1. Chapter 21
Magnetic Force

2. Magnetic Field of a Moving Charge
The Biot-Savart law for a moving charge
The Biot-Savart law for a short piece of wire:
How magnetic field affects other charges?

3. Magnetic Force on a Moving Charge
TV and CRT monitors:
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
Electrons are accelerated and deflected vertically by a magnetic field
q – charge of the particle
v – speed of the particle
B – magnetic field

4. Right Hand Rule for Magnetic Force
Electron charge = -e:
The magnetic force on a moving electron is in opposite direction to the direction of the cross product

5. Effect of B on the Speed of the Charge
What is the effect on the magnitude of speed?
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

6. Magnitude of the Magnetic Force
Move charged tape close to a magnet:
can you detect the magnetic force?
Single electron in television tube:
(v<<c)
e/me = C/kg
Magnetic forces on rubbed objects – always tiny!
q/m for charged tape: (10-8 C)/(10-4 kg)  10-4 C/kg

7. Exercise Net charge on invisible tape is ~ 10-8 C
Magnetic field near your bar magnet is ~0.1 T
What is the force on tape if the tape is moved in the direction perpendicular to the field with a speed of 10 m/s ?
Electric force between two charged tapes 10 cm apart:

8. Motion in a Magnetic Field
Confined area: deflection
What if we have large (infinite) area with constant Bv
No work is done on the charged particle

9. Circular Motion at any Speed
Any rotating vector:
…angular speed
Cyclotron Frequency

10. Circular Motion at Low Speed
if v<<c:
independent of v!
Alternative derivation:
Circular motion:
Period T:
Non-Relativistic

11. Determining the Momentum of a Particle
Position vector r:
Circular motion
Position r is a rotating vector!
valid even for relativistic speeds
Used to measure momentum in high-energy particle experiments

12. 1st Event at STAR Experiment
Looking along
collision direction
Looking at right
Angle to collision
direction Au

13. Determining e/m of an Electron

14. Joseph John Thomson (1856-1940)
1897: m/e >1000 times smaller than H atom

15. Clicker Question

16. Clicker Question

17. Exercise What if v is not perpendicular to B? Direction? Magnitude?
𝐹 𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐 =𝑞 𝑣 ⊥ + 𝑣 ∥ × 𝐵
𝑣 ∥ 𝑢𝑛𝑐ℎ𝑎𝑛𝑔𝑒𝑑; Trajectory: helix

18. Exercise: Circular Motion
Which direction an electron would go?
Clockwise or anti-clockwise?
Clockwise: -ev x B points inward

19. Exercise: B in TV set
How big is the magnetic field in a TV created by the electron beam deflection coil? B
Acceleration voltage: 15000V
Solution:
1. Electron speed:
v= m/s
Gamma = 1.03. 2.
Are we justified in using the nonrelativistic KE?

20. The Lorentz Force Can combine electric and magnetic forces:
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

21. A Velocity Selector
Is it possible to arrange E and B fields so that the total force on a moving charge is zero? E FE
What if v changes?
If velocity is less than E/B, magnetic force is less than electric force, leading (for a positive charge) to a greater deflection in +y direction. If velocity is greater than E/B, greater magnetic force (-y deflection). B FB

22. A negative charge is placed at rest in a magnetic field as shown below
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 Up
Down
Into the page
Out of the page
No force at all.
73 of 140

23. 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 B Up
Down
Into the page
Out of the page
No force at all.
72 of 140

24. 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 B
Left Up
Down
Into the page
Out of the page
Up out of page. v X B is into page but negative particle.
70 of 140

25. A Cyclotron Vequivalent ~ 108 V  v~c : need to adjust B or 
Used to study nucleus – accelerate particles (protons) and shoot at nucleus to observe nuclear reactions
Also create new particles – collide particles
Synchrotron – source of synchrotron radiation
D-shaped boxes (Dees)

26. Particle Accelerator Fermilab: Tevatron CERN: Large Hadron Collider
Final stage: circle radius ~ km
Proton speed up to c
To keep R constant B is increased up to 3 T (superconducting coils)

27. Magnetic Force on a Current-carrying Wire
Current: many charges are moving
Superposition: add up forces on individual charges
Number of moving charges in short wire: I
Total force:
𝐹 𝑚 = 𝑛𝐴∆𝑙 𝑞 𝑣 × 𝐵
= 𝑛𝑞𝐴 𝑣 ∆𝑙 × 𝐵
Electrons drift in a direction OPPOSITE to the conventional current.
The term qv has 2 minus which cancel
Force of a short wire:
In metals: charges q are negative. Will this equation still work?

28. Hall Effect When does it reach equilibrium? E h v 𝐸 𝑒⊥ = 𝑣 𝐵 B
𝐸 𝑒⊥ = 𝑣 𝐵 B
Electrons will be deflected toward the bottom surface
???? + -
V>0
∆𝑉= 𝐸 𝑒⊥ ℎ= 𝑣 𝐵 ℎ

29. Hall Effect for Opposite Chargesv B E
>0 + -
V>0
???? + - E v B
V<0

30. 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
( )

31. Hall Effect in a Metal
What is the magnitude of the Hall effect in a metal?
𝑣 = 𝐼 𝑞 𝑛𝐴
𝐼= 𝑞 𝑛𝐴 𝑣
∆𝑉= 𝐼𝐵ℎ 𝑞 𝑛𝐴
~5×1 0 −6 V
Measure ∆𝑉, know the charge (e)
Then we can find n
Sodium, copper
Last shown
Monovalent metals: n is the same as # of atoms per m3
Some metals: n is larger than
# of atoms per m3

32. Exercise What is VHall for the shown copper ribbon? 0.1mm
B=1T
5mm
0.1mm
I=20A
What is VHall for the shown
copper ribbon?
nCu= m3
What is potential difference along 5 mm of copper ribbon to sustain current 20 A? Cu = (A/m2)/(V/m2)
Must be carefull connecting voltmeter exactly across the strip! The potential difference along the I direction is much greater than that resulting from the Hall effect in the transverse direction.

33. Clicker Question Voltmeter 1 reading is POSITIVE
Mobile charges are:
A) Positive (holes)
B) Negative (electrons)
C) Not enough information
Must be electrons traveling to right. E points from right to left <---, making right end of bar higher potential than left. Electrons will be forced to top surface by B field, making top surface lower potential than bottom surface.