91影视

Write the expression for the power loss in an oscillatory magnetic dipole.

$\mathit{P}\mathbf{=}\mathit{I}{\left(t\right)}^{\mathbf{2}}\mathit{R}$ 鈥︹�� (1)

Here, I is the current, and R is the resistance.

Write the expression for the total radiated power.

$<P>\mathbf{=}\frac{{\mathbf{渭}}_{\mathbf{0}}{\mathbf{m}}_{\mathbf{0}}^{\mathbf{2}}{\mathbf{蝇}}^{\mathbf{4}}}{\mathbf{12}\mathbf{蟺}{\mathbf{c}}^{\mathbf{3}}}$ 鈥︹�� (2)

Here,${渭}_0$ is the magnetic constant,$m_0$ is the magnetic dipole moment,$蝇$ is the angular frequency, and c is the speed of light.

Write the expression for the current through a wire loop.

$I\left(t\right)=I_0\cos 蝇t$

Substitute $I\left(t\right)=I_0\cos 蝇t$in equation (1).

$$\begin{gathered} P={\left(I_0\cos 蝇t\right)}^{2}R \\ P=I_0^{2}R{\cos }^2蝇t \end{gathered}$$

The average value of${\cos }^2蝇t$is $\frac{1}{2}$.

Hence, the equation for P becomes,

$P=\frac{1}{2}{I}_0^{2}R$ 鈥︹�� (3)

Write the model for an oscillating magnetic dipole.

$$\begin{gathered} m\left(t\right)=蟺b^{2}I\left(t\right)\hat{z} \\ m\left(t\right)=m_0\cos 蝇t\hat{z} \end{gathered}$$

Here, $m_0=蟺b^2{I}_0$

Substitute $m_0=蟺b^2{I}_0$in equation (2).

鈥︹�� (4)

Equate equations (3) and (4).

Write the relation between angular frequency and wavelength.

$蝇=\frac{2蟺c}{位}$

Substitute $蝇=\frac{2蟺c}{位}$in equation (5).

$$\begin{gathered} R=\frac{{渭}_0{蟺}^2{b}^4{\left({\displaystyle \frac{2蟺c}{位}}\right)}^4}{6蟺c^3} \\ R=\frac{{渭}_0{蟺}^2{b}^4}{3蟺c^3}{\left(\frac{2蟺c}{位}\right)}^4 \\ R=\frac{8}{3}{蟺}^5{渭}_{0}c{\left(\frac{b}{位}\right)}^4........\left(6\right) \end{gathered}$$

Substitute${渭}_0=4蟺脳{10}^{-7}\mathrm{H}/\mathrm{m}$ and $c=3脳{10}^8\mathrm{m}/\mathrm{s}$in equation (6).

$$\begin{gathered} R=\frac{8}{3}{蟺}^5\left(4蟺脳{10}^{-7}\mathrm{H}/\mathrm{m}\right)\left(3脳{10}^8\mathrm{m}/\mathrm{s}\right){\left(\frac{b}{位}\right)}^4 \\ R=3.076脳{10}^5{\left(\frac{b}{位}\right)}^4惟鈮�3脳{10}^5\left(\frac{b}{位}\right)惟 \\ R=3脳{10}^5{\left(\frac{b}{位}\right)}^4惟 \end{gathered}$$

Therefore, the radiation resistance for the oscillating magnetic dipole is

$3脳{10}^5{\left(\frac{b}{位}\right)}^4惟$.