a charge Q of mass M is placed at distance D from infinite length of wire of lemda charge density .then its slightly displaced from equilibrium . its motion will be SHM or NOT.
We will denote a distance between a charge and a wire. Let's study the potential energy of wire:
, where is energy associated to the electromagnetical interaction between the wire and the charge, is a potential energy associated to gravity. Now to determine the behaviour of a charge due to a small displacement we need to consider the developpement of in a power series of until second order :
The 0 order term is a constant, therefore it does not affect the behaviour. First order term is zero, as the expression in the brackets is a total force acting on a charge, and charge is in an equilibrium.
The potential energy depends only on due to the symmetries of an infinite wire (rotation and translation parallel to wire). First derivative of is minus the force, therefore let's calculate the field of an infinite wire. We can apply Gauss theorem to find that . Therefore . Therefore this will be a simple harmonic motion if
. If we suppose that does not depend on a distance (which is a pretty good approximation), this will be a simple harmonic motion if , i.e. the charge and the wire's charge density are of the same sign (in this case charge is above the wire).
If the displacement is made in a direction that is parallel to wire, charge stays at equilibrium.
If the displacement have a non-zero component in a direction orthogonal to wire axis and distance to a wire, the equilibrium is unstable and motion will now be a SHM.
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