The cathode ray based e/M experiment and Milikan\'s oil drop experiment in 1909

Question

The cathode ray based e/M experiment and Milikan's oil drop experiment in 1909 enable us to calculate the specific charge (e/m) and electronic charge (e) of an elementary charge respectively. The following forces have effect on the cathode ray particles on above experiment: Accelerating potential (V_A) delivers kinetic energy to the electrons of mass (m), ejected continuously from the gun (heating filament): (-e) V_A = (1/2) m *v^2 Electrons move in circular path of radius, R, due to the uniform magnetic field strength, B, applied transversely to the Electric field intensity, E. It results in centripetal force on the electron that experiences Lorentz force in the electromagnetic field. mv^2/R = (-e) v times B Derive the steps of calculation necessary to calculate the specific charge of electron as given below? e/m = [2 V_A/(B^2R^2)]

Explanation / Answer

Electrons are thermally emitted from a surface and accelerated through a potential difference V. The kinetic energy of the accelerated electrons equals the energy they gain as a result of being accelerated through the potential difference. In other words:

(1/2)*m*v2 = eVA

and solving for velocity,

v = (2eVA/m)1/2 .

In this equation m is the mass of the electron and e is the charge of the electron.

The beam of electrons enters the region where a magnetic field B is set up by the Helmholz coils. The beam is deflected into a circular path of radius R by the magnetic force and undergoes a centripetal acceleration. This can be expressed as

evB = mv2/r

When the velocity is eliminated between the above two equations, then the charge to mass ratio can be written as

e/m = 2V/(B2r2)

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