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March 29, 2017

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Consider a ring of radius a containing q amount of negative charges. Derive an expression for the electric field strength at a point at a distance x on the axis of this charged ring. Hence prove that at a great enough distances, the charged ring behaves like a point charge.

  • Physics - ,

    dQ is portion of Q along a segment ds of the ring
    E = (1/4pieo) dQ/(x^2+a^2)
    the component of this in the axial direction is
    dEx = dE cos angle to x axis = dEx/sqrt(x^2+a^2)
    (component in direction perpendicular to axis is adds to zero by symmetry)
    so
    dEx = (1/4pieo) x dQ/(x^2+a^2)^1.5
    at a given constant x our integral is only over d Q and is Q
    so
    Ex = (1/4pieo)Q x/(x^2+a^2)^1.5

    if x is >> a
    then
    Ex = (1/4pieo)Q x/(x^2 + 0)^1.5
    =(1/4pieo)Q x/x^3
    = (1/4pieo)Q/x^2 which is point charge

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