91影视

Given data:

The equation is given as $p_0\cos 胃=p_0路\hat{r}$.

Write the expression for equation 11.14.

$\mathit{V}\left(r,胃,t\right)\mathbf{=}\mathbf{-}\frac{{\mathbf{p}}_{\mathbf{0}}\mathbf{蝇}}{\mathbf{4}\mathbf{螤}{\mathbf{蔚}}_{\mathbf{0}}\mathbf{c}}\left(\frac{cos胃}{r}\right)\mathit{s}\mathit{i}\mathit{n}\left[蝇\left(t-\frac{r}{c}\right)\right]$

Here, r is the radial distance, t is the time,$胃$ is the angle, c is the speed of light${蔚}_0$ is the susceptibility in free space, and$蝇$ is the angular frequency.

Substitute$p_0\cos 胃=p_0路\hat{r}$ in the above expression.

$V\left(r,t\right)=-\frac{蝇}{4蟺{蔚}_{0}c}\frac{p_0路\hat{r}}{r}\sin \left[蝇\left(t-\frac{r}{c}\right)\right]$

Write the expression for equation 11.17.

$A\left(r,胃,t\right)=-\frac{{渭}_0{p}_0蝇}{4蟺r}\sin \left[蝇\left(t-\frac{r}{c}\right)\right]\hat{z}$

Substitute$p_0\cos 胃=p_0路\hat{r}$ in the above expression.

$A\left(r,t\right)=-\frac{{渭}_0蝇}{4蟺}\frac{p_0}{r}\sin \left[蝇\left(t-\frac{r}{c}\right)\right]$

Write the expression for equation 11.18.

$$\begin{gathered} E=-鈭�V-\frac{鈭�A}{鈭�t} \\ E=-\frac{{渭}_0{p}_0{蝇}^2}{4蟺}\left(\frac{\sin 胃}{r}\right)\cos \left[蝇\left(t-\frac{r}{c}\right)\right]\widehat{胃} \end{gathered}$$

Substitute $p_0\cos 胃=p_0路\hat{r}$in the above expression.

$E=-\frac{{渭}_0{蝇}^2}{4蟺}\frac{\hat{r}脳\left(p_0脳\hat{r}\right)}{r}\cos \left[蝇\left(t-\frac{r}{c}\right)\right]$

Write the expression for equation 11.18.

$$\begin{gathered} B=鈭�脳A \\ B=-\frac{{渭}_0{p}_0{蝇}^2}{4蟺}\left(\frac{\sin 胃}{r}\right)\cos \left[蝇\left(t-\frac{r}{c}\right)\right]\widehat{蠒} \end{gathered}$$

Substitute$p_0\cos 胃=p_0路\hat{r}$ in the above expression.

$B=-\frac{{渭}_0{蝇}^2}{4蟺c}\frac{\left(p_0脳\hat{r}\right)}{r}\cos \left[蝇\left(t-\frac{r}{c}\right)\right]$

Write the expression for equation 11.21.

Substitute$p_0\cos 胃=p_0路\hat{r}$in the above expression.

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

$$\begin{gathered} V\left(r,t\right)=-\frac{蝇}{4蟺{蔚}_{0}c}\frac{p_0路\hat{r}}{r}\sin \left[蝇\left(t-\frac{r}{c}\right)\right] \\ A\left(r,t\right)=-\frac{{渭}_0蝇}{4蟺}\frac{p_0}{r}\sin \left[蝇\left(t-\frac{r}{c}\right)\right] \\ E=-\frac{{渭}_0{蝇}^2}{4\mathrm{蟺}}\frac{\hat{r}脳\left(p_0脳\hat{r}\right)}{r}\cos \left[蝇\left(t-\frac{r}{c}\right)\right] \\ B=-\frac{{渭}_0{蝇}^2}{4蟺c}\frac{\left(p_0脳\hat{r}\right)}{r}\cos \left[蝇\left(t-\frac{r}{c}\right)\right] \end{gathered}$$

Also,