Two positive charges Q are held fixed a distance d apart. A particle of negative charge -q and mass m is placed midway between them and then given a small displacement perpendicular to the line joining them and released. We have to show that the particle describes simple harmonic motion of period .

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It is given that two positive charges +Q are held fixed a distance d apart. A particle of negative charge -q and mass m is placed midway between them and given a small displacement perpendicular to the line joining them and is released. Let the displacement be y along the Y-axis. As the displacement is small we have the condition

By symmetry the net Coulomb force on the charge -q due to the charges Q will be in the direction and its magnitude will be

In the approximation we approximate F as

Equation of motion of the charge -q that has mass m will therefore be

Scalar form of this equation is

It is an equation of simple harmonic motion form. The period of SHM of the motion is

Or