Question

Show that the electric field of a (perfect) dipole (Eq. 3.103) can be written in the coordinate-free form

${\mathit{E}}_{\mathbf{d}\mathbf{i}\mathbf{p}}\left(r\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{0}}}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{0}}}\frac{\mathbf{1}}{{\mathbf{r}}^{\mathbf{3}}}\left[3\left(p\widehat{·}r\right)r-p\right]$

Short answer

Answer

The given relation is proved.

Step-by-step solution

Define functions

Write the expression for electric field.

${\mathit{E}}_{\mathbf{d}\mathbf{i}\mathbf{p}\mathbf{o}\mathbf{l}\mathbf{e}}\left(r,\theta \right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{0}}}\frac{\mathbf{1}}{{\mathbf{r}}^{\mathbf{3}}}\mathit{\rho }\left[2cos\hat{\theta }r+sin\hat{\theta }\theta \right]$ …… (1)

Here, $\rho$is the dipole moment, $\theta$is the orientation of dipole electric field and ${\epsilon }_0$is the permittivity for the free space.

Determine electric field

Write the expression for the electric field.

$$\begin{gathered} E_{dipole}\left(r,\theta \right)=\frac{1}{4\pi {\epsilon }_0}\frac{1}{r^3}\left[2p\cos \hat{\theta }r+p\sin \hat{\theta }\theta \right] \\ =\frac{1}{4\pi {\epsilon }_0}\frac{1}{r^3}\left[2p\cos \hat{\theta }r-p\cos \hat{\theta }r+p\sin \hat{\theta }\theta \right] \\ =\frac{1}{4\pi {\epsilon }_0}\frac{1}{r^3}\left[3p\cos \hat{\theta }r-\left(p\cos \hat{\theta }r+p\sin \hat{\theta }\theta \right)\right] \end{gathered}$$ …… (2)

Write the dipole moment vector.

$p=p\cos \hat{\theta }\theta$ …… (3)

$$\begin{gathered} p\widehat{·}r=\left(p\cos \hat{\theta }r-p\sin \hat{\theta }\theta \right)\widehat{·}r \\ =p\cos \theta \end{gathered}$$ …… (4)

Substitute $p\cos \hat{\theta }r-p\sin \hat{\theta }\theta$for $\rho$and $p\cos \theta$for $p\widehat{·}r$in equation (2).

$$\begin{gathered} E_{dipole}\left(r,\theta \right)=\frac{1}{4\pi {\epsilon }_0}\frac{1}{r^3}\left[3p\cos \hat{\theta }r-\left(p\cos \hat{\theta }r+p\sin \hat{\theta }\theta \right)\right] \\ {E}_{dipole}\left(r,\theta \right)=\frac{1}{4\pi {\epsilon }_0}\frac{1}{r^3}\left[3\left(p\widehat{·}r\right)r-p\right] \end{gathered}$$

Thus, the given relation is proved.