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♦ Relevant knowledge

Thomson carried out experiments using Thomson conducted experiments with the Cathode Ray Tube which resulted in the discovery of electrons. Thomson’s research also revealed characteristics of electrons like being negatively charged along with the attraction of positive charges and also its mass-to-charge ratio. Thomson utilized the Cathode-Ray Tube for his experiments. Thompson’s experiments led to the development of a new view of atoms modeling using the equations of law of physics to illustrate the motion of electrons with much accuracy and precision.

It is a given that

Mass of the particle [katex]=\mathrm{m}[/katex]

Mag field [katex]=\mathrm{B} 0[/katex]

Part (A). At what speed does the ion leave the acceleration region?

Let E1 be the initial energy of the particle, which is actually electirc energy. It can also be given by

[katex]E 1=q V[/katex]

The particle’s energy is converted to electricity when it moves. [katex]\mathrm{KE}[/katex]

[katex]\mathrm{E} 2=1 / 2 \mathrm{~m} \mathrm{u}^{2}[/katex]

We can save energy by conserving it.

[katex]q V=1 / 2 m u^{2}[/katex]

[katex]\mathrm{u}=\sqrt{(2 q V) / m}[/katex]

[katex]\operatorname{Part}(B)[/katex]

The cyclotron frequency is the (angular frequency) of orbital motion of the electron in the magnetic field. It is given by

[katex]\omega=q \mathrm{BO} / \mathrm{m}[/katex]

We know this by the definition og angular velocity.

[katex]\mathrm{u}=\mathrm{R} \omega \mathrm{So} \omega=\mathrm{u} / \mathrm{R}[/katex]

The two different expressions of [katex]\omega[/katex]

q [katex]\mathrm{BO} / \mathrm{m}=\mathrm{u} / \mathrm{R}[/katex]

Value of [katex]\mathrm{u}[/katex]

q [katex]\mathrm{BO} / \mathrm{m}=\sqrt{(2 q V) / m} / \mathrm{R}[/katex]

We get our sides quarrelled.

[katex]q^{2} B 0^{2} / m^{2}=(2 q V / m) / R^{2}[/katex]

[katex]m / q=R^{2} B 0^{2} /(2 V)[/katex]