A proton and an \(\alpha\) -particle, having kinetic energies \(K _{ p }\) and \(K _{\alpha},\) respectively, enter into \(a\) magnetic field at right angles. The ratio of the radii of trajectory of proton to that of \(\alpha\) -particle is \(2: 1 .\) The ratio of \(K _{ p }: K _{\alpha}\) is :

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Answer & Solution

Correct Answer

(D) \(4:1\)

Detailed explanation

\(r =\frac{ mv }{ qB }=\frac{ p }{ qB } \quad \frac{ m _{\alpha}}{ m _{ p }}=4\)

\(\frac{ r _{ p }}{ r _{\alpha}}=\frac{ p _{ p }}{ q _{ p }} \frac{ q _{\alpha}}{ p _{\alpha}}=\frac{2}{1}\)

\(\frac{ p _{ p }}{ p _{\alpha}}=\frac{2 q _{ p }}{ q _{\alpha}}=2\left(\frac{1}{2}\right)\)

\(\frac{ p _{ p }}{ p _{\alpha}}=1\)

\(\frac{ K _{ p }}{ K _{\alpha}}=\frac{ p _{ p }^{2}}{ p _{\alpha}^{ p }} \frac{ m _{\alpha}}{ m _{ p }}=(1)\) (4)

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