Question

A proton, a deuteron and an $\alpha $ particle are accelerated through same potential difference and they enter in a normal uniform magnetic field, the ratio of their kinetic energies will be:
(A) 2: 1: 3
(B) 1: 1: 2
(C) 1: 1: 1
(D) 1: 2: 4

Answer 1

Hint: The kinetic energy is gained by a charged particle when it is accelerated under a potential difference V. So, kinetic energy will be equal to potential energy.

Formula used: Here, we will use some basic formulas which include conservation of energy:
Potential energy = Kinetic energy of the particle
${\text{qV = KE}}$
Here, ${\text{q}}$ is the charge of the particle
${\text{V}}$ is the potential difference
${\text{KE}}$ is the kinetic energy of the particle

Complete step by step answer:
We will start by getting the proportionality equation,
For a given potential difference (${\text{V}}$), ${\text{KE}} \propto {\text{q}}$
Now, we already know the charge for the following particles:
Charge of Proton = 1
Since, deuteron is an isotope of hydrogen
Charge of Deuteron = 1 = (charge of proton)
Charge of Alpha particle =2(charge of proton)
${\text{K}}{{\text{E}}_{\text{p}}}:{\text{K}}{{\text{E}}_{\text{d}}}:{\text{K}}{{\text{E}}_\alpha }$
$ \Rightarrow $${{\text{q}}_{\text{p}}}:{{\text{q}}_{\text{d}}}:{{\text{q}}_\alpha }$
Now, we equate all the charges equal to charge of proton,
$ \Rightarrow {{\text{q}}_{\text{p}}}:{{\text{q}}_{\text{p}}}:{\text{2}}{{\text{q}}_{\text{p}}}$
$ \Rightarrow $1: 1: 2
So, we need to see from the above options, and select the correct value.

Thus, the correct answer is option B.

Additional Note: It should be known that a hydrogen atom has three isotopes. These are hydrogen, deuterium and tritium.
Similarly, carbon occurs naturally in three isotopes: ${{\text{C}}_{{\text{12}}}},{{\text{C}}_{{\text{13}}}},{{\text{C}}_{{\text{14}}}}$
We should always remember that for an alpha particle:
Mass ($\alpha $- particle) = 4 mass (proton)
Charge ($\alpha $- particle) = 2 charge (proton)

Note: The mistake that could be made here is in remembering the charge of particles. It should be always kept in mind that isotopes have the same number of protons, so they carry the same charge as seen here in case of deuterium which is an isotope of hydrogen and also has the same charge as hydrogen.
Also, we will always be cancelling the ratio if and only if the parameters are the same. Here we calculated all the charges of particles equal to the charge of protons to cancel further.