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The ratio of their time period is 1 : 2 : 2.
The momentum of a proton, a deuteron, and an alpha particle is the same in a uniform magnetic field.
The charge of a proton is q = e and the let m(p) be the mass of the proton.
Let m(d) = 2m(p) be the mass of the deuteron and the charge of a deuteron is q =e.
Now, an alpha particle is made up of 4 protons. Therefore the mass of an alpha particle is 4m(p) and the charge on an alpha particle is q = 2e.
The formula for the time period of a moving particle in a uniform magnetic field is given as:
[tex]T = \frac2 \pi m}{qB}[/tex] where B is the magnetic field, m is the mass and q is the charge of the particle.
So, the time period of a proton is:
[tex]T_{1} = \frac{2 \pi m(p)}{eB}[/tex]
The time period of a deuteron is:
[tex]T_{2} = \frac{2 \pi (2m(p))}{eB}[/tex]
And the time period of an alpha particle is:
[tex]T_{3} = \frac{2 \pi ( 4m(p) )}{2eB}[/tex]
Therefore, the ratio is:
[tex]T_{1}: T_{2} : T_{3} =\frac{2 \pi m(p)}{eB} : \frac{2 \pi 2m(p)}{eB} : \frac{2 \pi (4m(p))}{2eB}[/tex]
T₁ : T₂ : T₃ = 1 : 2 : 2
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