Question

A proton carrying 1 MeV kinetic energy is moving in a circular path of radius R in a uniform magnetic field. What should be the energy of an $\alpha$ particle to describe a circle of the same radius in the same field?
A) 2 MeV
B) 1 MeV
C) 0.5 MeV
D) 4 MeV

Answer 1

As the proton is moving along the field direction, thus no force acts on it and the velocity and the path will not change. It will keep on moving with the same speed along the field direction. For a charged particle's motion in a magnetic field, ${F_C} = {F_M}$ . Now by putting the required values into this solution we will find the answer.

Complete Step By Step Answer:
A proton is a subatomic particle, symbol p or p⁺ , with a positive electric charge of +1e elementary charge and a mass slightly less than that of a neutron. Protons and neutrons, each with masses of approximately one atomic mass unit, are jointly referred to as nucleons. The proton was discovered by Ernest Rutherford in the early 1900’s. During this period, his research resulted in a nuclear reaction which led to the first splitting of the atom, where he discovered protons. He named his discovery protons based on the Greek word protos which means first. It was also discovered that charged particles (protons and light ions) have a finite range in matter. The interaction probability to cause ionization increases as they lose velocity along their paths, so that a peak of deposited dose occurs at a depth proportional to the energy of the charged particle. Beyond this peak, no further dose is deposited. This scientific phenomenon was described by William Bragg at that time. In 1930, the American physicist Ernest O. Lawrence and his associates were the first to invent the cyclotron to accelerate protons to an energy high enough for cancer treatment applications. He invented the cyclotron in 1929 & developed it as a particle accelerator during the 1930s, winning the 1939 Nobel Prize for physics for this work. In 1931, he founded the Radiation Laboratory, later named the Lawrence Berkeley Laboratory . A decade later, his advanced version of the synchrocyclotron, which is 184 inches in diameter, is capable of producing 340 MeV protons.
For a charged particle's motion in a magnetic field,
${F_C} = {F_M}$
$\dfrac{{M{V^2}}}{R} = QVB$
We also know that,
${R_P} = \dfrac{{{M_P}{V_P}}}{{{Q_P}B}}$
= $\dfrac{{{P_P}}}{{{Q_P}B}} = \dfrac{{\sqrt {M{K_P}} }}{{{Q_P}B}}$
By further solving we get,
${R_\alpha } = \dfrac{{\sqrt {2(4M){K_\alpha }} }}{{2QB}}$
$\dfrac{{{R_P}}}{{{R_\alpha }}} = \sqrt {\dfrac{{{K_P}}}{{{K_\alpha }}}}$
But we also know that,
$\begin{gathered} {R_P} = {R_\alpha }........(given) \\\ {K_P} = {K_\alpha } = 1MeV \\\ \end{gathered}$ $$
Thus the energy of an $\alpha$ particle to describe a circle of the same radius in the same field = 1MeV.

Note:
Alpha particles, also called alpha rays or alpha radiation, consist of two protons and two neutrons bound together into a particle identical to a helium-4 nucleus. They are generally produced in the process of alpha decay, but may also be produced in other ways.