A proton and a deuteron both are having the same kinetic energy enter perpendicular into a uniform magnetic field B. For motion of proton and deutron on circular path of radius \[{R_p}\] and \[{R_d}\] respectively, the correct statement is
A. \[{R_d} = \sqrt 2 {R_p}\]
B. \[{R_d} = {R_p}/\sqrt 2 \]
C. \[{R_d} = {R_p}\]
D. \[{R_d} = 2{R_p}\]

Hint: In the given question, we need to determine the correct relation between \[{R_p}\] and \[{R_d}\]. For this, we need to use the concept that the centripetal force is balanced by the force due to the magnetic field to get the desired result.


Formula used:
The following formulae are used to solve the given question.
The centripetal force is balanced by the force due to the magnetic field is mathematically given by \[\dfrac{{m{v^2}}}{R} = qvB\].
Here, \[r\]indicates the radius of the circular path of a charged particle with mass \[m\] as well as charge \[q\] traveling at a speed \[v\] perpendicular to a magnetic field of intensity \[B\].


Complete answer:
We know that the centripetal force is balanced by the force due to the magnetic field.
Thus, we get
\[\dfrac{{m{v^2}}}{R} = qvB\]
Here, \[r\]indicates the radius of circular path of a charged particle with mass \[m\] as well as charge \[q\] traveling at a speed \[v\] perpendicular to a magnetic field of intensity \[B\].
For proton, the radius of circular path is \[{R_p} = \dfrac{{mv}}{{qB}}\]
That is \[{R_p} = {\dfrac{\sqrt{2{m_p}E}}{{qB}}} \]
Also, for deuteron, the radius of circular path is \[{R_d} = {\dfrac{\sqrt{2{m_d}E}}{{qB}}} \]
Now, take the ratio of \[{R_d}\]and \[{R_p}\].
Thus, we get
\[\dfrac{{{R_d}}}{{{R_p}}} = \dfrac{{ {\dfrac{\sqrt{2{m_d}E}}{{qB}}} }}{{ {\dfrac{\sqrt{2{m_p}E}}{{qB}}} }}\]
By simplifying, we get
\[\dfrac{{{R_d}}}{{{R_p}}} = \dfrac{{\sqrt {{m_d}} }}{{\sqrt {{m_p}} }}\]

As we know, a deuteron is the nucleus of a deuterium atom which contains one proton and one neutron.
So we can say that
\[{m_d} = 2{m_p}\]

Now put the values in the equation
\[\dfrac{{{R_d}}}{{{R_p}}} = \dfrac{{\sqrt {{2m_p}} }}{{\sqrt {{m_p}} }}\]

This gives
\[\dfrac{{{R_d}}}{{{R_p}}} = \sqrt 2 \]
\[{R_d} = \sqrt 2 {R_p}\]
Hence, For motion of proton and deutron on circular path of radius \[{R_p}\] and \[{R_d}\] respectively, the correct statement is \[{R_d} = \sqrt 2 {R_p}\].

Therefore, the correct option is (A).


Note:Many students make mistakes in the simplification part. That means, they may get wrong while taking the ratio of \[{R_p}\] and \[{R_d}\]. Also, it is necessary to analyze the desired relationship between \[{R_p}\] and \[{R_d}\].