Physics

Physics · electromagnetism

Problem

a) Show that the radial deviation of the electrons from the injection radius is finite. Given constants: A beam of electrons emitted by a point source

\mathrm{P}

enters the magnetic field

\vec{B}

of a toroidal coil (toroid) in the direction of the lines of force. The angle of the aperture of the beam

2 \cdot \alpha_{0}

is assumed to be small

\left(2 \cdot \alpha_{0}<<1\right)

. The injection of the electrons occurs on the mean radius

\mathrm{R}

of the toroid with acceleration voltage

\mathrm{V}_{0}

. Neglect any interaction between the electrons. The magnitude of

\vec{B}, B

, is assumed to be constant. <image_1> Data:

\frac{\mathrm{e}}{\mathrm{m}}=1.76 \cdot 10^{11} \mathrm{C} \cdot \mathrm{kg}^{-1} ; \quad \mathrm{V}_{0}=3 \mathrm{kV} ; \quad \mathrm{R}=50 \mathrm{~mm}

Context question: 1. To guide the electron in the toroidal field a homogeneous magnetic deflection field

\vec{B}_{1}

is required. Calculate

\vec{B}_{1}

for an electron moving on a circular orbit of radius

R

in the torus. Context answer:

1.48 \cdot 10^{-2}

Context question: 2. Determine the value of

\vec{B}

which gives four focussing points separated by

\pi / 2

as indicated in the diagram. Note: When considering the electron paths you may disregard the curvature of the magnetic field. Context answer:

0.37 \cdot 10^{-2}

Extra Supplementary Reading Materials: 3. The electron beam cannot stay in the toroid without a deflection field

\vec{B}_{1}

, but will leave it with a systematic motion (drift) perpendicular to the plane of the toroid. Note: The angle of aperture of the electron beam can be neglected. Use the laws of conservation of energy and of angular momentum.

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