Question

Assertion (A): A charged particle is moving with velocity \( v \) in \( x \)-\( y \) plane, making an angle \( \theta \) (\( 0<\theta<90^\circ \)) with \( x \)-axis. If a uniform magnetic field is applied in the region, along \( y \)-axis, the particle will move in a helical path with its axis parallel to \( x \)-axis.
Reason (R): The direction of the magnetic force acting on a charged particle moving in a magnetic field is along the velocity of the particle.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Both Assertion (A) and Reason (R) are false.

Hint:

The magnetic force on a charged particle is perpendicular to both the velocity and the magnetic field (\( \vec{F} = q (\vec{v} \times \vec{B}) \)). If the velocity has a component parallel to \( \vec{B} \), the particle moves in a helical path with the axis along \( \vec{B} \).

Answer 1

The Correct Option is D

Step 1: Evaluate Assertion (A).
Given velocity \( \vec{v} = v \cos \theta \hat{i} + v \sin \theta \hat{j} \) and magnetic field \( \vec{B} = B \hat{j} \), the magnetic force is calculated as: \[\vec{F} = q (\vec{v} \times \vec{B}) = q (v \cos \theta \hat{i} + v \sin \theta \hat{j}) \times (B \hat{j}) = q (v \cos \theta B) \hat{k}\]The calculated force acts along the \( z \)-axis. The velocity component perpendicular to \( \vec{B} \), which is \( v \cos \theta \hat{i} \), induces circular motion in the \( x \)-\( z \) plane. The velocity component parallel to \( \vec{B} \), \( v \sin \theta \hat{j} \), results in linear motion along the \( y \)-axis. Consequently, the particle follows a helical path with its axis aligned with the \( y \)-axis, contradicting the assertion that it is along the \( x \)-axis. Therefore, Assertion (A) is false.Step 2: Evaluate Reason (R).
The magnetic force is defined as \( \vec{F} = q (\vec{v} \times \vec{B}) \). This force is inherently perpendicular to the velocity vector \( \vec{v} \). An exception occurs only when \( \vec{v} \) is parallel to \( \vec{B} \), a condition not met in this scenario. Thus, Reason (R) is false.Step 3: Conclusion.
As both Assertion (A) and Reason (R) have been determined to be false, the appropriate selection is (D).