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Exercise:
Derive a formal expression for the force between a uniformly charged ring and a po charge on the ring's central axis. center includegraphicswidth.mm#image_path:charged-ring-# center

Solution:
We divide the ring o a large number N segments of equal length with charge dQsscQring/N. The force dF of one such segment on the po charge q is shown in the figure. center includegraphicswidth.mm#image_path:charged-ring-solution-# center The same amount of charge dQ'dQ on the opposite side of the ring exerts a force dF' on the po charge. The combined force sscdFnet is given by fracsscdFnet/dF fracdr Longrightarrow sscdFnet dF fracdr k_C fracdQ qr^ fracdr k_C fracsscQring q dN r^ k_C fracsscQring q dN leftR^ + d^right^/ There are N/ such pairs of opposite segments i.e. the total force on q is sscFnet fracN sscdFnet fracN k_C fracsscQring q dN leftR^ + d^right^/ k_C fracsscQring q dleftR^ + d^right^/ Remark: Coulomb's force law only yields the correct force between two po charges i.e. in the limit N to infty. Because of the symmetry of the charge distribution on the ring the correct result can be derived using segments of finite length.
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Exercise:
Derive a formal expression for the force between a uniformly charged ring and a po charge on the ring's central axis. center includegraphicswidth.mm#image_path:charged-ring-# center

Solution:
We divide the ring o a large number N segments of equal length with charge dQsscQring/N. The force dF of one such segment on the po charge q is shown in the figure. center includegraphicswidth.mm#image_path:charged-ring-solution-# center The same amount of charge dQ'dQ on the opposite side of the ring exerts a force dF' on the po charge. The combined force sscdFnet is given by fracsscdFnet/dF fracdr Longrightarrow sscdFnet dF fracdr k_C fracdQ qr^ fracdr k_C fracsscQring q dN r^ k_C fracsscQring q dN leftR^ + d^right^/ There are N/ such pairs of opposite segments i.e. the total force on q is sscFnet fracN sscdFnet fracN k_C fracsscQring q dN leftR^ + d^right^/ k_C fracsscQring q dleftR^ + d^right^/ Remark: Coulomb's force law only yields the correct force between two po charges i.e. in the limit N to infty. Because of the symmetry of the charge distribution on the ring the correct result can be derived using segments of finite length.
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Branches
Electrostatics
Tags
charge distribution, coulomb, electrostatic force
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Difficulty
(4, default)
Points
0 (default)
Language
ENG (English)
Type
Algebraic
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Decoration