MAGNETIC FIELDS AND MOTION IN MAGNETIC FIELDS

Magnetic fields and electric fields are both due to electrical charges. Historically, they were treated as distinct phenomena. They are, however, not different physical things. The electric fields we just finished discussing were due to stationary charges. Magnetic fields are due to currents (moving charges). We will study the generation of magnetic fields next week. This week we treat the more phenomenological problem of how charges move in magnetic fields (i.e., we look at the magnetic force).

As a comment, note that we are familiar with magnets already. For example, we have all played with bar magnets in sandboxes. We find that the iron filings in a sandbox form patterns of the form,

Although you cannot tell in this picture, the field lines, by definition, move from the North pole of the magnet to the South pole of the magnet. Further, if you placed two magnets near each other, the field lines would move from the North pole of the one magnet to the South pole of the other magnet.

Another comment is like poles repel and unlike poles attract.

The unit of the magnetic field is the Tesla, T. It is a large field. The Earth has a field of ~ 0.0001 T, while the largest fields one can generate in Terrestrial laboratories are on the order of 1,000 T.

MOTION IN A MAGNETIC FIELD

The effects on the motion of a particle in a magnetic field are a little odd. To see this recall that the electric force is

F = kQq/r^2 and so the electric force always pushes or pulls the affected charges along the line which connects the charges. This seems sensible as the electric force pulls the charges together or pushes them apart. How does the magnetic force work?

The direction in which the magnetic force pushes is perpendicular to the direction in which the particle moves and to the direction of the magnetic field. The exact direction also depends upon the sign of the charge of the particle. Note that the strength of the force depends upon the velocity of the particle as well as the charge of the particle and the strength of the magnetic field!

The magnetic force is given by

F = q ( v x B ) where the "x" means cross-product. The direction in which (v x B) points can be determined using the right hand rule.

More formally, we also have

v x B = (v_x,v_y,v_z) x (B_x,B_y,B_z) = (v_yB_z-v_zB_y,v_zB_x-v_xB_z,v_xB_y-v_yB_x)

Example

Discovery of the Positron (the anti-electron)

Anderson noted that cosmic rays would produce electrons which streamed thrugh his detector. The electrons having negative charge would veer off to the right. Protons having positive charge would veer off to the left. Being that protons are 1,836 times more massive than electrons, their paths are deflected much less than the electrons. Occasionaly Anderson noticed a particle would veer off to the left (like a proton) but that its path would be like the mirror image of an electron. This was puzzling. Anderson had discovered the anti-matter electron, the positron. A more modern example of this can be seen in the bubble chamber picture:

The tracks particles make can be easily understood. Recall that F = q (v x B) and so is always perpendicular to v (and B). And so,

At each step, the force is perpendicular to path of the particle ===> the path is curved (and in fact the force is such that the path is circular) ===> Work = 0 since the magnetic force is always perpendicular to the direction in which the particle is moving ===> the magnetic force does not perform work on the particle. The speed with which the particle moves is not changed by the magnetic field!

What are the properties of the motion?

gyro radius: F balanced by centrifual force, mv^2/r, ===> -qvB + mv^2/r = 0 ===> r = R(gyro) = mv/(qB)
Orbital Frequency (1/orbital period): P(orb) = 2 pi R(gyro)/v = 2 pi mv/(qvB) = 2 pi m/(qB). Define the cyclotron frequency as

Note that the cyclotron frequency doesn't depend upon the speed of the particle. Particles of all energies take the same time to perform 1 orbit. The gyro radius however, is larger the greater the energy (speed) of the particle.

Cyclotron (Lawrence 1932)

This property of motion in a magnetic field was utilized by Lawrence when he made a low-energy particle accelerator--a cyclotron machine. Lawrence made cylindrical cavity which he divided into 2 electrically insulated halves (called Dees) in which a magnetic field was imposed whose direction was parallel to the axis of the cylinder. The two halves were separated by a small gap. If a particle was injected into the Dee moving perpendicular to the axis of the cylinder, it traced out a circular path of radius r(gyro).

What Lawrence did was to put a source of particles (proton source) near the center of the acclerator. He then imposed a potential difference across the gap to acclerate the protons. When the protons entered the Dee, they orbited to the side and re-entered the gap. Lawrence's trick was then to reverse the polarity of the electric field so that the particle would be kicked-up in energy again. If he hadn't done this, the particle would have been de-celerated and there would be no net gain in energy. The particle then entered the other Dee and performed an orbit with a larger radius. When the particle re-entered the gap, Lawrence again switched the polarity of the field and the particle was kicked-up in energy again.

What made this accelerator nice was that the maximum energy to which you could accelerate a particle was simply how large of an orbit would fit inside of the Dee. Once the orbit (energy) of the proton exceeded this maximum size, it could not be trapped and so escaped. The maximum energy did not depend upon the size of the accelerating potential! In addition, because the cyclotron frequency is independent of the particle energy, it was easy to switch the polarity of the field correctly. It didn't matter what the energy of the particle was. If the frequency depended upon the energy of the particle then the frequency at which the electric field needed to be switched would change as the particle was accelerated.

Note that the above comments assume that the particles travel at speeds v much less than c. For high speeds, relativistic effects and radiation need to be taken into account.

FORCES ON CURRENT CARRYING WIRES

Charges moving through a medium (a wire) are also affected by magnetic fields. The force is given by

F = q ( v(d) x B) where v(d) is the drift velocity of the charge carrier. So, we see that

The amount of charge Q contained in the volume A x l is given by

Q = q(e) eta A l where q(e) = -1.6 x 10^(-19) C is the electron charge, and eta is the density of charge carriers (the electrons). Recall that the current is given by

I = q(e) eta A v(d) The current is in the positive z-direction because the the electron charge is < 0 and the electrons drift downward. The force is then given by

F = Q ( v(d) x B ) = q(e) eta A l v(d) x B = l (I x B) The size of the force is

F = lIB sine theta where theta is the angle made by the current and the magnetic field.

Example

Loop

The current flows through the loop in the Clockwise (CW) direction and the magnetic field points out of the paper. Consider

Side 1: I x B points to the right
Side 2: I x B points down
Side 3: I x B points to the left
Side 4: I x B points up

In this configuration all of the forces balance and the loop will remain still.

Suppose we now tilt the loop so that its face makes an angle of 90 degrees with the magnetic field.

We have

Side 1: I x B points to the right
Side 2: I x B = 0 since I is parallel to B
Side 3: I x B points to the left
Side 4: I x B = 0 since I is parallel to B

Again, the forces all cancel so that the loop will not start moving to the left, right or up and down. However, we see that the loop will move in that it will start to rotate about an axis drawn through the centers of sides 2 and 4. That is, the forces on the top and bottom of the loop exert a torque on the loop which causes it to start turning. The size of the torque is (as usual)

tau = r x F The vector r makes an angle of 90 degrees with the force F and so, the torque due to the force on the top of the loop is

tau = (l/2) lIB sine 90 degrees = (1/2) l^2 I B The torque on the loop due to the bottom branch is of the same size and so the total torque is

tau = l^2 I B = (area of the loop x I) B

A generalization of this result:

Let us define a quantity known as the magnetic moment, mu. Define mu = I x A. The mu is actually a vector. How do we define its direction?
Consider a current moving in a circular path on a flat surface. The moment is pi r^2 I. Its direction is normal to the face of the circle, that is, it is normal to the flat surface. The direction is determined using the right hand rule. Curl your fingers in the direction of the current flow. Your thumb then points in the direction of the magnetic moment.
Recall that there was no torque on the loop when the face of the loop was parallel to the magnetic field and that there was a torque when the field was perpendicular to the face of the loop. This smacks of a cross-product. We, in fact, have
tau = mu x B

Example
What is the torque on a circular loop which makes an angle of 30 degrees with a magnetic field, B. Let I = 5 Amp, B = 1 T, and the radius of the loop = 20 cm.

The moment of the current loop is mu = pi r^2 I = pi 0.2 m)^2 5 A = 0.65 A-m^2
Since the loop makes an angle of 30 degrees with the magnetic field, we have that mu makes an angle of 60 degrees with the magnetic field and so the torque is tau = 0.65 A-m^2 x 1 T x sine 60 degrees = 0.56 N-m