Far-field approximation

The far-field approximation (Fraunhofer approximation) is a technique used in analysis of electromagnetic fields at observation points that are large distances away from the sources. The idea is that at the near field (small $r$), the field is complicated in terms of geometry and magnitude. At the far field, we have simplifying assumptions since the field gets weaker and the geometry simplifies.

Take an electric dipole, for example. As we move our observation point $P$ further away from the dipole, the vectors from $q$ to $P$ will become parallel. From this, we can define the distance with respect to an angle.

$$R_{\pm }=R\mp \frac{d}{2}\cos \theta$$ $$V(P)=\frac{q}{4\pi {ϵ}_0{r}_{+}}-\frac{q}{4\pi {ϵ}_0{r}_{-}}=\frac{q}{4\pi {ϵ}_0}(\frac{1}{r_{+}}-\frac{1}{r_{-}})$$ $$=\frac{q}{4\pi {ϵ}_0}\left(\frac{r_{-}-r_{+}}{r_{+}{r}_{-}}\right)=\frac{q}{4\pi {ϵ}_0}\left(\frac{r_{-}-r_{+}}{r^2-(\frac{d}{2}\cos \theta {)}^2}\right)$$

Since $r\gg d$:

$$V=\frac{q}{4\pi {ϵ}_0}\frac{d\cos \theta }{r^2}$$