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## Homework Statement

Two infinite line charges, of charge density λ Cm

^{-1}, are aligned parallel to one another as shown in the diagram below. The line charges are a distance 2a apart. An electron is placed at location (x, 0, 0), x << a and released. Show that it will execute simple harmonic motion about the origin if the sign of the charge density is chosen correctly. Find an expression for the frequency of the motion.

http://img15.imageshack.us/img15/1507/45768328.jpg [Broken]

## Homework Equations

The integral form of Gauss’s law: [itex]\oint \vec{E} \ . \ \hat{n} da = \frac{q_{enc}}{\epsilon_0}[/itex]

SHM equation: [itex]\frac{d^2x}{dt^2}=- \frac{k}{m} x[/itex]

The relationships: [itex]f=\frac{1}{T} = \frac{kv}{2 \pi} = \frac{v}{\lambda}[/itex]

## The Attempt at a Solution

So using Gauss's law the electric field due to the two line charges at any point would be:

[itex]E= \frac{\lambda}{2 \pi \epsilon_0 s} \hat{s}[/itex]

Since the electron has negative charge and it is closer to the left-hand line charge, I think the left line charge must also be negative so they repel and the electron stays somewhere in the middle. Is this right?

So, how can I show that the motion would be SHM? I know that the solution to the SHM equation is given by x(t) = A cos (ωt-ϕ). But how do I relate this to the electric fields and the motion of the electron?

Any help would be greatly appreciated.

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