A) \[\sqrt{\frac{{{m}_{p}}}{{{m}_{e}}}}\]
B) \[\sqrt{\frac{{{m}_{e}}}{{{m}_{p}}}}\]
C) \[\frac{{{m}_{e}}}{{{m}_{p}}}\]
D) \[1\]
Correct Answer: B
Solution :
Radius of circular path in a uniform magnetic field \[r=\frac{mv}{qB}\] Kinetic energy of the charged-particle \[K=\frac{1}{2}m{{v}^{2}}\] \[{{v}^{2}}=\frac{2K}{m}\] \[\Rightarrow \] \[v=\sqrt{\frac{2K}{m}}\] \[\therefore \] \[r=\frac{m}{qB}\sqrt{\frac{2K}{m}}\] \[r=\frac{\sqrt{2Km}}{qB}\] For the same value of\[K\]and B \[r\propto \frac{\sqrt{m}}{q}\] \[\therefore \] \[\frac{{{r}_{e}}}{{{r}_{p}}}=\frac{\sqrt{{{m}_{e}}}}{\sqrt{{{m}_{p}}}}\times \frac{{{q}_{p}}}{{{q}_{e}}}\] \[\frac{{{r}_{e}}}{{{r}_{p}}}=\sqrt{\frac{{{m}_{e}}}{{{m}_{p}}}}\times \frac{e}{e}=\sqrt{\frac{{{m}_{e}}}{{{m}_{p}}}}\]
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