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A point charge q is situated at a distance d from one end of a thin non-conducting rod of length L having a charge Q (uniformly distributed along its length) as shown in fig. Then the magnitude of the electric force between them is:

Options

  1. A1 4 π ε 0 q Q 2 d ( d + L )
  2. B1 4 π ε 0 2 q Q d ( d + L )
  3. C1 4 π ε 0 q Q 3 d ( d + L )
  4. D1 4 π ε 0 q Q d ( d + L )

Correct answer

D. 1 4 π ε 0 q Q d ( d + L )

Step-by-step solution

F = ∫ d d + L k q d q x 2 = ∫ d d + L k q Q L d x x 2 = k q Q L ∫ d d + L 1 x 2 d x F = k q Q d ( d + L ) = 1 4 π ε 0 q Q d ( d + L ) Or The linear charge density of rod λ = Q L ;   d q = λ d x Force due to small element d x to charge q d F = 1 ( q ) ( λ ) d x 4 π ϵ 0 x 2 Total force in-between point charge and rod F n e t = q Q L ∫ d L + d x - 2 d x 4 π ϵ 0 = q Q 4 π ϵ 0 L - 1 x d L + d = q Q 4 π ε 0 L 1 d - 1 L + d =

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