Deflection of an electron beam in a magnetic field. Applications.

Hello friends of Steemit!

In this publication, it will be demonstrated in the laboratory how an electron beam is curved due to the action of a magnetic field.

In principle we will make a brief review of the theoretical and experimental studies on which this article is based.

Figure 1 - Vector representation of the magnetic force on a moving charge
Image Source

When an electric charge enters a region in which a magnetic field exists, it experiences a magnetic force given by the expression:

In magnitude the magnetic force is expressed in the form:

According to the vectorial properties, equation (1) shows that the direction of the magnetic force that this charge undergoes will always be perpendicular to the plane where the velocity and magnetic field vectors lie and also shows a direct relation of the force with respect to the charge, speed and magnetic field (see figure 1).

It is important to note that equation (1), like Coulomb's Law and other laws of electromagnetism, is an expression obtained from analysis and experimental observations.

In the figure 2, we observe the particular case of an electric charge which enters a magnetic field region with a perpendicular velocity to said field.

Due to the deflection of the magnetic force over the electric charge, the movement will be described by a circular movement in which the magnetic force will point to the center of the circle.

Figure 2 – Circular movement of an electric charge due to a uniform magnetic field
Image Source

Given that:

Where:

a_r ➞ radial acceleration of the charge
r ➞ radius of the circle
θ ➞
m ➞ mass of the particle

From Equations (4) and (5) it’s obtained:

Equation (6) shows the radial behavior described by the charge in movement, as a function of the charge, mass, velocity, and the magnetic field.

The kinetic energy acquired by the electrons due to the difference of potential “v” applied, allow determining the velocity of the beam as a function of said potential.

The direction of the movement of the charge in the magnetic field will depend of the sign associated to the electric charge. (see figure 2)

To demonstrate the curvature of an electron beam due to a magnetic field we will use the model SE-9638 PASCO equipment commonly used for the determination of the relation (Charge/mass) of the electron.

Figure 3 – Pasco equipments used in the lab

The electron beam is generated in the tube which is provided with a cathode and anode that works as electron cannon when the cathode is warmed due to the application of a difference of potential. The helium gas in the tube allows seeing the electron beam.

The Helmholtz coils generate a magnetic field perpendicular to the direction of the electron beam provoking the curvature of its trajectory just as expressed in equation (6), which is visualized in a scale collocated in the equipment.

The magnetic field generated by the Helmholtz coils, in the configuration of the equipment shown, it's obtained through the equation.

N ➞ number of turns in the Helmholtz coils
I ➞ Total current generated by the Helmholtz coils
μ₀ ➞ magnetic permeability constant
a ➞ radius of Helmholtz coils*

Figure 4

The photography in figure 5 was taken without illumination, for a greater visualization of the curvature of the beam. In the center of the image it’s observed in blue tone the curvature of the beam.

Figure 5

Experimental data

Calculation and results

Substituting (7) and (8) in (6) it’s obtained:

Radius of the curvature in the scale of the equipment:

The obtained results demonstrate the curvature effect generated by the magnetic field over the electron beam.

Applications

One of the most important applications of the deflection of charged particles due to the action of a magnetic field it’s the spectrometer of masses that it’s illustrated in figure (5).

Figure 6
Image Source

The ions sent through the velocities selector to the magnetic field “B” are deflected in semicircular trajectories to a detector o photographic plate, given that every registered radius correspond to a mass “m”, it’s possible to obtain the mass of every ion through this method.

Magnetic fields play a fundamental role in different areas of science so nowadays many investigations point in the study of its properties and applications that allow the creation of new technologies.

References

• Física para Ciencias e Ingenierías Vol.2 Segunda edición / Gettys, Keller, Skove.

• Física para la Ciencia y la Tecnología Vol.2 Quinta Edición / Tipler, Mosca.

• Física Universitaria Vol.2 undécima edición / Sears, Zemansky, Young, Freedman.

• Física para Ciencias e Ingeniería Vol 2. John P. MccKelvey, Howard Grotch. Primera edición.

• http://www.manualsdir.com/manuals/340885/pasco-se-9638-e-m-apparatus.html?page=6