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# Consider a circular ring of radius 'r' uniformly charged with a linear charge density lemda .Find the electric potential at a point on the axis at a distance x from the centre of the ring .Using this expression for the potential, find the electric field at this poi

Eshan
askIITians Faculty 2095 Points
3 years ago
Dear student,

Distance of each infinitesimally small charge element on the ring from the given point is

$\dpi{80} d=\sqrt{x^2+r^2}$
Hence total electric potential at that point=$\dpi{80} V=\dfrac{1}{4\pi\epsilon_0}\dfrac{\int dq}{\sqrt{x^2+r^2}}=\dfrac{1}{4\pi\epsilon_0}\dfrac{\lambda( 2\pi r)}{\sqrt{x^2+r^2}}=\dfrac{\lambda r}{2\epsilon_0\sqrt{x^2+r^2}}$
Hence$\dpi{80} E=-\dfrac{dV}{dx}=\dfrac{1}{2}\dfrac{\lambda r}{2\epsilon_0 (x^2+r^2)^{3/2}}.2x=\dfrac{\lambda rx}{2\epsilon_0(x^2+r^2)^{3/2}}$