Motion of a Charged Particle Between Two Parallel Currents

Problem Statement

In a three-dimensional Cartesian coordinate system in vacuum, two infinitely long straight wires are placed parallel to the $z$z-axis. One wire passes through $(d, 0, z)$(d,0,z) and the other through $(-d, 0, z)$(−d,0,z). Both wires carry a constant current $I$I in the $+z$+z direction.

At time $t$t, a charged particle of mass $m$m and charge $q$q ($q > 0$q>0) is placed at the origin $(0,0,0)$(0,0,0) and given an initial velocity $\vec{v} = (v_1, 0, v_0)$v=(v1​,0,v0​). Here $v_1$v1​ is much smaller than $v_0$v0​ ($v_1 \ll v_0$v1​≪v0​), and the $x$x-displacement of the particle is always much smaller than $d$d ($|x| \ll d$∣x∣≪d).

The particle moves in the $z$z-direction while performing small oscillations in the $x$x-direction. The $z$z-component of velocity may be approximated as constant at $v_0$v0​. Find the distance $L$L the particle travels in the $z$z-direction during one period of the $x$x-oscillation.

Constraints

• Wire position parameter: $d = 0.50\ \mathrm{m}$d=0.50 m
• Current in wires: $I = 50\ \mathrm{A}$I=50 A
• Particle charge: $q = 2.0 \times 10^{-4}\ \mathrm{C}$q=2.0×10−4 C
• Particle mass: $m = 4.0 \times 10^{-5}\ \mathrm{kg}$m=4.0×10−5 kg
• Initial velocity in $z$z-direction: $v_0 = 1.0 \times 10^2\ \mathrm{m/s}$v0​=1.0×102 m/s
• Vacuum permeability: $\mu_0 = 4\pi \times 10^{-7}\ \mathrm{N/A^2}$μ0​=4π×10−7 N/A2
• Ignore gravity and electromagnetic radiation from the particle.

Input Format

The distance $L$L is expressed as $A\pi\ (\mathrm{m})$Aπ (m). Give the value of positive integer $A$A.