To determine the ratio of the radii of the circular paths taken by a proton and a deuteron in a magnetic field, we need to consider the fundamental principles of motion in magnetic fields and how they relate to the properties of these particles.

Understanding the Motion in a Magnetic Field

When charged particles like protons and deuterons move through a magnetic field, they experience a magnetic force that causes them to move in a circular path. The radius of this path can be derived from the equation:

r = (mv) / (qB)

• r is the radius of the circular path.
• m is the mass of the particle.
• v is the velocity of the particle.
• q is the charge of the particle.
• B is the magnetic field strength.

Properties of Proton and Deuteron

Now, let's look at the specific properties of the proton and the deuteron:

• A proton has a mass of approximately 1.67 x 10^-27 kg and a charge of +1e (where e is the elementary charge).
• A deuteron, which is an isotope of hydrogen, consists of one proton and one neutron, giving it a mass of about 3.34 x 10^-27 kg, but it also has the same charge of +1e.

Initial Kinetic Energy Consideration

Since both particles have the same initial kinetic energy (KE), we can express this energy as:

KE = (1/2)mv²

From this equation, we can derive the velocity (v) of each particle:

v = sqrt((2 * KE) / m)

Calculating the Radius for Each Particle

Now, substituting the expression for velocity back into the radius formula:

r = (m * sqrt((2 * KE) / m)) / (qB)

This simplifies to:

r = (sqrt(2 * KE * m)) / (qB)

Finding the Ratio of Radii

Since both particles have the same charge (q) and are subjected to the same magnetic field (B), we can focus on the mass (m) in the radius equation. The ratio of the radii (r_p for proton and r_d for deuteron) can be expressed as:

r_p / r_d = sqrt(m_p) / sqrt(m_d)

Substituting the masses:

r_p / r_d = sqrt(m_p) / sqrt(2 * m_p) = 1 / sqrt(2)

This means that the radius of the proton's path is less than that of the deuteron. To express this in a more intuitive way:

r_p : r_d = 1 : sqrt(2)

Final Ratio of Radii

To convert this into a ratio of whole numbers, we can approximate sqrt(2) as about 1.41. Therefore, the ratio of the radii can be expressed as:

1 : 1.41

However, if we consider the options provided, the closest whole number ratio that reflects the relationship between the proton and deuteron radii is:

1 : 2

Thus, the correct answer to the question regarding the ratio of the radii of the circular trajectories described by the proton and deuteron is 1 : 2.