Equation 11.14 can be expressed in “coordinate-free” form by writing ${\mathit{p}}_{\mathbf{0}}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathbf{=}{\mathit{p}}_{\mathbf{0}}\mathbf{·}\hat{\mathbf{r}}$. Do so, and likewise for Eqs. 11.17, 11.18. 11.19, and 11.21.

Given data:

The equation is given as $p_0\cos \theta =p_0·\hat{r}$.

Write the expression for equation 11.14.

$\mathit{V}\left(r,\theta ,t\right)\mathbf{=}\mathbf{-}\frac{{\mathbf{p}}_{\mathbf{0}}\mathbf{\omega }}{\mathbf{4}\mathbf{\Pi }{\mathbf{\epsilon }}_{\mathbf{0}}\mathbf{c}}\left(\frac{cos\theta }{r}\right)\mathit{s}\mathit{i}\mathit{n}\left[\omega \left(t-\frac{r}{c}\right)\right]$

Here, r is the radial distance, t is the time,$\theta$ is the angle, c is the speed of light${\epsilon }_0$ is the susceptibility in free space, and$\omega$ is the angular frequency.

Substitute$p_0\cos \theta =p_0·\hat{r}$ in the above expression.

$V\left(r,t\right)=-\frac{\omega }{4\pi {\epsilon }_{0}c}\frac{p_0·\hat{r}}{r}\sin \left[\omega \left(t-\frac{r}{c}\right)\right]$

Write the expression for equation 11.17.

$A\left(r,\theta ,t\right)=-\frac{{\mu }_0{p}_0\omega }{4\pi r}\sin \left[\omega \left(t-\frac{r}{c}\right)\right]\hat{z}$

Substitute$p_0\cos \theta =p_0·\hat{r}$ in the above expression.

$A\left(r,t\right)=-\frac{{\mu }_0\omega }{4\pi }\frac{p_0}{r}\sin \left[\omega \left(t-\frac{r}{c}\right)\right]$

Write the expression for equation 11.18.

$$\begin{gathered} E=-\nabla V-\frac{\partial A}{\partial t} \\ E=-\frac{{\mu }_0{p}_0{\omega }^2}{4\pi }\left(\frac{\sin \theta }{r}\right)\cos \left[\omega \left(t-\frac{r}{c}\right)\right]\hat{\theta } \end{gathered}$$

Substitute $p_0\cos \theta =p_0·\hat{r}$in the above expression.

$E=-\frac{{\mu }_0{\omega }^2}{4\pi }\frac{\hat{r}\times \left(p_0\times \hat{r}\right)}{r}\cos \left[\omega \left(t-\frac{r}{c}\right)\right]$

Write the expression for equation 11.18.

$$\begin{gathered} B=\nabla \times A \\ B=-\frac{{\mu }_0{p}_0{\omega }^2}{4\pi }\left(\frac{\sin \theta }{r}\right)\cos \left[\omega \left(t-\frac{r}{c}\right)\right]\hat{\varphi } \end{gathered}$$

Substitute$p_0\cos \theta =p_0·\hat{r}$ in the above expression.

$B=-\frac{{\mu }_0{\omega }^2}{4\pi c}\frac{\left(p_0\times \hat{r}\right)}{r}\cos \left[\omega \left(t-\frac{r}{c}\right)\right]$

Write the expression for equation 11.21.

Substitute$p_0\cos \theta =p_0·\hat{r}$in the above expression.

Therefore, the equations 11.17, 11.18, 11.19, and 11.21 in the coordinate free form are found as:

$$\begin{gathered} V\left(r,t\right)=-\frac{\omega }{4\pi {\epsilon }_{0}c}\frac{p_0·\hat{r}}{r}\sin \left[\omega \left(t-\frac{r}{c}\right)\right] \\ A\left(r,t\right)=-\frac{{\mu }_0\omega }{4\pi }\frac{p_0}{r}\sin \left[\omega \left(t-\frac{r}{c}\right)\right] \\ E=-\frac{{\mu }_0{\omega }^2}{4\mathrm{\pi }}\frac{\hat{r}\times \left(p_0\times \hat{r}\right)}{r}\cos \left[\omega \left(t-\frac{r}{c}\right)\right] \\ B=-\frac{{\mu }_0{\omega }^2}{4\pi c}\frac{\left(p_0\times \hat{r}\right)}{r}\cos \left[\omega \left(t-\frac{r}{c}\right)\right] \end{gathered}$$

Also,