1. Physics Electricity and Magnetism Lecture 09 - Charges & Currents in Magnetic Fields Y&F Chapter 27, Sec
What Produces Magnetic Field?
Comparing 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
Motor Effect: Force on a Current-Carrying Wire
Torque on a Current Loop
The Magnetic Dipole Moment
Summary

2. Magnetic Field
Electrostatic & Gravitational forces act through vector fields
Now: Magnetic force on moving charge is due to
“Permanent Magnets”: North- and South- seeking poles
Natural magnets interact via B field
Compass – Earth has a magnetic field
Dipoles align to an external magnetic field
Can make magnets - ferromagnetic materials
cooled in B field (Fe, Ni, Co)
Materials like plastic, copper, wood,… only
slightly affected (para- and dia- magnetism)
REPELS
ATTRACTS
Field Lines
UNIFORM N S
Force law: A given B field exerts force on moving charges or currents
Next week: how to create B field using…
Currents in loops of wire or moving charges
Intrinsic spins of e-,p+  elementary currents  magnetic dipoles
Spins can align permanently to form natural magnets
Not covered: B & E fields  Maxwell’s equations, electromagnetic waves

3. Similarities between electric and magnetic fields
Electric Field
Force acts at a distance through
vector electric field
Source: stationary or moving electric charge.
Positive (+) & negative (-) charge.
Opposite charges attract
Like charges repel.
Field lines show the direction and magnitude of E.
Magnetic Field
New force acts at a distance through new vector magnetic field
Source: moving electric charge (i.e., current)
North pole (N) and south pole (S)
Opposite poles attract
Like poles repel.
Field lines show the direction and magnitude of B.

4. Differences between magnetic & electrostatic field
Test charge feels electric field
Single electric poles exist
Force parallel to E-field
There is no “magnetic charge” p such that
No single N or S magnetic poles have ever
been seen. Magnets are always dipoles.
Complicated force on charge in B field.
Cut up a bar magnet
small, complete magnets
Electrostatic Gauss Law
Magnetic Gauss Law
There is no magnetic monopole dipoles are the basic units
Magnetic field exerts force on moving charges (current) only
Magnetic flux through each and every Gaussian surface = 0
Magnetic field lines are closed curves - no beginning or end

5. Magnetic force on a moving, charged particle
Point charge particle with velocity v in magnetic field B
Typical Magnitudes:
Earth’s field: 10-4 T.
Bar Magnet: 10-2 T.
Electromagnet: 10-1 T.
Units:
1 “GAUSS” = 10-4 Tesla
See Lecture 01 for cross product
FB is proportional to speed v, charge q, and field B.
FB depends on complex geometry using cross product
FB = 0 if v is parallel to B
FB is normal to plane of both v and B (use right hand rule)
FB reverses sign if charge sign reverses.
B field created by moving charge also depends on qv [current x length].
Electrostatic force can do work on a charged particle…BUT…
… Magnetic force cannot do work on moving particles since FB.v = 0.

6. Examples with simple geometries:y z
charge +q
B along –z direction
v along +x direction
CONVENTION
Head Tail q
More simple examples
charge +q
Fm is down
|Fm| = qv0B0 x
charge -q
B into page
Fm is still down
Fm is out of
the page
charge +q or -q
v parallel to B
|Fm| = qv0B0sin(0) +q -q

7. A numerical example Single electron moving in plane of sketch
force for +e
Single electron moving in plane of sketch
v = 107 m/s along +x
B = 10-3 T. out of page along +y
z axis is down
a) Find the magnetic force on the electron :
Negative sign means force is opposite to result of using the RH rule
b) Find the electron’s acceleration (mechanics still applies) :
Direction is the same as that of the force: Fov = 0

8. When is magnetic force = zero?
9-1: Suppose that a particle in a magnetic field has zero magnetic force on it. Which situation below could not be the case?
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.

9. 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 does the magnetic force point along the +x axis ?
Hint: Use Right Hand Rule. x z y B v A C D E

10. Demonstration Source: Pearson Study Area - VTD
Chapter 28
Magnet and Electron Beam
Discussion:
You will need the right hand rule.
How did the demo create the electron beam?
In what way does the magnetic force act on the beam?
Why is the beam visible with or without the magnet?

11. Magnetic Units and Field Line Examples
SI unit of magnetic field: Tesla (T)
1T = 1 N/[Cm/s] = 1 N/[Am] = 104 gauss
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
Magnetic field lines – Similarities to E
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. Strong field where the lines are close together.
Differences from E field lines
B field direction is not direction of force on charges
B field lines have no beginning or end – closed curves
But magnetic dipoles will align with B field

12. Charged Particles Circle at Constant Speed in a Uniform Magnetic Field: Centripetal Force
Uniform B along z. v in x-y plane tangent to path.
FM is normal to both v & B ... so Power = F.v = 0.
Magnetic force cannot change particle’s speed or KE.
Force direction does change (rotates)
A charged particle moving in a plane perpendicular
to a B field circles in the plane with constant speed v
FM is a centripetal force (Uniform Circular Motion) x y z v FM B
+ charge CW
rotation
Set
Radius of the path
Period
Cyclotron
angular frequency
tc and ωc 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.
Positive and negative particles rotate in opposite directions.
If v has a component along B the particle spirals around B

13. Cyclotron particle accelerator
Early nuclear physics research, Biomedical applications
gap
magnetic
field + -
Inject charged particles in center at S
Charged plates (“Dees”) reverse polarity
of E field in gap with frequency wc.
This accelerates particles crossing gap.
Particles spiral out along field lines as
they gain KE until they enter the beam.
Frequency of Dee polarity reversal can
be constant! It does not depend on
speed or radius of path. So particles
can gain energy every time they cross
the gap..

14. 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).
Solar wind particles spiral around the Earth’s magnetic field lines producing the Aurora at high latitudes.
They can also be trapped.

15. Circulating Charged Particle
9-3: The 5 figures show circular paths of two particles having the same speed in a uniform magnetic field B, which points into the page. One particle is a proton; the other is an electron (much less massive). Which figure is physically reasonable? A B C D E

16. 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 the velocity produces zero net force? E A v v D B v B v C

17. Force on a straight wire carrying current in a B field
Current flows opposite to electron’s drift velocity vd
Lorentz force on an electron = - evd x B (normal to wire)
Electrons transmit force to the wire structure
Motor effect: wire is pushed rightward by the charges
dq = idt is the charge moving in wire segment whose length dx = -vddt. dx is along current direction. iL
uniform B Fm
Integrate along the length L of the wire (assume B is uniform )

18. Demonstration Source: Pearson Study Area - VTD
Chapter 27
Current-Carrying Wire in Magnetic Field
Discussion:
What expression tells us the force magnitude and
direction on the wire?
The field exerts force directly on conduction electrons.
How does it get transmitted to the lattice of ions in the wire?

19. Motor effect: direction of the pull on a wire?B
F = 0
F is into slide
F is out of slide
What is the effect on a current loop in a magnetic field? Answer:
zero net force …but
a torque about an axis
in space

20. Torque on a current loop in magnetic fieldz x
Restoring Torque:
Maximum torque:
Rotational oscillation about
Current loop in B field behaves like electric
dipole in E field – MAGNETIC DIPOLE
maximum
torque
zero

21. Current loops are basic magnetic dipolesq
Represents a current loop as a vector y
z (B) -x q
Magnetic dipole moment m measures:
strength of response to external B field
strength as source of a dipole field
Torque on a loop in external
magnetic field is normal to both m and B.

22. Comparison: Magnetic & Electric Dipole Moments
TORQUE
POTENTIAL
ENERGY
MAGNETIC
DIPOLE
ELECTRIC
MOMENT
SAMPLE DIPOLE VALUES [J / T]
m ~ 1.4×10-26 J/T
Proton (intrinsic)
m ~ 9.3×10-24 J/T
Electron (intrinsic)
m ~ 8.0×1022 J/T
Earth
m ~ 5 J/T
Small bar magnet
Magnetic dipole moments are basic
building blocks
…..but….
Electric dipoles are composed of
smaller components (charges)

23. Devices using the Motor Effect
MOVING COIL GALVANOMETER
Basis of most 19th & 20th century
analog instruments: voltmeter,
ohmmeter, ammeter, speedometer,
gas gauge, ….
Coil spring calibrated to balance
torque at proper mark on scale
ELECTRIC MOTOR:
Reverse current I when torque
t changes sign (q = 0 )
Use mechanical “commutator”
for AC or DC
Multiple turns of wire increase torque
N turns assumed in flat, planar coil
Real motors use multiple coils (smooth torque)

24. 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 magnitude?
9-6: In which configuration (see below) does the potential energy of the dipole have it’s minimum value? B A C D E

25. Supplementary Material

26. Charged particle in both E and B fields
Add the electrostatic
and magnetic forces
Velocity Selector: Crossed E and B fields
B out of paper
E up and normal to B
q is + charge
v normal to both E & B
CONVENTION
OUT IN
Free Body Diagram for q E B v Fe Fm
OPPOSED
FORCES
Can have equilibrium if...
Independent of charge q
Charges with speed v are un-deflected
Can select particles with a particular velocity

27. Measuring e/m ratio for the electron (J. J. Thompson, 1897)+ - e-
CRT
screen y
electron
gun L
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)
Need vx to find flight time t: adjust B until beam deflection y = 0
flight time t = L / vx ( vx is constant)

28. Mass spectrometer - Another cyclotron effect device:
Separates molecules with different charge/mass ratios
Ionized particle paths have separate radii, forming “mass spectrum”
Molecular mass
Ionized molecules or isotopes are accelerated
varying masses & q/m
same potential V & KE
some types use velocity selector here
Mass spectrum of a peptide shows the isotopic distribution.
Relative abundances are plotted as functions of the ratio of mass m to charge z.

29. Torque on a rectangular current loop in a magnetic fieldz x
Upper sketch: y-axis toward viewer
Lower sketch: y-axis toward right
Loop can rotate about y-axis
B field is along z axis
Apply
to each of 4 sides of the loop
|F1|= |F3| = iaB:
Forces cancel but net torque is not zero!
|F2|= |F4| = i.b.Bcos(q): Forces cancel,
Same line of action  zero torque
z into paper x y
Moment arms for F1 & F3 equal b.sin(q)/2
Force F1 produces CW torque equal to
t1 = i.a.B.b.sin(q)/2
Same for F3
Torque vector is along –y (rotation axis)

30. The DC Motor Maximum Torque CCW Zero Torque Cross Commutator Gap
(Reversed)