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Question

Derive an expression for electric field due electric dipole at a point on an equatorial line.


Solution

Electric field at A due to the charges $$q$$ and $$-q$$ is shown in the above figure. We resolve $$E$$ into horizontal and vertical components. The vertical components $$(E \sin\theta)$$ cancel out each other and only the horizontal components survive to give net electric field at A as $$2E\cos\theta$$.
Electric field at point A, $$E_A = 2E \cos\theta$$
where $$E = \dfrac{q}{4\pi \epsilon_o (r^2 + a^2)}$$
We get $$E_A = \dfrac{2q }{4\pi \epsilon_o (r^2 + a^2)} \cos\theta$$
From figure, we have $$\cos\theta = \dfrac{a}{\sqrt{r^2 + a^2}}$$
$$\therefore$$ $$E_A = \dfrac{2q a}{4\pi \epsilon_o (r^2 + a^2)^{3/2}} $$
We know dipole moment $$P = 2qa$$
$$\implies$$ $$E_A = \dfrac{P}{4\pi \epsilon_o (r^2 + a^2)^{3/2}} $$
For $$a<<r$$, we can neglect $$a^2 $$ compared to $$r^2$$.
$$\implies$$ $$E_A = \dfrac{P}{4\pi \epsilon_o r^3} $$

658734_623499_ans_cc220208469a43d0b0ed75e4c4e6d89c.png

Physics

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