Download Electron motion in electric and magnetic fields

Motion in electric and magnetic fields
Magnetic fields
The force (F) on a wire of length L carrying a current I in a magnetic field of strength B is
given by the equation: F = BIL .
But Q = It and since Q = e for an electron and v = L/t you can show that :
Magnetic force on an electron = BIL = B[e/t][vt] = Bev where v is the electron velocity
In a magnetic field the force is always at right angles to the
motion of the electron (Fleming's left hand rule) and so the
resulting path of the electron is circular (Figure 1).
electron
Therefore :
Force = Bev
magnetic force = Bev = mv2/r = centripetal force
magnetic field
perpendicular to figure
[Ber]/m = v
Figure 1
and so you can see from these equations that as the electron
slows down the radius of its orbit decreases.
Charged particles move in circles at a constant speed if projected into a magnetic field at
right angles to the field.
Charged particles move in straight lines at a constant speed if projected into a magnetic field
along the direction of the field.
Figure 2 shows a 3D diagram of and electron moving at right
angles to a uniform magnetic field.
If the electron enters the field at an angle to the field direction
the resulting path of the electron (or indeed any charged
particle) will be helical as shown in figure 3. Such motion
occurs above the poles of the earth where charges particles
from the Sun spiral through the Earth’s field to produce the
aurorae.
Figure 2
electron
magnetic field
Figure 3
helical motion
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Path of the electron in an electric field
We next consider the case of an electron entering a uniform electric field between two
parallel plates (Figure 4). The potential difference between the plates is V and the plates are
aligned along the x direction and the electron enters the field at right angles to the field lines:
The force on the electron is given by the equation:
F = eE = eV/d = ma
Electron paths
Figure 4
But since there is a force the electron must accelerate in the y direction and the acceleration
is given by a = 2y/t2. (From the equation s = y = ut + ½ at2)
Therefore if we combine these to equations F = m2s/t2 and at right angles to then field x = vt
so the equation for the path of the electron is:
eV/d = m2y/t2 = 2myv2/x2 or:
Electron path: y = [eV/2dmv2]x2
this is the equation of a parabola since for a given electron velocity y is proportional to x2
Notice that if the electron is moving at right angles to the field then the path in the field is
independent of the distance of the original direction from either plate.
Charged particles move in parabolas if projected into an electric field in a direction at right
angles to the field.
Charged particles move in straight lines and accelerate (or decelerate) if projected into an
electric field along the direction of the field.
In an electric field the electron moves at a constant velocity at right angles to the field but
accelerates along the direction of the field.
Example problem
An electron is accelerated from rest through a potential difference of 5000 V and then enters a magnetic
field of strength 0.02 T acting at right angles to its path. Calculate the radius of the resulting electron
orbit.
Bev = mv2/r so r = mv/Be = 9.1x10-31x4.2x107/0.02x1.6x10-19 = 1.2 x10-2 m = 1.2 cm.
As the electrons orbit they accelerate and so lose energy by radiation and therefore slow
down and their orbit decreases.
It must be remembered that the electric force acts along the line of the electric field direction
while the magnetic force acts at right angles to the field direction. Also a charged particle at
rest experiences a force in an electric field but none in a magnetic field.
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