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
90^{o} 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]
The charge is usually the electronic charge, 1.6 ´ 10^{-19} C.
Question 1 |
An electron accelerated to 6.0 ´ 10^{6} 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? |
Path
of charged particles in a Magnetic Field
We have seen that the force always acts on the wire at 90^{o}, and that gives us the condition for circular motion.
We can combine the
relationship
a = v^{2}/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 × 10^{7} 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 = mv^{2}/r ̃ r = mv Bq |
r
= 9.11 × 10^{-31} kg × 3.7 × 10^{7} 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 ´ 10^{7} m/s. What is the mass of the particle? How does it compare to the mass of an electron (9.11 ´ 10^{-31} kg)? |
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.
̃ 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: