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Derived formula of extended charge bodies like disc .
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We determine the field at point P on the axis of the ring. It should be apparent from symmetry that the field is along the axis. The field dE due to a charge element dq is shown, and the total field is just the superposition of all such fields due to all charge elements around the ring. The perpendicular fields sum to zero, while the differential x-component of the field is
This gives the predicted result. Note that for x much larger than a (the radius of the ring), this reduces to a simple Coulomb field. This must happen since the ring looks like a point as we go far away from it. Also, as was the case for the gravitational field, this field has extrema at x = +/-a.
Electric Field on the Axis of a Uniformly Charged Disk
[Note from ghw: This is a local copy of a portion of Stephen Kevan's lecture on Electric Fields and Charge Distribution of April 8, 1996.]
Using the above result, we can easily derive the electric field on the axis of a uniformly charged disk, simply by invoking superposition and summing up contributions of a continuous distribution of rings, as shown in the following figure from Tipler:
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