Ratio of Curvature Radii for Moving Charge Near Current Wires

To find the ratio of the radii of curvature $\frac{R_1}{R_2}$, we analyze the magnetic field at the position of the moving charge in both cases.

Let the two parallel wires lie in the plane of the paper, separated by a distance $X_0$. The point charge moves with speed $u$ in the same plane at a distance $X_1$ from the first wire, which means its distance from the second wire is: $X_2 = X_0 - X_1$

Given that $\frac{X_0}{X_1} = 3$, we have $X_0 = 3X_1$, and therefore: $X_2 = 3X_1 - X_1 = 2X_1$

The magnetic field $\vec{B}$ produced by a long, straight wire carrying current $I$ at a perpendicular distance $r$ is given by: $B = \frac{\mu_0 I}{2\pi r}$ Since the wires lie in the plane of the paper, the magnetic field at any point in the plane is perpendicular to the plane of the paper.

Case 1: Currents flow in the same direction

Between the two wires, the magnetic fields produced by the two currents are in opposite directions. Thus, the net magnetic field $B_{\text{net}, 1}$ is: $B_{\text{net}, 1} = \left| \frac{\mu_0 I}{2\pi X_1} - \frac{\mu_0 I}{2\pi X_2} \right| = \frac{\mu_0 I}{2\pi} \left( \frac{1}{X_1} - \frac{1}{2X_1} \right) = \frac{\mu_0 I}{4\pi X_1}$

Case 2: Currents flow in opposite directions

Between the two wires, the magnetic fields produced by the two currents are in the same direction. Thus, the net magnetic field $B_{\text{net}, 2}$ is: $B_{\text{net}, 2} = \frac{\mu_0 I}{2\pi X_1} + \frac{\mu_0 I}{2\pi X_2} = \frac{\mu_0 I}{2\pi} \left( \frac{1}{X_1} + \frac{1}{2X_1} \right) = \frac{3\mu_0 I}{4\pi X_1}$

Radius of Curvature

Since the charge moves with speed $u$ in the plane of the paper, its velocity is perpendicular to the magnetic field $\vec{B}$. The magnetic force acts as the centripetal force: $F = q u B_{\text{net}} = \frac{m u^2}{R} \implies R = \frac{m u}{q B_{\text{net}}}$

The radius of curvature is inversely proportional to the net magnetic field ($R \propto \frac{1}{B_{\text{net}}}$). Therefore, the ratio of the radii of curvature is: $\frac{R_1}{R_2} = \frac{B_{\text{net}, 2}}{B_{\text{net}, 1}} = \frac{\frac{3\mu_0 I}{4\pi X_1}}{\frac{\mu_0 I}{4\pi X_1}} = 3$