Question

A proton and a deuteron moving with equal kinetic energy enter perpendicularly into a magnetic field. What will be the ratio of radii of the circular path of the proton to that of the deuteron?

• A. 1
• B. 2
• C. 1/2
• D. \frac{1}{\sqrt{2}}

Hint:

When two particles with equal kinetic energy enter a magnetic field, the ratio of the radii of their paths is determined by the square root of their mass ratio. For a proton and deuteron, this results in a ratio of 1.

Answer 1

The Correct Option is A

Step 1: Understanding the formula for the radius of a charged particle in a magnetic field.
The radius \( r \) of the circular path of a charged particle moving perpendicular to a magnetic field is given by the formula:
\[ r = \frac{mv}{qB} \]
Where:
\( m \) 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.
Step 2: Relation between kinetic energy and velocity.
The kinetic energy \( K \) of a particle is given by:
\[ K = \frac{1}{2} mv^2 \]
For both the proton and the deuteron, the kinetic energy is equal, so we can express the velocity \( v \) in terms of kinetic energy as:
\[ v = \sqrt{\frac{2K}{m}} \]
Thus, both particles have the same kinetic energy, and the expression for the velocity will be similar for both.
Step 3: Radius of the proton and the deuteron.
Substituting for \( v \) into the formula for radius:
\[ r = \frac{m \sqrt{\frac{2K}{m}}}{qB} = \frac{\sqrt{2Km}}{qB} \]
Since both the proton and the deuteron have the same kinetic energy, their radii will depend on their masses and charges. The charge of both the proton and the deuteron is the same (both carry the charge of a proton, \( e \)), but the mass of the deuteron is twice that of the proton.
Step 4: Ratio of radii.
Let the mass of the proton be \( m_p \) and the mass of the deuteron be \( m_d = 2m_p \). The radius of the proton \( r_p \) and the radius of the deuteron \( r_d \) can be written as:
\[ r_p = \frac{\sqrt{2K m_p}}{eB}, \quad r_d = \frac{\sqrt{2K (2m_p)}}{eB} \]
Thus, the ratio of the radii is:
\[ \frac{r_p}{r_d} = \frac{\sqrt{m_p}}{\sqrt{2m_p}} = \frac{1}{\sqrt{2}} \times \sqrt{2} = 1 \]
Therefore, the ratio of the radii of the proton's circular path to the deuteron's circular path is 1.
Step 5: Conclusion.
The correct answer is 1, which corresponds to option (1).