1. Magnetic Flux Density Df DA
Magnetic flux lines are continuous and closed.
Direction is that of the B vector at any point.
Flux lines are NOT in direction of force but ^.
When area A is perpendicular to flux:
The unit of flux density is the Weber per square meter.

2. Calculating Flux Density When Area is Not Perpendicularq a B
The flux penetrating the area A when the normal vector n makes an angle of q with the B-field is:
The angle q is the complement of the angle a that the plane of the area makes with the B field. (Cos q = Sin a)

3. Origin of Magnetic Fields
Recall that the strength of an electric field E was defined as the electric force per unit charge.
Since no isolated magnetic pole has ever been found, we can’t define the magnetic field B in terms of the magnetic force per unit north pole. + E
We will see instead that magnetic fields result from charges in motion—not from stationary charge or poles. This fact will be covered later. +
B v v ^

4. Magnetic Force on Moving Charge
Imagine a tube that projects charge +q with velocity v into perpendicular B field. N S B v F
Experiment shows:
Upward magnetic force F on charge moving in B field.
Each of the following results in a greater magnetic force F: an increase in velocity v, an increase in charge q, and a larger magnetic field B.

5. Direction of Magnetic Force
The right hand rule:
With a flat right hand, point thumb in direction of velocity v, fingers in direction of B field. The flat hand pushes in the direction of force F. B v F B v F N S
The force is greatest when the velocity v is perpendicular to the B field. The deflection decreases to zero for parallel motion.

6. Force and Angle of Path S N S N S N
Deflection force greatest when path perpendicular to field. Least at parallel. S N S N B v F
v sin q q S N

7. Definition of B-field Experimental observations show the following:
By choosing appropriate units for the constant of proportionality, we can now define the B-field as:
Magnetic Field Intensity B:
A magnetic field intensity of one tesla (T) exists in a region of space where a charge of one coulomb (C) moving at 1 m/s perpendicular to the B-field will experience a force of one newton (N).

8. Example 1. A 2-nC charge is projected with velocity 5 x 104 m/s at an angle of 300 with a 3 mT magnetic field as shown. What are the magnitude and direction of the resulting force?
Draw a rough sketch.
v sin f v
300 B v F B
q = 2 x 10-9 C v = 5 x 104 m/s B = 3 x 10-3 T q = 300
Using right-hand rule, the force is seen to be upward.
Resultant Magnetic Force: F = 1.50 x 10-7 N, upward

9. Forces on Negative Charges
Forces on negative charges are opposite to those on positive charges. The force on the negative charge requires a left-hand rule to show downward force F. N S N S F B v
Left-hand rule for negative q B v F
Right-hand rule for positive q

10. Indicating Direction of B-fields
One way of indicating the directions of fields perpen-dicular to a plane is to use crosses X and dots · :
A field directed into the paper is denoted by a cross “X” like the tail feathers of an arrow.
X X X X X X X X X X X X X X X X
· · · ·
A field directed out of the paper is denoted by a dot “ ” like the front tip end of an arrow.

11. Practice With Directions:
What is the direction of the force F on the charge in each of the examples described below? Up F + v
X X X X X X X X X X X X X X X X
X X X X X X X X X X X X X X X X v +
Left F
· · · · Up F
· · · · - v - v F
Right
negative q

12. Zero deflection when FB = FE
Crossed E and B Fields
The motion of charged particles, such as electrons, can be controlled by combined electric and magnetic fields.
Note: FE on electron is upward and opposite E-field.
x x x x x x x x + - e- v
But, FB on electron is down (left-hand rule). B v FB - B v FE E e- -
Zero deflection when FB = FE

13. The Velocity Selector -
This device uses crossed fields to select only those velocities for which FB = FE. (Verify directions for +q)
When FB = FE :
x x x x x x x x + - +q v
Source of +q
Velocity selector
By adjusting the E and/or B-fields, a person can select only those ions with the desired velocity.

14. Example 2. A lithium ion, q = +1
Example 2. A lithium ion, q = +1.6 x C, is projected through a velocity selector where B = 20 mT. The E-field is adjusted to select a velocity of 1.5 x 106 m/s. What is the electric field E?
x x x x x x x x + - +q v
Source of +q V
E = vB
E = 3.00 x 104 V/m
E = (1.5 x 106 m/s)(20 x 10-3 T);

15. Circular Motion in B-field
The magnetic force F on a moving charge is always perpendicular to its velocity v. Thus, a charge moving in a B-field will experience a centripetal force.
X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X
Centripetal Fc = FB + R Fc
The radius of path is:

16. x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
Mass Spectrometer +q R + -
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
x x x x x x x x
Photographic plate m1 m2
slit
Ions passed through a velocity selector at known velocity emerge into a magnetic field as shown. The radius is:
The mass is found by measuring the radius R:

17. x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
Example 3. A Neon ion, q = 1.6 x C, follows a path of radius 7.28 cm. Upper and lower B = 0.5 T and E = 1000 V/m. What is its mass? +q R + -
x x x x x x x x
Photographic plate m
slit
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
v = 2000 m/s
m = 2.91 x kg

18. Summary
The direction of forces on a charge moving in an electric field can be determined by the right-hand rule for positive charges and by the left-hand rule for negative charges. N S B v F
Right-hand rule for positive q N S F B v
Left-hand rule for negative q

19. Summary (Continued) F B v q v sin q
For a charge moving in a B-field, the magnitude of the force is given by:
F = qvB sin q

20. Summary (Continued) The velocity selector: - The mass spectrometer: -
x x x x x x x x + - +q v V +q R + -
x x x x x x x x m
slit
x x x x x x x x x x x x x x x x x x x x x x x x x x
The mass spectrometer: