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.

Calculating Flux Density When Area is Not Perpendicular q 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)

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 ^

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.

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.

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

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).

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

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

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. ·

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

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

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.

Example 2. A lithium ion, q = +1 Example 2. A lithium ion, q = +1.6 x 10-16 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);

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:

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:

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 10-19 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 10-24 kg

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

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

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: