Repeat, using the relativistic equations of motion, the deriviation in class of the cyclotron frequency. Show the relativistic cyclotron angular frequency is qB ym Show the radius of the cyclotron motion Bc r = qB / ym

Repeat, using the relativistic equations of motion, the deriviation in class of the cyclotron frequency. Show the relativistic cyclotron angular frequency is qB ym Show the radius of the cyclotron motion Bc r = qB / ym

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Question

Repeat, using the relativistic equations of motion, the deriviation in class of the
cyclotron frequency. Show the relativistic cyclotron angular frequency is
qB
ym
Show the radius of the cyclotron motion
Вс
r =
qB/ ym

Expert Solution

Step 1

Relativistic momentum is given by,

$p = \frac{m v}{\sqrt{1 - {(\frac{v}{c})}^{2}}} = γ m v (1) w h e r e, γ = \frac{1}{\sqrt{1 - {(\frac{v}{c})}^{2}}}$

The circular motion in cyclotron is brought about b magnetic force, so the centripetal force and magnetic force should be equal in magnitude, this condition leads to,

$\frac{m}{r} {(\frac{p}{m})}^{2} = q (\frac{p}{m}) B \frac{p}{r} = q B r = \frac{p}{q B} r = \frac{γ m v}{B} (2)$

Set ß=v/c in (2) gives,

$r = \frac{ß c}{\frac{q B}{γ m}} (3)$

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