We are all familiar with magnetic fields, or at least with magnets. Magnets always come with pairs of opposing poles, identified as north and south. This differs in an important way from electric charges which do not have to come in opposing pairs. Isolated + or - charges are common. Like poles will repel each other while unlike poles will attract.

Historical note--the north pole of a magnet is that end which will point north when allowed to swing freely. Since opposite poles attract, what does this mean about the magnetic polarity of the place where Santa lives?

We know a magnetic field is present in space if a small test magnet, such as a compass, feels a torque tending to line it up in a particular direction. Whip out your pocket compass and see which way it points--that is the direction of the local magnetic field.

You are probably familiar with the magnetic field around a N-S magnetic dipole:

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It was discovered in the 19th century that if an electrically charged particle such as an electron moved through a magnetic field, that a force would be exerted on it. The strange thing is that the force is NOT in the direction of the magnetic field, NOT in line with the motion of the particle, but at right angles to both!

If the particle sits without moving, it feels no force. If it travels parallel to the field lines, it feels no force. ONLY when it is traveling across field lines will a force be felt. When it is moving at right angles to a field, the force is given by:

F = qvB where q, v, and B are the charge, velocity, and field strength.

This is really the defining equation of magnetic field. A 1 coulomb charge moving at 1 m/sec will feel a force of 1 newton if the magnetic field strength is 1 weber/m². (This odd unit will be explained gradually.)

Be careful with this equation. If v and B are NOT at right angles, the force will be equal to qv_¬B where v_¬ is the component of relatively perpendicular to the field B.

1. An electron travels at 200 m/sec through a 0.1 web/m² magnetic field. What force is exerted on it? What is its acceleration?

F = qvB F = (1.6 * 10-19)(200)(0.1) F = 3.20 * 10-18 nt

Confused about what a weber is? Be patient, the time to find out hasn't come yet.

2. A particle of mass m, charge q, and velocity v moves in a magnetic field of strength B. If its motion is perpendicular to the field, what is the shape of its path? (be quantitative)

Since the force is always perpendicular to the velocity, it changes the particle's direction, but not its speed. Thus the particle travels in a circle such that: qvB = mv2/r r = mv/qB

Before going further, you should be familiar with the conventions on drawing magnetic fields. First, you must study this arrow:

If you keep the arrow in mind you'll see that this is how you draw a field pointing
.

IMAGE	IMAGE	IMAGE	IMAGE
up	down	left	right

Here is how you draw a field going away from you into the screen.	IMAGE
IMAGE	Here is how you draw a field coming toward you, out of the screen.

3. A proton orbits in a magnetic field of 0.2 web/m² at 30 m/sec. What is the radius of its orbit?

mv2/r = qvB r = mv/qB r = [(1.67 * 10-27)(30)]/[(1.6 * 10-19)(.2)] r = 1.57 * 10-6 m

4. A particle of mass m and charge q orbits with velocity v in a magnetic field B. What is the period of its orbit?

mv2/r = qvB v = qBr/m t = s/v t = [2(3.14)r]/(qBr/m) t = [2(3.14)m]/qB note: Ther period is independent of velocity. This turns out to be of great importance.

Until now I have neglected to show you how to find the direction of the force on a charged particle. By being perpendicular to both the field and the direction of travel, the direction has been narrowed down, but not completely defined. Here is how you finish the job:

Study your hands. Which one would most likely represent positive, and which negative? Of course: In a right-handed world, right is positive. (Don't be gauche about this, left-handers.)

Study them again. What is it about a hand that most clearly would represent field lines? There's only one reasonable choice, your fingers.	IMAGE
IMAGE	What, then, would represent velocity? All you hitch-hikers know that!

If you want to exert a force on something with your hand, what part of it do you use? The palm, of course. You're now all set. Choose positive or negative hand, line up your fingers in the direction of the field, your thumb with the velocity, and your palm will push in the direction of the force. Try it.

5. What force is exerted on the particle in each case?

a) Use your left(-) hand, thumb to the right(v), fingers up(B), and your palm into the screen(F).

b) Right(+) hand, fingers out(B) of the screen, and thum(v) up. The force is to the right.

c) No force because the particle has no component of veloctiy perpendicular to B.

6. An electron enters a region in which an electric field and a magnetic field exist at right angles. If the E field is 3.5 * 10³ nt/coul, and the B-field is 0.2 web/m², at what speed will the electron pass through in a straight line? (i.e. - at what sped will the E and B forces be in equilibrium?)

FE = FB qE = qvB v = E/B v = (3.5 * 103)/(.2) v = 1.75 * 104 m/sec

7. A charge, hanging from a spring, is moved horizontally through a field B at speed v. How much does the spring stretch?

qvB = kx x = qvB/k

8. A mass , bearing charge q is fired horizontally through a magnetic field B. If the acceleration of gravity is g, at what speed will the mass maintain a constant height without dropping?

qvB = mg v = mg/qB

9. A mass m with charge +q is dragged across a table top with friction µ.
a) Will the presence of the magnetic field increase, or decrease the normal force?
b) At velocity v, what is the tension in this string?

a) Using the right(v) hand, thumb to the right, fingers pointing out of the screen, +q is forced against the computer screen. Thus the normal force will increase.

b) T = µN T = µ(mg + qvB)

10.

a) Which way is the force? (up, down, left, right, in or out)	IMAGE	b) What B-field will force the particle in the curve shown?	IMAGE
c) Is the charge positive or negative?	IMAGE	d) How should a B-field be directed to balance the force of the E-field?	IMAGE

a) in

b) in

c) negative

d) in

11. A 6 gm particle has a charge of +2 * 10^-5 coulombs. If it travels from left to right at 5 * 10³ m/sec, what strength magnetic field will keep it from falling to Earth? In what direction should the field be directed?

Fg = FB mg = qvB B = mg/qv B = [(.006)(9.8)]/[(2 * 10-5)(5 * 103)] B = 0.588 web/m2 note: The field is horizontal and into the screen.

12*. A proton (m = 1.67 * 10^-27 kg, q = same as electron, but positive) enters a 0.8 web/m² B-field and is observed to be diverted through an angle of 9°. If the field is 3 cm wide, what is the speed of the proton? (Use small angle approximation.)

Since 9° is a small angle the arc length of the curved path is very nearly 3 cm. The radius of the cirlce, then, is: r = s/Ø r = (.03 m)/[9° * [2(3.14)]/(360)] r = 5.82 * 10-5 m mv2/r = qvB v = qBr/m v = [(1.6 * 10-19)(.8)(5.82 * 10-5)]/(1.67 * 10-27) v = 1.46 * 107 m/sec

13. A mass spectrometer is a device for accurately determining atomic mass. Atoms are ionized, accelerated to a known velocity, and curved in a B-field until they strike a photographic plate at P. If triply ionized atoms (that is, q = 3 q_e) are accelerated through a potential V and passed into a afield B, they strike a plate at distance d. What is their mass?

In the particle gun: PE = KE qV = (1/2)mv2 v = (2qv/m)1/2 mv2/r = qvB qB = mv/r qB = [m(2qv/m)1/2]/(d/2) qB = (2/d)(2qvm)1/2 m = qB2d2/8v m = 3eB2d2/8v note: Remember that q = 3e