Directions : In the following question more than one of the answers given may be correct. Select the correct answer and mark it according to the code:
Two identical charged particles enter a uniform magnetic field with same speed but at angles\[30{}^\circ \]and\[60{}^\circ \]with field. Let a, b and c be the ratio of their time periods, radii and pitches of the helical paths then (1) \[abc>1\] (2) \[abc=1\] (3) \[abc<1\] (4) \[a=bc\]A) 1, 2 and 3 are correct
B) 1 and 2 are correct
C) 2 and 4 are correct
D) 1 and 3 are correct
Correct Answer: C
Solution :
\[T=\frac{2\pi m}{Bq}\] \[\therefore \] \[a=\frac{{{T}_{1}}}{{{T}_{2}}}=1\] \[r=\frac{mv\,\sin \theta }{qB}\] \[\therefore \] \[b=\frac{{{r}_{1}}}{{{r}_{2}}}=\frac{\sin {{30}^{o}}}{\sin {{60}^{o}}}=\frac{1}{\sqrt{3}}\] \[p=(T)(v\cos \theta )\] \[\therefore \] \[c=\frac{{{p}_{1}}}{{{p}_{2}}}=\frac{\cos {{30}^{o}}}{\cos {{60}^{o}}}=\sqrt{3}\] Therefore, \[abc=1\times \frac{1}{\sqrt{3}}\times \sqrt{3}=1\] and \[a=bc\]
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