Introduction:
When charged particles move through a uniform magnetic field perpendicular to their velocity, they experience a force that causes them to move in a circular path. The radius of this circular path is determined by the mass, charge, and kinetic energy of the particle.

Comparison of Radii:
The radii of the circular paths for the proton, deuteron, and alpha particle can be compared by considering their respective masses, charges, and kinetic energies.

Proton:
- Mass: The mass of a proton is approximately 1.67 × 10^(-27) kg.
- Charge: The charge of a proton is +1.6 × 10^(-19) C.
- Kinetic Energy: Let's assume the kinetic energy of the proton is K.

Deuteron:
- Mass: A deuteron is a nucleus consisting of a proton and a neutron. The total mass of a deuteron is approximately 3.34 × 10^(-27) kg.
- Charge: The charge of a deuteron is the same as that of a proton, +1.6 × 10^(-19) C.
- Kinetic Energy: Since the deuteron has the same kinetic energy as the proton, its kinetic energy is also K.

Alpha Particle:
- Mass: An alpha particle is a helium nucleus consisting of two protons and two neutrons. The total mass of an alpha particle is approximately 6.64 × 10^(-27) kg.
- Charge: The charge of an alpha particle is twice that of a proton, +3.2 × 10^(-19) C.
- Kinetic Energy: Since the alpha particle has the same kinetic energy as the proton and deuteron, its kinetic energy is also K.

Explanation:
The radius of the circular path followed by a charged particle in a magnetic field can be determined using the equation:
r = (mv) / (qB)
where r is the radius, m is the mass of the particle, v is the velocity of the particle, q is the charge of the particle, and B is the magnetic field strength.

Since the kinetic energies of all three particles are the same, their velocities will also be the same. Therefore, we can compare the radii of their circular paths by comparing their masses and charges.

Using the above equation, we can observe that the radius is directly proportional to the particle's mass and velocity, and inversely proportional to the charge and magnetic field strength.

Comparison:
- Proton: r_proton = (m_proton * v) / (q_proton * B)
- Deuteron: r_deuteron = (m_deuteron * v) / (q_deuteron * B)
- Alpha Particle: r_alpha = (m_alpha * v) / (q_alpha * B)

Since the velocities and magnetic field strengths are the same for all three particles, we can compare their radii by comparing their masses and charges.

- Comparing the proton and deuteron:
r_proton / r_deuteron = (m_proton * q_deuteron) / (m_deuteron * q_proton)

- Comparing the proton and alpha particle:
r_proton / r_alpha = (m_proton * q_alpha) / (m_alpha * q_proton)

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