Before You Begin

Question #43382

Difficulty: 8/10 | 🟢 Free

The Question

A long straight wire fixed along the $z$ -axis carries a steady current $I$ . A particle of mass $m$ and positive charge $q$ is projected from a point at a radial distance $r_0$ from the wire. The initial velocity of the particle is $\vec{v}_0 = v_0 \hat{r}$ , where $\hat{r}$ is the unit vector in the radial direction pointing perpendicularly away from the wire. Neglecting gravity and electromagnetic radiation, what is the maximum radial distance $r_{max}$ from the wire reached by the particle during its subsequent motion?

Options

A.
$r_0 \exp\left( \frac{2\pi m v_0}{\mu_0 q I} \right)$
B.
$r_0 \left( 1 + \frac{2\pi m v_0}{\mu_0 q I} \right)$
C.
$r_0 \exp\left( \frac{\mu_0 q I}{2\pi m v_0} \right)$
D.
$r_0 \left( 1 + \frac{\mu_0 q I}{2\pi m v_0} \right)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Trajectory of Charged Particle|Impulsive Torque and Angular Momentum|Electric Field Line Geometry

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze motion in a non-uniform magnetic field with a radial gradient ( $B \propto 1/r$ ).

A common trap is to locally apply the constant-field cyclotron radius formula $R = mv/qB$ at the initial position, incorrectly assuming the particle follows a circular arc to a maximum distance of $r_0 + R$ (Option B).

However, because the field varies, the Lorentz force is not purely centripetal with a constant radius.

By recognizing that the torque about the current axis is zero, students can deduce that the particle remains in a constant azimuthal plane.

Integrating the z-component of Newton's Second Law yields a conserved canonical momentum, which, when combined with the Work-Energy Theorem (constant speed), reveals an exponential 'Magnetic Mirror' reflection point, forcing the abandonment of circular assumptions.

Notice that the trap answer is exactly the first-order Taylor expansion of the correct exponential answer, highlighting how the circular assumption only holds for infinitesimally small trajectories.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Trajectory of Charged Particle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free