Magnetic Effects Of Current Question 406
Question: An alternating electric field, of frequency v, is applied across the dees (radius=R) of a cyclotron that is being used to accelerate protons (mass=m). The operating magnetic field used in the cyclotron and the kinetic energy (K) of the proton beam, produced by it, are given by:
Options:
A) $ B=\frac{mv}{e} $ and $ K=2m{{\pi }^{2}}v^{2}R^{2} $
B) $ B=\frac{2\pi mv}{e} $ and $ K=m^{2}\pi vR^{2} $
C) $ B=\frac{2\pi mv}{e} $ and $ K=2m{{\pi }^{2}}v^{2}R^{2} $
D) $ B=\frac{mv}{e} $ and $ K=m^{2}\pi vR^{2} $
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Answer:
Correct Answer: C
Solution:
[c] Time period of cyclotron is $ T=\frac{1}{v}=\frac{2\pi m}{eB} $ ; $ B=\frac{2\pi m}{e}v $ ; $ R=\frac{mv}{eB}=\frac{P}{eB} $
$ \Rightarrow ,P=eBR=e\times \frac{2\pi mv}{e}R=2\pi mvR $ $ K.E.=\frac{P^{2}}{2m}=\frac{{{(2\pi mvR)}^{2}}}{2m}2{{\pi }^{2}}mv^{2}R^{2} $
BETA
