Question

A proton and an alpha-particle moving with the same velocity enter a uniform magnetic field with their velocities perpendicular to the magnetic field. The ratio of radii of their circular paths is

• A. 1 : 4
• B. 4 : 1
• C. 1 : 2
• D. 2 : 1

Answer 1

The Correct Option is C

Step 1: Recall the formula for the radius of a charged particle moving in a magnetic field: 

When a charged particle with mass \(m\), charge \(q\), and velocity \(v\) enters a uniform magnetic field \(B\) perpendicular to the velocity, the radius (\(r\)) of its circular path is given by:

\[ r = \frac{mv}{qB} \]

Step 2: Define parameters clearly:

We have two particles:

• Proton: mass \(m_p\), charge \(+e\)
• Alpha-particle (He nucleus): mass \(4m_p\), charge \(+2e\)

Given: Both have the same velocity \(v\).

Step 3: Calculate radii separately:

- Radius for proton (\(r_p\)): \[ r_p = \frac{m_p v}{e B} \]

- Radius for alpha particle (\(r_\alpha\)): \[ r_\alpha = \frac{4 m_p v}{2 e B} = \frac{2 m_p v}{e B} \]

Step 4: Compute the ratio of radii:

\[ \frac{r_p}{r_\alpha} = \frac{\frac{m_p v}{e B}}{\frac{2 m_p v}{e B}} = \frac{1}{2} \]

Thus, the ratio of radii (proton : alpha-particle) is 1 : 2.

Answer 2

The magnetic force on a charged particle moving perpendicular to a magnetic field provides the centripetal force for the circular motion. The radius \( r \) of the circular path is given by the equation: \[ r = \frac{mv}{qB} \] 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.

For a proton, the charge is \( +e \) and mass is \( m_p \). For an alpha particle (which consists of 2 protons and 2 neutrons), the charge is \( +2e \) and mass is approximately \( 4m_p \). The ratio of radii \( r_p \) for a proton and \( r_{\alpha} \) for an alpha particle is: \[ \frac{r_{\alpha}}{r_p} = \frac{m_{\alpha} q_p}{m_p q_{\alpha}} = \frac{4m_p \cdot e}{m_p \cdot 2e} = 2 \] Thus, the ratio of the radii of the proton to the alpha particle is \( 1 : 2 \).