Physics 121 - Electricity and Magnetism
Lecture 09 - Charges & Currents in Magnetic Fields
Y&F Chapter 27, Sec. 1 - 8
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What Produces Magnetic Field?
Properties of Magnetic versus Electric Fields
Force on a Charge Moving through Magnetic Field
Magnetic Field Lines
A Charged Particle Circulating in a Magnetic Field –
Cyclotron Frequency
The Cyclotron, the Mass Spectrometer, the Earth’s Field
Crossed Electric and Magnetic Fields
The e/m Ratio for Electrons
Magnetic Force on a Current-Carrying Wire
Torque on a Current Loop: the Motor Effect
The Magnetic Dipole Moment
Summary
Copyright R. Janow Fall 2015
Magnetic Field


E and g
Electrostatic & Gravitational forces act through
 vector fields
Now: Magnetic force on moving charge
B  magnetic field
Force law first: Effect of a given B field on charges & currents
Next Lecture: How to create B field
• Currents in loops of wire
• Intrinsic spins of e-,p+  elementary currents  magnetic dipole moment
• Spins can align permanently to form natural magnets
B & E fields  Maxwell’s equations, electromagnetic waves (optional)
“Permanent Magnets”: North- and South- seeking poles
•
•
•
•
Natural magnets known in antiquity
Compass - Earth’s magnetic field
Ferromagnetic materials magnetize
when cooled in B field (Fe, Ni, Co) - Spins align into“domains” (magnets)
Other materials (plastic, copper, wood,…)
slightly or not affected (para- and
dia- magnetism)
ATTRACTS
REPELS
UNIFORM
S
N
Copyright R. Janow Fall 2015
DIPOLES ALIGN IN B FIELD
Electric Field versus Magnetic Field
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Electric force acts at a distance
through electric field.
Vector field, E.
Source: electric charge.
•
Positive (+) & negative (-) charge.
Opposite charges attract
Like charges repel.
Field lines show the direction and
magnitude of E.
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Magnetic force acts at a distance
through magnetic field.
Vector field, B
Source: moving electric charge
(current, even in permanent
magnets).
North pole (N) and south pole (S)
Opposite poles attract
Like poles repel.
Field lines show the direction and
magnitude of B.
Copyright R. Janow Fall 2015
Differences between magnetic & electrostatic field
Test charge and electric field


F
E E
q
Single electric poles exist
Cut up a bar magnet
Test monopole and magnetic field ?

B

FB
p
Single magnetic poles have never been
Found. Magnetic poles always occur as
dipole pairs..
small, complete magnets
There is no magnetic monopole...
...dipoles are the basic units

 q


Electrostatic
Magnetic
enc
E  dA 

S B  dA  0
0
S
Gauss Law
Gauss Law
Magnetic flux through each and every Gaussian surface = 0
Magnetic field exerts force on moving charges (current) only
Magnetic field lines are closed curves - no beginning or end
Copyright R. Janow Fall 2015
Magnetic Force on a Charged Particle
Define B by the magnetic force FB it exerts on a charged particle
moving with a velocity v
Typical Magnitudes:

Earth’s field: 10-4 T.
| FB |  qvBsin()
Bar Magnet: 10-2 T.
Electromagnet: 10-1 T.

 
FB  qv  B
“LORENTZ FORCE”
Units:
 1 Tesla  
 Newtons 
Coulombm / s
1 “GAUSS” = 10-4 Tesla
See Lecture 01 for cross product definitions and examples
• FB is proportional to speed v, charge q, and field B.
• FB is geometrically complex – depends on cross product
o F = 0 if v is parallel to B.
o F is normal to plane of both v and B.
o FB reverses sign for opposite sign of charge
o Strength of field B also depends on qv [current x length].
• Electric force can do work on a charged particle…BUT…
… magnetic force cannot do work on moving particles since FB.v = 0.
Copyright R. Janow Fall 2015
Magnetic force geometry examples:

A simple geometry
B  B0k̂
 
v  v0 î
î  k̂   ĵ
y
• charge +q
• B along –z direction
• v along +x direction

Fm

 
 Fm  qv  B
 q v0 B0 ĵ

B
CONVENTION
x

v
z
Head Tail
î
ĵ
k̂
More simple examples

Fm

F
• charge +q
• Fm is down
• |Fm| = qv0B0

v0

B

v0

B
• charge +q
• Fm is out of
the page
• |Fm| = qv0B0
x
x
x
x
x
x
 x v0 x
Fm
x

 x
B
x

v0
x
•
•
•
•
charge -q
B into page
Fm is still down
|Fm| = qv0B0
• charge +q or -q
• v parallel to B
• |Fm| = qv0B0sin(0)

 Fm  0
Copyright R. Janow Fall 2015
A numerical example
• Electron beam moving in plane of sketch
• v = 107 m/s along +x
• B = 10-3 T. out of page along +y
• z axis is down

B

Fm
e

v
force for +e
a) Find the magnetic force on an electron :

 
Fm  - e v  B  1.6x1019 x107 x103 x sin(90o ) k̂

Fm   1.6x1015 k̂
Negative sign means force is opposite to result of using the RH rule
b) Acceleration of electron :

 Fm  1.6 x 1015 N
 15
a

k̂

1.76
x 10
k̂

31
me
9 x 10 Kg
2
[m/s ]
Direction is the same as that of the force: Fov = 0
Copyright R. Janow Fall 2015
When is magnetic force = zero?
9-1: A particle in a magnetic field is found to have zero magnetic
force on it. Which of the situations below is NOT possible?
A.
B.
C.
D.
E.
The particle is neutral.
The particle is stationary.
The motion of the particle is parallel to the magnetic field.
The motion of the particle is opposite to the magnetic field.
All of them are possible.
FB  q vB sin( )
Copyright R. Janow Fall 2015
Direction of Magnetic Force
9-2: The figures show five situations in which a positively charged particle
with velocity v travels through a uniform magnetic field B. For which
situation is the direction of the magnetic force along the +x axis ?
Hint: Use Right Hand Rule.
A
B
y
y
v
C
y
B
B
B
x
x
v
z
z
z
D
E
y
v
y
v
B
v
z
B
x
x
z
Copyright R. Janow Fall 2015
x
Magnetic Units and Field Line Examples
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SI unit of magnetic field: Tesla (T)
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Magnetic field lines – Similarities to E
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1T = 1 N/[Cm/s] = 1 N/[Am] = 104 gauss
The tangent to a magnetic field line at any point gives
the direction of B at that point (not direction of force).
The spacing of the lines represents the magnitude of B
– the magnetic field is stronger where the lines are
closer together, and conversely.
At surface of neutron star
108 T
Near big electromagnet
1.5 T
Inside sunspot
10-1 T
Near small bar magnet
10-2 T
At Earth’s surface
10-4 T
In interstellar space
10-10 T
Differences from E field lines
–
–
B field direction is not that of the force on charges
Field lines have no end or beginning – closed curves
But magnetic dipoles will align with field
Copyright R. Janow Fall 2015
Charged Particles Circle at Constant Speed in a Uniform
Magnetic Field: Centripetal Force
y
 Choose B uniform along z, v in x-y plane tangent to path
 FB is normal to both v & B ... so Power = F.v = 0.
 Magnetic force cannot change particle’s speed or KE
 FB is a centripetal force, motion is UCM
 A charged particle moving in a plane perpendicular
to a B field circles in the plane with constant speed v
mv 2
Set
FB  qvB 
r
mv
Radius of
r
the path
qB
Period
2r 2m
c 

v
qB
Cyclotron
2 qB



angular
c
c
m
frequency
v
CW
rotation
Fm
v

 
FB  qv  B
B
Fm
x
Fm
B
v
B
z
If v not normal to B
particle spirals around B

v  causes circling
v para is cons tan t
  and ω do not depend on velocity.
 Fast particles move in large circles and slow ones in small circles
 All particles with the same charge-to-mass ratio have the same period.
Copyright R. Janow Fall 2015
 The rotation direction for a positive and negative particles is opposite.
Cyclotron particle accelerator
Early nuclear physics research, Biomedical applications
magnetic
field
gap
• Inject charged particles in center at S
• Charged plates (“Dees”) reverse polarity
of E field in gap to accelerate particles as
they cross, using c.
• Particles spiral out in magnetic
field as they gain KE and are detected.
mv
r
qB
• Frequency of polarity reversal can be
constant! It does not depend on speed
or radius of path.
c 
2 qB

c
m
Copyright R. Janow Fall 2015
Earth’s field shields us from the Solar
Wind and produces the Aurora
Earth’s magnetic field deflects charged
solar wind particles (via Lorentz force)
. Protects Earth
. Makes life possible (magnetosphere).
Some solar wind particles
spiral around the Earth’s
magnetic field lines producing
the Aurora at high latitudes.
They can also be trapped.
Copyright R. Janow Fall 2015
Circulating Charged Particle
9-3: The figures show circular paths of two particles having the same speed in a
uniform magnetic field B, which is directed into the page. One particle is a proton;
the other is an electron (much less massive). Which figure is physically reasonable?
r
B
A
D
C
E
Copyright R. Janow Fall 2015
mv
qB
Charged particle in both E and B fields
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


 
Add the electrostatic
Ftot  ma
Ftot  qE  qv  B
also
and magnetic forces
Does F = ma change if you observe this while moving at constant velocity v’?
Velocity Selector: Crossed E and B fields

B

E
OUT
• B out of paper
• E up and normal to B
• + charge
• v normal to both E & B

v


Fe  qE  (up)

Fm  qv  B (down)
Equilibrium
when...

Ftot  0
CONVENTION
qE  qvB
v  E /B
• Independent of charge q
• Charges with speed v are un-deflected
Copyright
R. Janow
• Can select particles with a particular
velocity
IN
FBD of q
E
Fe
B
v
Fm
OPPOSED
FORCES
Fall 2015
Measuring e/m ratio for the electron (J. J. Thompson, 1897)
+
+
+

E
+
CRT
screen
+
L
y
e-
electron
gun
-
For E = 0, B = 0 beam hits center of screen
Add E field in –y direction…
 y = deflection of beam from center
Add a crossed B field (into page, E remains)
FM points along –y, opposite to FE (negative charge)
-
FE,y  qE  ma y
-
-
-
e ay

m E
q-e
y  21 ayt 2  ay  2y/t2

 
FM   ev  B


FE  eE
Adjust B until beam deflection = 0 (FE cancels FM) to find vx and time t
when y  0
BL
t
E
vx  E / B
2y
E2
 a y  2  2y 2 2
t
BL
flight time t = L / vx ( vx is constant)
e
E
 2y 2 2  1.76 x 1011 C/kg
m
BL
Copyright R. Janow Fall 2015
Mass spectrometer - Another cyclotron effect device:
Separates molecules with different charge/mass ratios
v
r
some types use
velocity selector here
2qV
m
mv
qB
Ionized molecules or
isotopes are accelerated
• varying masses & q/m
• same potential V
ionized particles follow
separate radii, forming a
“mass spectrum”
1 2mV
r
B
q
q
2V
 2 2
m Br
Molecular mass
Mass spectrum of a peptide
shows the isotopic
distribution.
Relative abundances are
plotted as functions of the
ratio of mass m to charge z.
Copyright R. Janow Fall 2015
Force due to crossed E and B fields
9-4: The figure shows four possible directions for the velocity
vector v of a positively charged particle moving through a
uniform electric field E (into the page) and a uniform magnetic
field B (pointing to the right). The speed is E/B.
Which direction of velocity produces the greatest magnitude of
the net force?
E
A
D
v
v
v
v
B
C


 
Fnet  qE  qV  B
Copyright R. Janow Fall 2015
B
Force on a straight wire carrying current in a B field
Free electrons (negative) flowing:
• Drift velocity vd is opposite to current (along wire)
• Lorentz force on an electron = - evd x B (normal to wire)
Motor effect: wire is pushed or pulled by the charges

vd is opposite to current direction
CONVENTION

OUT
IN
dFm  force due to charges in wire length dx



dq is the charge moving in wire
dFm  vd  B dq
segment whose length dx = -vddt
Recall: dq = - i dt, Note: dx is in direction of current (+ charges)



 
dFm   i (vddt)  B  i dx  B
Integrate along
the whole length L of the wire (assume B is constant )

L  length of wire,
parallel to current

 
Fm  i L  B
iL
Fm
uniform BCopyright R. Janow
Fall 2015
Motor effect on a wire: which direction is the pull?

 
Fm  i L  B

F

iL
replaces

qv

F
F is into slide
F=0
F=0
A current-carrying loop experiences
A torque (but zero net force)
i
B
F is out of slide
Copyright R. Janow Fall 2015
Torque on a current loop in a magnetic field
• Upper sketch: y-axis toward viewer
• Lower sketch: y-axis toward right
• Loop can rotate about y-axis
• B field is along z axis
x

 
• Apply Fm  i L  B
z
to each side of the loop
• |F1|= |F3| = iaB:
• Forces cancel but net torque is not zero!
• Lower sketch:
• |F2|= |F4| = i.b.Bcos(): Forces cancel,
same line of action  zero torque
Net force on loop  0
z into paper
x
y
• Moment arms for F1 & F3 equal b.sin()/2
• Force F1 produces CW torque equal to
1 = i.a.B.b.sin()/2
• Same for F3
• Torque vector is down into paper along –y
rotation axis
Net torque
  iabBsin()
Copyright R. Janow Fall 2015
Motor Effect: Torque on a current loop in B field
x

z
z (B)
y
-x
Torque   iabBsin()


  i A n̂  B A  area of loop

Maximum torque: max  iAB n̂  B
Restoring Torque: ()  max sin()

Oscillates about B

B
Current loop in B field is like electric dipole
maximum
torque
Field PRODUCED BY a current loop is a magnetic dipole field
Copyright R. Janow Fall 2015
zero
torque
ELECTRIC MOTOR:
MOVING COIL GALVANOMETER
• Reverse current I when torque • Basis of most 19th & 20th century
analog instruments: voltmeter,
 changes sign ( = 0 )
• Use mechanical “commutator”
ohmmeter, ammeter, speedometer,
for AC or DC
gas gauge, ….
• Coil spring calibrated to balance


torque at proper mark on scale
• Multiple turns of wire increase torque
• N turns assumed in flat, planar coil
• Real motors use multiple coils (smooth torque)


  NiA n̂  B A  area
of loop
 ab
or
  NiA Bsin()
Copyright R. Janow Fall 2015
Current loops are basic magnetic dipoles

Represent loop
Magnetic dipole moment  m  N i A n̂
as a vector
Newton - m Joule
Dimensions m   ampere - m 

Tesla
Tesla
N  number of turns in the loop
2

B

  
  m B
A  area of loop

B

m


  m Bsin()
Magnetic dipole moment m measures
• strength of response to external B field
• strength of loop as source of a dipole field
ELECTRIC
DIPOLE
MAGNETIC
DIPOLE
MOMENT
p  qd
m  NiA
TORQUE
 

e  p  E

 
m  m  B
POTENTIAL
ENERGY
 
Ue  p  E
SAMPLE VALUES [J / T]
Small bar magnet
m ~ 5 J/T
Earth
m ~ 8.0×1022 J/T
Proton (intrinsic)
m ~ 1.4×10-26 J/T
Electron (intrinsic)
m ~ 9.3×10-24 J/T
UM   d   mBsin()d
 
Um   m  BCopyright R. Janow

B
Fall 2015
The DC Motor
 
  m B
Maximum Torque CCW
Zero Torque
Cross Commutator Gap
Maximum Torque CCW
(Reversed)
Copyright R. Janow Fall 2015
Torque and potential energy of a magnetic dipole
9-5: In which configuration (see below) does the torque on the
dipole have it’s maximum value?

 
m  m  B
9-6: In which configuration (see below) does the potential energy of
the dipole have it’s smallest value?
 
Um   m  B
A
B
C
B
D
E
Copyright R. Janow Fall 2015