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An electron sits exactly midway between two long, fixed, parallel line charges, each with positive linear charge density +λ, both lying in the xy-plane (ignore gravity). The line charges run parallel to the y-axis, separated along x, and the perpendicular z-axis is out of the plane. Which statement about the electron's equilibrium is correct?

1. The equilibrium is unstable along the x-direction
2. The equilibrium is neutral along the y-direction
3. The equilibrium is stable along the z-direction
4. The equilibrium is stable along the y-direction

Correct answer: The equilibrium is neutral along the y-direction

Solution

Each wire is parallel to y, so the field has no y-component and the situation is translationally symmetric along y; moving the electron along y changes nothing, giving neutral equilibrium in y. For an electron (negative) between two positive wires, displacing along x toward one wire increases attraction to it -> unstable in x; by symmetry/zero-divergence considerations the z-direction is stable. The clearly correct statement here is neutral equilibrium along y.

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