Moving Charges and Magnetism - Result Question 3
3. 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 $(B)$ used in the cyclotron and the kinetic energy $(K)$ of the proton beam, produced by it, are given by :
[2012]
(a) $B=\frac{m v}{e}$ and $K=2 m \pi^{2} v^{2} R^{2}$
(b) $B=\frac{2 \pi m v}{e}$ and $K=m^{2} \pi v R^{2}$
(c) $B=\frac{2 \pi m v}{e}$ and $K=2 m \pi^{2} v^{2} R^{2}$
(d) $B=\frac{m v}{e}$ and $K=m^{2} \pi v R^{2}$
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Answer:
Correct Answer: 3. (c)
Solution:
- (c) Time period of cyclotron is
$T=\frac{1}{v}=\frac{2 \pi m}{e B} ; B=\frac{2 \pi m}{e} v ; R=\frac{m v}{e B}=\frac{p}{e B}$
$\Rightarrow p=e B R=e \times \frac{2 \pi m v}{e} R=2 \pi m v R$
K.E. $=\frac{p^{2}}{2 m}=\frac{(2 \pi m v R)^{2}}{2 m}=2 \pi^{2} m v^{2} R^{2}$
