Magnetic Fields Tutorial 3 - Force on a Charge

 Contents

Interaction of Charged Particles with a Magnetic Field

We know that a magnetic field and an electric current interact to produce a force.  Since a current is a flow of charge, it is reasonable to suppose that a magnetic field exerts a force on individual charge carriers.

We find that in a magnetic field, the force acts on a stream of electrons always at 90o to the direction of the movement.  Therefore the path is circular.

Consider a charge q moving through a magnetic field B at a constant velocity v.  The charge forms a current that moves a certain distance, l, in a time t.

We know:

• Velocity = distance ¸ time

• Current = charge ¸ time

•  F = BIl

So we can substitute into this relationship to give us:

F = B ´ (q/t) ´ vt = Bqv

So the formula now becomes:

F = Bqv

[ F- force in N; B – field strength in T; q – charge in C; v – speed in m s-1]

The charge is usually the electronic charge, 1.6 ´ 10-19 C.

If the magnetic field is at an angle q to the magnetic field, the equation is modified to:

F = Bqv sin q

If no angle is mentioned in the question, assume that the angle is 90o.

 An electron accelerated to 6.0 ´ 106 m s-1 is deflected by a magnetic field of strength 0.82 T.  What is the force acting on the electron?  Would it be any different for a proton?

 Remember that the direction of the electrons’ movement is in the opposite direction to the conventional current. So if the electrons are going from left to right, the conventional current is going from right to left. When using Fleming’s Left Hand Rule, the current is conventional.

Path of charged particles in a Magnetic Field

We have seen that the force always acts on the wire at 90o, and that gives us the condition for circular motion.

We can combine the relationship a = v2/r with Newton II to give us:

Therefore:

The v on the left cancels to get rid of the v2 term on the right:

This rearranges to give us:

 Worked Example An electron passes through a cathode ray tube with a velocity of 3.7 × 107 m s-1.  It enters a magnetic field of flux density 0.47 mT at a right angle.  What is the radius of curvature of the path in the magnetic field? Combine F = Bqv and F = mv2/r to give: r = 9.11 × 10-31 kg × 3.7 × 107 m s-1 = 0.39 m = 39 cm        0.47 × 10-3 T × 1.6 × 10-19 C

 Question 2 In a particle physics experiment, a detector is placed in a magnetic field of 0.920 T.  A particle is found to produce a circular track of radius 0.500 m.  Other experiments have shown that the particle carries a charge of +1.60 ´ 10-19 C and that its speed was 3.00 ´ 107 m s-1.    What is the mass of the particle?  How does it compare to the mass of an electron (9.11 ´ 10-31 kg)?

The Hall Effect

Magnetism Tutorial 3B is for students for the Welsh Board and Eduqas Syllabuses.  Students studying other syllabuses are, of course, welcome.

The cyclotron is a particle accelerator that relies on this idea.  The machine’s main components are two D-shaped electrodes ("Dees") in an evacuated chamber, placed between the poles of a large electromagnet.

From the top it looks like this:

Notice that the beam of particles is not circular, but a spiral.  This is because the particles are being accelerated by the electric field between each D-shaped electrode (called a dee).  As their speed increases, so does the radius of the curved path.

If a particle of charge q enters one of the dees with a speed v, it will move in a semi-circular path of radius r.

Þ rearranging gives us

We can work out from t = s/v what time it takes for the charge to travel:

For 1 revolution:

s = 2pr

Therefore:

The r terms cancel:

Since f = 1/t:

Rearranging gives us:

 A cyclotron has magnets of flux density 1.50 T and the polarity changes with a frequency of 2.00 MHz.  A large charged particle which has a charge of +2e is inserted into the machine and is accelerated.  Calculate the mass of the particle.

Click HERE to see an animation by Stephen Lucas, a past student of mine.  I am most grateful for his permission to use it.

The largest particle accelerators are synchrotrons.  The earliest machines had a racetrack arrangement as shown:

The machine works like this:

• Charged particles are accelerated from the particle source.  If they are electrons, they are produced by an electron gun (see Particle Physics 4).  If the particles are protons, they are produced by ionising hydrogen atoms.   Note that the particles must be moving before they enter the synchrotron.  The synchrotron cannot accelerate a stationary particle.

• The particles from the source fly down the injection tube

• The particles enter the synchrotron into the flight tube, which has a vacuum in it to reduce collisions with other particles.

• The particles are accelerated by the accelerator in the same way as a linear accelerator (see Particle Physics 4).

• The particles are deflected in a circular path by the guiding magnets.

• As the particle speed increases, the time to race around the track decreases.  Therefore the frequency at which they pass the accelerator increases.

• Therefore the acceleration frequency has to go up.

• When the correct speed has been reached, the magnetic field in the guide magnet at the bottom right is reduced.

• Therefore the particles fly down the target tube to hit the target.  The target tube may also lead to another accelerator, or a storage ring.

The particles are in small groups rather than a continuous beam.  The idea is that the particles get accelerated at the same point as they pass that point.  For example the electrons are going around the track 15 000 times a second, the accelerator must give the little brutes a kick up the backside 15 000 times a second. So the acceleration is synchronised with the passage of the particles, hence the name synchrotron.

A synchrotron that accelerates electrons is called an electron synchrotron, and (what a surprise) a synchrotron that accelerates protons is called a proton synchrotron.

More modern machines are circular and have an arrangement like this:

The particles may come from an electron gun, or a proton source, or may have been accelerated by a linear accelerator or another synchrotron.

Although the diagram shows the flight tube as circular, some machines have a polygonal flight tube with the electromagnets being rectangular.  The path of the particles is circular.  There may be several flight tubes to different targets which leave the ring tangentially as shown above.  An extraction electromagnet may also be used to remove the particles from the ring.

A storage ring is similar, although there are no acceleration cavities (or they are turned off).  The particles travel at the same speed in a storage ring, and are extracted magnetically.

As the particles accelerate faster, relativistic effects become important, as the particles gain mass as they approach the speed of light.