Before someone puts it on 'hold' , this is not a homework question, I doubt you'll find it somewhere as I thought of it. I wish to find force of electrostatic attraction between 2 uniform rings of equal charge and radii. I can not find a solution to this without going to the integral of a 3D angle which is beyond my power. I do not wish a calculated answer, just an approach
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3$\begingroup$ What is the relative orientation of the two rings? Are the parallel and on the same axis (like Helmholtz coil pair)? Or do they rather have an arbitrary relation to each other? $\endgroup$– Martin UedingMay 1, 2016 at 20:44
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1 Answer
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Assuming that the rings share a central axis A:
You'll have a double integral, with the outer integral being a sum of the component of force parallel to the axis A on each tiny chunk dQ of the first ring. The inner integral finds that force component on dQ by adding up the component parallel to A of the force between dQ and each tiny chunk dq of the second ring.
You can use a parameter to eliminate most of those pesky angles from your inner integral. There might be a more elegant option, but one obvious choice would be the angle about A between dQ and dq. You'd write an expression for the distance between the two chunks in terms of that angle, then use that to find the force component, which is the expression to go in the inner integral.
Since we applied a symmetry argument and only included the force components parallel to A in this approach, the outer integral is laughably simple, just adding up the scalar result of the inner integral, rather than integrating the full force vectors in 3D.
Hope that helps!
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$\begingroup$ I did the same thing stated above but I can't eliminate the angles in the inner integral. Any tips? $\endgroup$ May 2, 2016 at 3:00
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$\begingroup$ What is your expression for the distance between the two chunks of charge? $\endgroup$ May 2, 2016 at 4:01