Harpell/Physics 4

Magnetic Fields and Forces

Most of you know about Magnets and Magnetic Forces….

An even more dramatic example is the Aurora Borealis or Northern Lights (there is also an Aurora Australis). Here Charged particles from the sun and stars strike the Earth's magnetic field. Due to the force we will describe in some detail below, they are deflected and spiral down towards the poles where they eventually scintillate the upper atmosphere. On a clear dark night, you might see a shimmering curtain of light like the one above (with Comet Hale Bop for good measure)..…..

The Earth's magnetic field has a north and south pole, and extends well out into space:

Where the field lines enter the Earth is called the South Magnetic Pole…this is up in Hudson's bay Canada. Where they leave the Earth is the North magnetic pole…near the coast of Antarctica. Without this field, the Earth would be a shooting gallery for deadly radiation….high energy charged particles that cause genetic damage at the very least. Note that there are periods when the Earth's magnetic field is "off" just prior to reversing polarity. These instances have been linked to subsequent periods of increased variation in mutations and new species!

Most of the time, however, the charged particles spiral around magnetic field lines, loosing energy in collisions, until the strike the atmosphere above the Poles…

this photo is from a satellite looking down!

First things first…..

Remember that electric forces were due to charges interacting. Well, magnetic forces are due to moving charges interacting. To begin, assume a magnetic field B exists and points to the right…i.e. B is a vector in the y direction. If a positively charged particle is moving in the x-direction, then the particle will experience a force F in the z direction:

In Fact, the magnitude of this force will be given by the product:

F = qvB

Where q is the charge on the particle, v is the speed of the particle, and B is the magnitude of the magnetic field strength.

This simple relation gives us a chance to "define" magnetic field strength.

B = F/qv

Suppose a one coulomb charge is moving with a velocity of one meter per second experiences a magnetic force of 1 Newton. Then B = 1 Newton/(coulomb-meter/sec) or 1 N-s/C-m. This magnetic field strength is also known as a 1 Tesla.

You may remember that A coulomb is a lot of charge, but one amp, which is one coulomb/sec is not so much current. In any case, current in a wire represents moving charges

Imagine a piece of wire of length dl. A current of magnitude I flows through the wire. Imagine a length of wire dl such that dl is the distance charges travel in a time dt. If the charges are moving perpendicular to the magnetic field as before with velocity v, then the force dF is given by:

…dF = (dq)vB

Where dq is the amount of charge contained in the length dl.

Now the current I is related to dq by:

I = dq/dt so that

…dq = Idt

and..

dF = (Idt)vB

Since vdt is the length dl, then

..dF = IdlB, .

Integrating over a length of wire L, the magnetic force is given by:

F = ILB.

Since it's pretty easy to measure currents and forces, we determine magnetic fields strength from:

B = F/IL

Thus a one Tesla field creates a force of one Newton on a amp current flowing though each one meter of wire within the magnetic field.

In practice, a one-meter field is a large region to have a magnetic field…

The Earth's field is about 10^{-4 }Tesla for example. In fact 10^{-4 }Tesla is known as a "Gauss"

Question: What Force would the Earth's magnetic field exert on 10 cm of wire carrying 100 mA?

..

F = (.1A)(.1m)(10^{-4} Tesla) = 10^{-6} Newtons…much less than the weight of the wire!

Question: Since a wire has both positive and negative charges, would it actually experience a force in a magnetic field?

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Answer: of course! Only the charge carriers are in motion (the electrons). …

Since the magnetic force is always directed perpendicular to the velocity of the particle and the magnetic field, we can represent it as

**F** = q(**v** x **B**)

What happens if a particle enters a magnetic field with its velocity perpendicular to the magnetic field?

Consider this JAVA applet Charged particles Moving in a Magnetic Fields © Copyright 1997, Sergey Kiselev and Tanya Yanovsky-Kiselev

If the Velocity is slow enough, and/or the B field is strong enough, it will actually orbit: Its pretty easy to calculate the orbital speed, momentum, Energy, and angular frequency

Note that relation (3) implies that the angular frequency of an orbiting particle is independent of the particles speed or radius. This idea wasn't lost on Ernest Lawrence, who seems to have much to do with our existence out here…

If you can inject a charged particle near the center of the hollow "Dees" shown below then the magnetic force will deflect it in a circle.

The only "trick" is to change the voltage and the direction of the electric field as the particle comes around to give it a kick each time. By equation (3) above, the angular or "cyclotron frequency:" is qB/m which is independent of what radius and speed the particle has. Thus an voltage generator can be tuned to have the proper frequency for q and B. Eventually when the particle comes to the proper radius orbit, it leaves the cyclotron with kinetic energy given by equation #2 above.

Lawrence's first cyclotron :

Of course, to get more energy, you want a larger cyclotron. To produce 1MeV protons for example, you can have a cyclotron about a meter or two in diameter depending on the magnetic field strength. This is same energy a proton would have it if was accelerated across a D.C. potential difference of 1 million volts.

If you want much more energy than that you run into both engineering and physics problems:

- You need bigger cyclotrons (bigger radius), and thus you need to have a large region magnetic field…takes lots of current or lots of magnets!
- Charged particles radiate energy as they "turn". Since an orbit is a continuous turn, they are constantly radiating away energy. At a certain energy. As the particles move faster, they radiate away a larger fraction of their energy. Eventually, you can't make the particles move any faster…
- Einstein's special theory of relativity says that mass increases with speed, thus the cyclotron frequency becomes a function of speed as the mass changes. Things get a lot trickier..

this can be used to an advantage…the SSRL at SLAC for example…..

So far we've only discussed when the velocity is perpendicular to the magnetic field direction. What if it isn't? For example, a particle with initial velocity components in x,y, z…..

In this case, the particle will spiral around the magnetic field lines…can you see why?

In general:

One last application of this perpendicular force is the Hall effect…

In this case, the applied voltage causes the charge carriers to move from on end of the material (the rectangle) to the other. A magnetic field into the paper then deflects the charge carriers either up or down depending on their sign. If the are positive, then they are deflected up, charging the upper (blue) edge positive. The lack of positive charge on the bottom end will case it to be negative. This charge separation will cause an electric field which deflects the charges downward. Eventually, the electric force and magnetic force and equal and no more deflection occur.

…qE = qvB

But since the voltage V across the material is related to E by:

E = V/d, then

V/d = vB. What good is this:

- by measuring the Voltage and Magnetic field, you can determine the speed v of the charge carriers. Note that this is in turn related to the resitivity, and density of charge carriers…
- By calibrating this device for known Magnetic fields, it becomes a magnetic field sensor, so that V = constant x B where the constant is measured " in the factory"
- If you know the magnetic field strength, and drift velocity, you can use the "hall probe" to determine the charge on the charge carriers.
- The "Hall Voltage" V answers the age-old question: what sign are the charge carriers. If they positive, moving from left to right, then the top will be higher potential than the bottom. If they are electrons moving right to left, then the top will be at lower potential..

That’s all for now….