An electric dipole that has dipole separation \(d\) is aligned along the \(y\) axis of an \(x y\) coordinate system, pointed in the positive \(y\) direction.

(a) Show that for \(x\) and \(y\) much greater than \(d\), the \(x\) and \(y\) components of the electric field of the dipole are given by

\[E_{x}=\frac{3 k p x y}{\left(x^{2}+y^{2}\right)^{5 / 2}} ; E_{y}=\frac{k p\left(2 y^{2}-x^{2}\right)}{\left(x^{2}+y^{2}\right)^{5 / 2}} .\]

(b) Show that this general result includes the special results derived in Section 23. 6.

Data from Section 23. 6

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Transcribed Image Text:

Next we examine the electric field due to a permanent electric dipole. Figure 23.24 on the next page shows a dipole that consists of a particle car- rying a charge +q, at x = 0, y = +d, and another particle carrying a charge -qp at x= 0, yd, where d is the distance between the two particles. The charge qp of the positively charged pole is called the dipole charge, and the distance d is called the dipole separation. Each particle creates an electric field at all positions in space, so the two fields overlap