Question

A charged particle when enters a uniform magnetic field moves in a helical path. If its angular velocity is \( 4\pi \times 10^6 \) rad s\(^{-1} \) and its velocity in the direction of magnetic field is \( 3 \times 10^5 \) ms\(^{-1} \) then the pitch of the helix is

• A. \( 5 \) cm
• B. \( 10 \) cm
• C. \( 15 \) cm
• D. \( 20 \) cm

Hint:

The pitch of a helix is the distance covered along the magnetic field in one time period of the circular motion. Calculate the time period using the angular velocity. Then multiply the parallel component of velocity by the time period to find the pitch. Ensure consistent units throughout the calculation.

Answer 1

The Correct Option is C

When a charged particle enters a uniform magnetic field at an angle to the field, its velocity can be resolved into two components: one parallel to the magnetic field (\( v_{\parallel} \)) and one perpendicular to the magnetic field (\( v_{\perp} \)).
The perpendicular component causes the particle to move in a circular path, and the parallel component causes it to move along the magnetic field lines.
The combination of these two motions results in a helical path.
The angular velocity \( \omega \) of the circular motion is given as \( 4\pi \times 10^6 \) rad s\(^{-1} \).
The velocity of the particle in the direction of the magnetic field (parallel component) is \( v_{\parallel} = 3 \times 10^5 \) ms\(^{-1} \).
The pitch of the helix is the distance traveled by the particle along the direction of the magnetic field during one complete revolution in the circular path.
The time period \( T \) of one revolution is related to the angular velocity by \( T = \frac{2\pi}{\omega} \).
$$ T = \frac{2\pi}{4\pi \times 10^6} = \frac{1}{2 \times 10^6} \text{ s} = 0.
5 \times 10^{-6} \text{ s} $$ The distance traveled along the magnetic field during this time period (which is the pitch \( p \)) is given by: $$ p = v_{\parallel} \times T $$ $$ p = (3 \times 10^5 \text{ ms}^{-1}) \times (0.
5 \times 10^{-6} \text{ s}) $$ $$ p = 1.
5 \times 10^{-1} \text{ m} $$ $$ p = 0.
15 \text{ m} $$ Convert the pitch to centimeters: $$ p = 0.
15 \text{ m} \times \frac{100 \text{ cm}}{1 \text{ m}} = 15 \text{ cm} $$ The pitch of the helix is 15 cm.