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two concentric rings of radii r and 2rar...

two concentric rings of radii r and `2r`are placed with centre at origin. Tow charges ` +q` each are fixed at the diametrically opposite points of the rings as shown in figure . Smallerring is now rotated by an angle ` 90^@` about Z-axis then it is again rotated by ` 90^@` about Y-axis . Find the work done by electrostatic forces in each strop . If finally larger ring is rotated by `90^@` about X-axis , find the total work required to perform all three steps .
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Text Solution

Verified by Experts

The correct Answer is:
` W_(first step ) = ( 8/ 3 - 4/( sqrt 5)) (Kq^2)/r , W _(secibd steo) =0`.
`U_i = (Kq^2)/r + (Kq^2)/(3r) + (Kq^2)/(4r) + (Kq^2)/(2r) + (Kq^2)/(3r) + (Kq^2)/r`
`U_i = (2Kq^2)/r + (2Kq^2)/(3r) + (Kw^2)/(2r) + (Kq^2)/(4r)`
` (##NG_PHY_C01_E01_122_S01.png" width="80%">
when smaller ring is rotated by `90^@`
`U_f = (Kq^2) /(sqrt 5r) + (Kq^2)/(sqrt 5r) + (Kq^2)/(4r) + (Kq^2)/(2r) + (Kq^2)/(sqrt5r) + (Kq^2)/(sqrt 5r)`
`U_f = (4Kq^2)/(sqrt 5 r) + (Kq^@)/(4 r) + (Kq^3)/(2r)`
` (##NG_PHY_C01_E01_122_S02.png" width="80%">
` W_("elecrtrostatice") = U_i - U_f = ( 2Kq^2 )/r + (2Kq^2)/(3r) - (4Kq^2)/(sqrt 5r)`
`W_("electrostatic") = (8/3- 4/(sqrt5)) (Kq^@)/r`
(b) When it rotaed by ` 90^@` about `y` ais then intial `&` final configration are same . Therefore its final and inital energy are same so `WD` is zero .
`C` If again ring is rotated by `90^@` then the initial configruation is achive which is givien in the question Therefore net work done will be zero . (`:.` electrostatic is conservative force field so `WD` does not depend upon path . If initial final posiyions are same `WD `=0`).
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