Magnetic Force and Angle Dependency - Detailed Solution

Magnetic Force and Angle Dependency

Question: When a charged particle moving with velocity \( \vec{v} \) is subjected to a magnetic field of induction \( \vec{B} \), the force on it is non-zero. This implies that:

(a) The angle is either \( 0^\circ \) or \( 180^\circ \).

(b) The angle is necessarily \( 90^\circ \).

(c) The angle can have any value other than \( 90^\circ \).

(d) The angle can have any value other than \( 0^\circ \) and \( 180^\circ \).

Detailed Step-by-Step Explanation

Step 1: Magnetic Force Formula
The magnetic force \( \vec{F} \) on a charged particle moving with velocity \( \vec{v} \) in a magnetic field \( \vec{B} \) is given by the Lorentz force law: \[ \vec{F} = q (\vec{v} \times \vec{B}) \] where \( q \) is the charge.

Step 2: Dependency on the Angle
The magnitude of the force is: \[ F = qvB \sin \theta, \] where \( \theta \) is the angle between \( \vec{v} \) and \( \vec{B} \).

Step 3: Condition for a Non-Zero Force
For the force \( F \) to be non-zero, the term \( \sin \theta \) must be non-zero. This means: \[ \sin \theta \neq 0. \] The sine of an angle is zero when the angle is \( 0^\circ \) or \( 180^\circ \).

Therefore, if the force is non-zero, the angle \( \theta \) must be any value other than \( 0^\circ \) or \( 180^\circ \).

Final Answer: The angle between \( \vec{v} \) and \( \vec{B} \) can have any value other than \( 0^\circ \) and \( 180^\circ \), which corresponds to option (d).