Question

A proton, an electron and an \(\alpha\)-particle enter at right angles to a uniform magnetic field with the same velocity. If R\(_p\), R\(_e\) and R\(_\alpha\) are the radii of circular paths of these particles, then

• A. R\(_\alpha\) = R\(_p\) = R\(_e\)
• B. R\(_\alpha\) \textgreater R\(_p\) \textgreater R\(_e\)
• C. R\(_\alpha\) \textless R\(_p\) \textless R\(_e\)
• D. R\(_\alpha\) \textgreater R\(_p\) = R\(_e\)

Hint:

For particles moving with the same velocity in a magnetic field, the radius of the path depends solely on the specific mass (mass per unit charge). Particles with a higher "specific mass" have more inertia relative to the magnetic force and thus form larger circular paths.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We must compare the radii of curvature for three distinct charged particles injected perpendicularly into a uniform magnetic field at equal speeds.
Step 2: Key Formula or Approach:
The radius \( R \) of a circular orbit for a particle with mass \( m \) and charge \( q \) moving with velocity \( v \) perpendicular to a uniform magnetic field \( B \) is given by the centripetal force condition:
\[ R = \frac{mv}{qB} \]
Since velocity \( v \) and magnetic induction \( B \) are identical for all three particles, the radius is directly proportional to the mass-to-charge ratio:
\[ R \propto \frac{m}{q} \]
Step 3: Detailed Explanation:
Let's analyze the \( m/q \) ratios for the given particles:
1. Electron: Mass \( m_e \), Charge magnitude \( q_e = e \).
\( R_e \propto \frac{m_e}{e} \)
2. Proton: Mass \( m_p \), Charge \( q_p = e \).
\( R_p \propto \frac{m_p}{e} \)
Since \( m_p \approx 1836 \times m_e \), we conclude that \( R_p \textgreater R_e \).
3. \(\alpha\)-particle (Helium nucleus): Mass \( m_{\alpha} \approx 4m_p \), Charge \( q_{\alpha} = 2e \).
\( R_{\alpha} \propto \frac{4m_p}{2e} = 2 \left( \frac{m_p}{e} \right) \)
This shows that \( R_{\alpha} = 2R_p \), therefore \( R_{\alpha} \textgreater R_p \).
Synthesizing the results, we get the following order: \( R_{\alpha} \textgreater R_p \textgreater R_e \).
Step 4: Final Answer:
The correct order of radii is R\(_\alpha\) \textgreater R\(_p\) \textgreater R\(_e\).