Motion in magnetic and electric fields
Path of an electron in a magnetic field
The force (F) on 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).
Therefore :
Magnetic force = Bev = mv2/r = centripetal force
v = [Ber]/m
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.
Path of an electron in an electric field
We will consider next the case of an electron entering a uniform electroc
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
But since there
is a force the electron must accelerate in the y direction and the acceleration is given by a =
2y/t
2. (From the equation s = y = ut + ½ at
2)
Therefore if we combine these to equations F = m2s/t
2 and at right angles to then field x = vt so the
equation for the path of the electron is:
eV/d = m2y/t
2 = 2myv
2/x
2 or:
Electron path: y = [eV/2dmv2]x2
this is the equation of a parabola since for a given electron velocity y is
proportional to x
2.
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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