91影视

Write the expression for the power radiated by an ideal electric dipole.

$\mathit{P}\mathbf{=}\mathbf{鈭�}{\mathit{S}}_{\mathbf{r}}\mathbf{鈥�}\mathit{d}\mathit{a}$ 鈥︹�� (1)

Here, $S_r$is the Poynting vector of an ideal dipole which is given as:

$S_r=\frac{1}{{渭}_0}\left(E_r脳B_r\right)$ 鈥︹�� (2)

Here,$S_r$ is the electric field of the ideal dipole and is the magnetic field of the ideal dipole.

Write the expression for a non-static ideal dipole in an electric field.

$E\left(r,t\right)=-\frac{{渭}_0}{4蟺}\left\{\frac{{\displaystyle \stackrel{,,}{p}}{\displaystyle \stackrel{}{-\hat{r}}}\left(\hat{r}-{\displaystyle \stackrel{,,}{p}}\right)}{r}+C^2\frac{\left[p+\left({\displaystyle \frac{r}{c}}\right){\displaystyle \stackrel{,}{p}}\right]\left(\hat{r}鈥�\left[p+{\displaystyle \frac{r}{c}}\stackrel{,}{p}\right]\right)}{r}\right\}$

Here, ${渭}_0$is the permeability of free space, p is the dipole moment, r is the distance of the source, and c is the speed of light.

Write the expression for a non-static ideal dipole in the magnetic field.

$B\left(r,t\right)=-\frac{{渭}_0}{4蟺}\left\{\frac{\hat{r}脳\left[{\displaystyle \stackrel{,}{p}}+\left({\displaystyle \frac{r}{c}}\right){\displaystyle \stackrel{,,}{p}}\right]}{r^2}\right\}$

Determine the total power radiated over the sphere with the radius r is,

Write the radiation for the fixed time at the origin $t=t-\frac{r}{c}$from the electric and magnetic dipole moment.

$$\begin{gathered} E_r=-\frac{{渭}_0}{4蟺}\left\{\frac{{\displaystyle \stackrel{,,}{p-\hat{r}}}\left(\hat{r}鈥�{\displaystyle \stackrel{,,}{p}}\right)}{r^2}\right\} \\ {B}_r=-\frac{{渭}_0}{4蟺}\left\{\frac{\hat{r}脳{\displaystyle \stackrel{,,}{p}}}{rc}\right\} \end{gathered}$$

Substitute the value of $E_r$and $B_r$in equation (2).

$$\begin{gathered} S_r=\frac{1}{{渭}_0}\left(-\frac{{渭}_0}{4蟺}\left\{\frac{{\displaystyle \stackrel{,,}{p}}-\hat{r}\left(\hat{r}鈥�{\displaystyle \stackrel{,,}{p}}\right)}{r}\right\}脳-\frac{{渭}_0}{4蟺}\left\{\frac{\hat{r}鈥�\stackrel{,,}{p}}{rc}\right\}\right) \\ {S}_r=\frac{{渭}_0}{16{蟺}^{2}c{r}^2}\left[\left(\stackrel{,,}{p}-\hat{r}\left(\hat{r}鈥�\stackrel{,,}{p}\right)\right)脳\left(\hat{r}鈥�\stackrel{,,}{p}\right)\right] \\ {S}_r=\frac{{渭}_0}{16{蟺}^{2}c{r}^2}\left[\left(\stackrel{,,}{p}-\hat{r}\left(\hat{r}鈥�\stackrel{,,}{p}\right)\right)-\left(\hat{r}鈥�\stackrel{,,}{p}\right)\left[\hat{r}脳\left(\hat{r}鈥�\stackrel{,,}{p}\right)\right]\right] \\ {S}_r=\frac{{渭}_0}{16{蟺}^{2}c{r}^2}\left[{\stackrel{,,}{p}}^2-\left(\hat{r}鈥�\stackrel{,,}{p}\right)\right]\hat{r} \end{gathered}$$

Substitute the value of$S_r$in equation (1).

$$\begin{gathered} P=鈭�\frac{{渭}_0}{16{蟺}^{2}c{r}^2}\left[{\stackrel{,,}{p^2-\left(\hat{r}鈥�\stackrel{,,}{p}\right)}}^2\right]\hat{r}-da \\ P=鈭�\frac{{渭}_0}{16{蟺}^{2}c{r}^2}\left[{\stackrel{,,}{p^2-\left(\hat{r}鈥�\stackrel{,,}{p}\right)}}^2\right]\hat{r}-\left(r^2\sin 胃d胃d蠒\right) \end{gathered}$$

Here,$\left[{\stackrel{,,}{p^2-\left(\hat{r}鈥�\stackrel{,,}{p}\right)}}^2\right]=\stackrel{,,}{p^2{\sin }^2胃}$

Hence, the equation becomes,

$$\begin{gathered} P=\frac{{渭}_0}{16{蟺}^{2}c{r}^2}\stackrel{..}{p}鈭�\frac{1}{r^2}{\sin }^2胃r^2\sin 胃d胃d蠒 \\ P=\frac{{渭}_0}{16{蟺}^{2}c{r}^2}\stackrel{..}{p^2}\left(4蟺\right){鈭�}_0^{蟺}\left({\sin }^3胃d胃\right) \\ P=\frac{{渭}_{0}p{\displaystyle \stackrel{..}{p^2}}}{6蟺c} \end{gathered}$$ 鈥︹�� (3)

Write the expression for the electric dipole moment in the case of sinusoidal time dependence.

$p=p_0\cos 蝇t$

Take the double differentiation of the above equation.

$$\begin{gathered} \stackrel{}{\stackrel{.}{p}=-}蝇p_0\sin 蝇t \\ \stackrel{}{\stackrel{..}{p}=-{蝇}^2{p}_0}\cos 蝇t \end{gathered}$$

Squaring on both sides,

$\stackrel{}{\stackrel{..}{p^2}=-{蝇}^4{p^2}_0}{\cos }^2蝇t$

Take the time average of .

Substitute the value of $\stackrel{.{.}^2}{p}$in equation (3).

$$\begin{gathered} P=\frac{{渭}_0\left({\displaystyle \frac{1}{2}}{蝇}^4{p}_0^2\right)}{6蟺c} \\ P=\frac{{渭}_0{p}_0^2{蝇}^4}{12蟺c} \end{gathered}$$

Hence, the result is consistent with equation 11.22.

Therefore, the power radiated by an ideal electric dipole at the origin is, $P=\frac{{渭}_0{p}_0^2{蝇}^4}{12蟺c}$0and the obtained result is consistent with equation 11.22.