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: Analyze Assertion (A).
The velocity of the particle is \( \vec{v} = v \cos \theta \hat{i} + v \sin \theta \hat{j} \), and the magnetic field is \( \vec{B} = B \hat{j} \). The magnetic force is: \[ \vec{F} = q (\vec{v} \times \vec{B}) = q (v \cos \theta \hat{i} + v \sin \theta \hat{j}) \times (B \hat{j}) = q (v \cos \theta B) \hat{k} \] The force is along the \( z \)-axis. The velocity component perpendicular to \( \vec{B} \) (\( v \cos \theta \hat{i} \)) causes circular motion in the \( x \)-\( z \) plane, while the component parallel to \( \vec{B} \) (\( v \sin \theta \hat{j} \)) causes linear motion along the \( y \)-axis. This results in a helical path with the axis along the \( y \)-axis, not the \( x \)-axis as stated. Thus, Assertion (A) is false. Step 2: Analyze Reason (R).
The magnetic force \( \vec{F} = q (\vec{v} \times \vec{B}) \) is perpendicular to the velocity \( \vec{v} \), not along it, unless \( \vec{v} \parallel \vec{B} \), which is not the case here. Thus, Reason (R) is false. Step 3: Conclusion.
Since both Assertion (A) and Reason (R) are false, the correct option is (D).