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In a free space, a thin rod carrying uni...

In a free space, a thin rod carrying uniformly distributed negative charge `-q` is placed symmetrically along the axis of a tin ring of radius R varrying uniformly dostributed charge Q. The mass of the rod is m and length is `l=2R`. The ring is fixed and the rod is free to move. The rod is displaced slightly along the axis of the ring and then released. Find the period T to the small amplitude oscillations of the rod.

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To solve the problem of finding the period \( T \) of small amplitude oscillations of a charged rod placed along the axis of a charged ring, we can follow these steps: ### Step 1: Understand the System We have a thin rod of length \( l = 2R \) carrying a uniformly distributed negative charge \( -q \). The rod is placed symmetrically along the axis of a fixed tin ring of radius \( R \) that carries a uniformly distributed positive charge \( Q \). The rod is free to move along the axis of the ring. ### Step 2: Displacement and Forces When the rod is slightly displaced by a distance \( y \) along the axis, it experiences a restoring force due to the electric field created by the charged ring. The electric field \( E \) at a distance \( x \) from the center of the ring along its axis is given by: \[ ...
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