Question

# Drive an expression for Electric field due to an Electric dipole at a point on the axial line.

Solution

## Let the electric field at an axial point P at an distance $$r$$ from the center of the dipole be $$E_p$$.Electric field at P, $$E_p = E_q - E_{-q}$$Or $$E_p = \dfrac{kq}{(r-a)^2} - \dfrac{kq}{(r+a)^2}$$ where $$k = \dfrac{1}{4\pi \epsilon_o}$$Or $$E_p = kq[\dfrac{1}{(r-a)^2} - \dfrac{1}{(r+a)^2}]$$Or $$E_p = kq[\dfrac{(r+a)^2 - (r-a)^2}{(r+a)^2 (r-a)^2}]$$Or $$E_p = kq[\dfrac{4ar}{(r^2-a^2)^2 }]$$Or $$E_p = \dfrac{2kP r}{(r^2-a^2)^2 }$$ where $$P = 2qa$$For $$r>>a$$, we can neglect $$a^2$$ compared to $$r^2$$.We get $$E_p = \dfrac{2kP r}{r^4 }$$$$\implies$$ $$E_p = \dfrac{2kP }{r^3 }$$Physics

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