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Hello, friends! I read that, if a dipole is centred on the origin, with the ##+q>0## charge in ##a>0## and the negative ##-q<0## charge in ##-a##, the field in a point ##(x,y)## of the plane is $$\mathbf{E}=k\frac{3pxy}{(x^2+y^2)^{5/2}}\mathbf{i} +k\frac{p(2y^2-x^2)}{(x^2+y^2)^{5/2}}\mathbf{j}$$ where ##k## is Coulomb's constant.

If I have correctly calculated, the exact electric field in ##(x,y)## should be ##\mathbf{E}=kq\Big(\frac{x}{(x^2+(y-a)^2)^{3/2}}-\frac{x}{(x^2+(y+a)^2)^{3/2}}\Big)\mathbf{i}## ##+kq\Big(\frac{y-a}{(x^2+(y-a)^2)^{3/2}}-\frac{y+a}{(x^2+(y+a)^2)^{3/2}}\Big)\mathbf{j}##, but I have got serious troubles in finding a way to see that this is desired result. Nevertheless, I suspect that an approximation for ##|x|\gg a## and ##|y|\gg a## represents the formula given by my book, but I cannot find a way to prove the approximation.

I heartily thank you everybody for any answer.

If I have correctly calculated, the exact electric field in ##(x,y)## should be ##\mathbf{E}=kq\Big(\frac{x}{(x^2+(y-a)^2)^{3/2}}-\frac{x}{(x^2+(y+a)^2)^{3/2}}\Big)\mathbf{i}## ##+kq\Big(\frac{y-a}{(x^2+(y-a)^2)^{3/2}}-\frac{y+a}{(x^2+(y+a)^2)^{3/2}}\Big)\mathbf{j}##, but I have got serious troubles in finding a way to see that this is desired result. Nevertheless, I suspect that an approximation for ##|x|\gg a## and ##|y|\gg a## represents the formula given by my book, but I cannot find a way to prove the approximation.

I heartily thank you everybody for any answer.

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