A) \[{{r}_{a}}={{r}_{P}}<{{r}_{d}}\]
B) \[{{r}_{a}}>{{r}_{d}}<{{r}_{P}}\]
C) \[{{r}_{a}}={{r}_{d}}>{{r}_{P}}\]
D) \[{{r}_{P}}={{r}_{d}}<{{r}_{a}}\]
Correct Answer: A
Solution :
Given that \[{{K}_{P}}={{K}_{d}}={{K}_{a}}=K\] (say) We know that \[{{q}_{P}}=e,\,{{q}_{d}}=e\]and \[{{q}_{a}}=2e\]and \[{{m}_{P}}=m,\]\[{{m}_{d}}=2\,m\]and \[{{m}_{a}}=4\,m\] Further, \[r=\frac{mv}{qB}=\frac{\sqrt{2mK}}{qB}\] Hence, \[{{r}_{p}}=\frac{\sqrt{2mK}}{eB}\] \[{{r}_{d}}=\frac{\sqrt{2(2m)K}}{eB}=\sqrt{2}\,{{r}_{P}}\] and \[{{r}_{a}}=\sqrt{\frac{2(4m)K}{(2e)B}}={{r}_{P}}\] Hence, the correction option is (a).
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