infinite) area with constant Bv

dp
 qvB
dt
Circular Motion at any Speed

 
Fmagnetic  qv  B
Any rotating vector:


dX
  X …angular speed
dt


dp


d
p
 p
 qv  B  q vB sin 90o
dt
dt

mv
1 v / c
2
2
 q vB
qB

1  v2 / c2
m
Cyclotron Frequency
Circular Motion at Low Speed
qB

1  v2 / c2
m
qB
if v<<c:  
m
independent of v!
Alternative derivation:
F  ma
2
v
q vBsin 90o  m
R
q B  m
Period T:
2

T
Circular motion:
v2
v
a

R
R
qB

m
m
T  2
qB
Non-Relativistic
Determining the Momentum of a Particle

dr
 v  r
dt
Position vector r:

dp
v
 p    p
dt
r
v

r
Circular motion

dp
 q vB
dt
v
  p  q vB
r
p  q Br
valid even for relativistic speeds
Used to measure momentum in high-energy particle experiments
Determining e/m of an Electron
p  q Br
𝑣 = 𝐵𝑟 𝑞 /𝑚
2
mv
 q V
2
q
v  2V
m
2
𝑞
𝑣2
=
𝑚 2∆𝑉
𝑞
𝐵𝑟 𝑞 /𝑚
=
𝑚
2∆𝑉
𝑞
2∆𝑉
=
𝑚
𝐵𝑟 2
e 2V
 2 2
m Br
2
Joseph John Thomson (1856-1940)
1897: m/e >1000 times smaller than H atom
Clicker Question
Clicker Question
Exercise

v||

v
What if v is not perpendicular to B?

 
Fmagnetic  qv  B
Direction?
Magnitude?
𝐹𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐 = 𝑞 𝑣⊥ + 𝑣∥ × 𝐵
Fmagnetic  qv B
Fmagnetic  qvBsin
𝑣∥ 𝑢𝑛𝑐ℎ𝑎𝑛𝑔𝑒𝑑; Trajectory: helix
Exercise: Circular Motion

 
Fmagnetic  qv  B
Which direction is electron
going?
The Lorentz Force
Can combine electric and magnetic forces:

 
Fm  qv  B


Fe  qE


 
F  qE  qv  B
Coulomb law and Biot-Savart law have coefficients 1/(40) and
0/(4) to make the field and force equations consistent with
each other
A Velocity Selector


 
F  qE  qv  B
Is it possible to arrange E and B fields so that the total force
on a moving charge is zero?
𝐹 = 𝑞(𝐸 + 𝑣 × 𝐵)

 
E  v  B
E
E  vB
FE
What if v changes?
FB
B
E
v
B
A negative charge is placed at rest in a magnetic field as
shown below. What is the direction of the magnetic force
on the charge?
B
A.
B.
C.
D.
E.
Up
Down
Into the page
Out of the page
No force at all.
A negatively charged particle is moving horizontally to the right
in a uniform magnetic field that is pointing in the same direction
as the velocity. What is the direction of the magnetic force on
the charge?
𝒗
A.
B.
C.
D.
E.
Up
Down
Into the page
Out of the page
No force at all.
B
Now, another negatively charged particle is moving upward and
to the right in a uniform magnetic field that points in the
horizontal direction. What is the direction of the magnetic force
on the charge?
𝒗
A.
B.
C.
D.
E.
Left
Up
Down
Into the page
Out of the page
B
Magnetic Force on a Current-carrying Wire

 
Fm  qv  B
Current: many charges are moving
Superposition: add up forces on individual charges
Number of moving charges in short wire:
nAl
Total force:
I
𝐹𝑚 = 𝑛𝐴∆𝑙 𝑞𝑣 × 𝐵 = 𝑛𝑞𝐴𝑣 ∆𝑙 × 𝐵
 

Force of a short wire: Fm  Il  B
In metals: charges q are negative. Will this equation still work?
Hall Effect
F = qE + qv ´ B
When does it reach equilibrium?
Fe^ + FB = 0
E
v
h
eEe^ = -ev ´ B
𝐸𝑒⊥ = 𝑣 𝐵
B
V>0
????
-
∆𝑉 = 𝐸𝑒⊥ ℎ = 𝑣 𝐵 ℎ
+
Hall Effect for Opposite Charges
Fm = qv ´ B
E
v
E
B
B
V>0
>0
-
+
v
V<0
????
-
+
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
Edwin Herbert Hall
(1855 - 1938)
Hall Effect in a Metal
What is the magnitude of the Hall effect in a metal?
DV = Ee^h = v Bh
𝐼 = 𝑞 𝑛𝐴𝑣
𝐼
𝑣=
𝑞 𝑛𝐴
𝐼𝐵ℎ
∆𝑉 =
~5 × 10−6 V
𝑞 𝑛𝐴
Measure ∆𝑉, know the charge (e)
Then we can find n
Monovalent metals: n is the
same as # of atoms per m3
Some metals: n is larger than
# of atoms per m3
Clicker Question
Voltmeter 1 reading is POSITIVE
Voltmeter 2 reading is POSITIVE
Mobile charges are:
A) Positive (holes)
B) Negative (electrons)
C) Not enough information