Magnetic Fields Tutorial 3 - Force on a Charge

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:       

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]

 

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

 

Question 1 

An electron accelerated to 6.0 106 m/s 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?

Answer

 

 

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:        

                                                           

 

This rearranges to give us:

 

 Worked Example

An electron passes through a cathode ray tube with a velocity of 3.7 107 m/s.  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?

F = Bqv and F = mv2/r r = mv

   Bq

r = 9.11 10-31 kg 3.7 107 m/s = 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 place in a magnetic field of 0.92 T.  A particle is found to produce a track of radius 0.5 m.  Other experiments have shown that the particle carries a charge of +1.6 10-19 C and that its speed was 3.0 107 m/s.  What is the mass of the particle?  How does it compare to the mass of an electron (9.11 10-31 kg)? 

Answer

 

The cyclotron is a particle accelerator that relies on this idea.  The machines 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. 

 

 

 

rearranging gives us

 

           

 

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

 

 

The voltage will change twice in each alternating current cycle, therefore in the period of one cycle, the time is 2t.  Since f = 1/T, we can write the expression as: