A) \[B=\frac{mv}{e}\]and \[K=2m{{\pi }^{2}}{{v}^{2}}{{R}^{2}}\]
B) \[B=\frac{2\pi mv}{e}\] and \[K={{m}^{2}}\pi v{{R}^{2}}\]
C) \[B=\frac{2\pi mv}{e}\] and \[K=2m{{\pi }^{2}}{{v}^{2}}{{R}^{2}}\]
D) \[B=\frac{mv}{e}\] and \[K={{m}^{2}}\pi v{{R}^{2}}\]
Correct Answer: C
Solution :
[c] Time period of cyclotron is \[T=\frac{1}{v}=\frac{2\pi m}{eB}\]; \[B=\frac{2\pi m}{e}v\]; \[R=\frac{mv}{eB}=\frac{P}{eB}\] \[\Rightarrow \,\,\,P=eBR=e\times \frac{2\pi mv}{e}R=2\pi mvR\] \[K.E.=\frac{{{P}^{2}}}{2m}=\frac{{{(2\pi mvR)}^{2}}}{2m}2{{\pi }^{2}}m{{v}^{2}}{{R}^{2}}\]
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