A proton, a deuteron and an \(\alpha\) particle are moving with same momentum in a uniform magnetic field. The ratio of magnetic forces acting on them is.......... and their speed is.................. in the ratio.

\(\Rightarrow F \propto \frac{ q }{ m }\)

thus \(F _{1}: F _{2}: F _{3}=\frac{ q _{1}}{ m _{1}}: \frac{ q _{2}}{ m _{2}}: \frac{ q _{3}}{ m _{3}}\)

\(=\frac{ e }{ m _{ p }}: \frac{ e }{2 m _{ p }}: \frac{2 e }{4 m _{ p }}\)

\(=\frac{1}{1}: \frac{1}{2}: \frac{2}{4}\)

\(=2: 1: 1\)

Now for speed calculation

\(P = constant \Rightarrow v \propto \frac{1}{ m }\)

thus \(v _{1}: v _{2}: v _{3}=\frac{1}{ m _{ p }}: \frac{1}{2 m _{ p }}: \frac{1}{4 m _{ p }}\)

\(=\frac{1}{1}: \frac{1}{2}: \frac{1}{4}\)

\(=4: 2: 1\)