1. Physics Electricity and Magnetism Lecture 09 - Charges & Currents in Magnetic Fields Y&F Chapter 27, Sec
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

2. Magnetic Field
Electrostatic & Gravitational forces act through vector fields
Now: Magnetic force on moving charge acts through
Force law: Effect of a given B field on charges & currents in wires
Next week: How to create B field.
Currents in loops of wire, moving charges
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 (not covered)
“Permanent Magnets”: North- and South- seeking poles
Natural magnets interact via field
Compass – Earth has a magnetic field
Ferromagnetic materials magnetize
when cooled in B field (Fe, Ni, Co) -
- Spins align into“domains” (magnets)
Other materials (plastic, copper, wood,…)
at most slightly affected (para- and
dia- magnetism)
REPELS
ATTRACTS
DIPOLES ALIGN IN B FIELD
UNIFORM N S

3. Similarities: Electric Field versus Magnetic Field
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.
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.

4. Differences between magnetic & electrostatic field
Test charge and electric field
Single electric poles exist
Magnetic “charge” p and magnetic field ?
Single N or S magnetic poles have never
been seen.
Magnets always occur as dipole pairs..
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
Particle moves with a velocity v through a magnetic field B
Typical Magnitudes:
Earth’s field: 10-4 T.
Bar Magnet: 10-2 T.
Electromagnet: 10-1 T.
“LORENTZ FORCE”
Units:
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 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.
FB reverses sign for opposite sign of charge
Strength of B field created 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. Magnetic force geometry examples:
A simple geometry x y z
charge +q
B along –z direction
v along +x direction
CONVENTION
Head Tail
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)

7. A numerical example Electron beam moving in plane of sketch
force for +e
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
a) Find the magnetic force on an electron :
Negative sign means force is opposite to result of using the RH rule
b) Find the electron’s acceleration :
Direction is the same as that of the force: Fov = 0

8. 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 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. 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 – the magnetic field is stronger where the lines are closer together, and conversely.
Differences from E field lines
B field direction is not force direction on charges
B field lines have no beginning or end – closed curves
But magnetic dipoles will align with B field

11. 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
FB is normal to both v & B ... so Power = F.v = 0.
Magnetic force cannot change particle’s speed or KE.
Force direction does change
A charged particle moving in a plane perpendicular
to a B field circles in the plane with constant speed v
FB is a centripetal force (Uniform Circular Motion) x y z v Fm B 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 not normal to B
particle spirals around B

12. 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 in magnetic
field as they gain KE and are detected.
Frequency of Dee polarity reversal can
be constant! It does not depend on
speed or radius of path.

13. 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.

14. 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? A B C D E

15. Charged particle in both E and B fields
Add the electrostatic
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 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
Equilibrium when...
Independent of charge q
Charges with speed v are un-deflected
Can select particles with a particular velocity

16. 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)
Adjust B until beam deflection y = 0 (FE cancels FM) to find vx and time t
flight time t = L / vx ( vx is constant)

17. 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.

18. 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

19. Force on a straight wire carrying current in a B field
Current is flowing opposite to electron’s drift velocity vd
Lorentz force on electron = - evd x B (normal to wire)
Motor effect: wire is pushed right 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 )

20. Motor effect on a wire: which direction is the pull?B
F = 0
F is into slide
F is out of slide
A current-carrying loop experiences a torque
(but zero net force)

21. Torque on a 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 side 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)

22. Torque on a current loop in B fieldz x
Restoring Torque:
Maximum torque:
Oscillates about
Current loop in B field is like electric dipole
maximum
torque
zero

23. 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

24. 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

25. 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)

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

27. 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?
9-6: In which configuration (see below) does the potential energy of the dipole have it’s smallest value? B A C D E