電動力學每日一題 2021/10/12

(a) To make the EM field trapped inside a perfectly electric conducting cavity, the tangential component of the E-field must vanish on the internal surface:
$$\begin{gathered} E_0{J}_0(w_{0}R/c)\sin (w_{0}t)=0 \\ {J}_0(w_{0}R/c)=0 \end{gathered}$$

This means the radius of cylinder must be chosen such that $w_{0}R/c$ is a zero of Bessel function $J_0(?)$.

(b) Maxwell 1st equation delievers the boundary condition of E-field
$$\begin{gathered} {\text{D}}_{\perp }({\text{r}}_2,t)?{\text{D}}_{\perp }({\text{r}}_1,t)={\sigma }_S \\ ?{?}_0{E}_0{J}_0(w_0{r}_{||}/c)\sin (w_{0}t)={\sigma }_S(r_{||},?,\pm L/2,t) \end{gathered}$$

(c) Maxwell 2nd equation delievers the boundary condition of H-field
$$\begin{gathered} {\text{H}}_{||}({\text{r}}_2,t)?{\text{H}}_{||}({\text{r}}_1)={\text{J}}_{free}({\text{r}}_0,t)\times \hat{n} \\ {\text{J}}_S(R,?,z,t)=?(E_0/Z_0)J_1(w_{0}R/c)\cos (w_{0}t)\hat{z} \\ {\text{J}}_S(r_{||},?,\pm L/2,t)=\pm (E_0/Z_0)J_1(w_0{r}_{||}/c)\cos (w_{0}t)\hat{z} \end{gathered}$$

(d) The charge-currrency continuinity equation is
$$??{\text{J}}_S(r_{||},?,\pm L/2,t)+\frac{?}{?t}{\sigma }_S(r_{||},?,\pm L/2,t)=0$$

Evaluate
$$\begin{array}{rl}??{\text{J}}_S& =\pm (E_0/Z_0)\frac{?}{r_{||}?r_{||}}{r}_{||}{J}_1(w_0{r}_{||}/c)\cos (w_{0}t)\\ & =\pm (E_0/Z_0)\frac{J_1(w_0{r}_{||}/c)+(w_0{r}_{||}/c)J_1^{\prime }(w_0{r}_{||}/c)}{r_{||}}\cos (w_{0}t)\\ & =\pm (E_0/Z_0)\frac{(w_0{r}_{||}/c)J_0(w_0{r}_{||}/c)}{r_{||}}\cos (w_{0}t)\\ & =\pm {?}_0{E}_0{w}_0{J}_0(w_0{r}_{||}/c)\cos (w_{0}t)\end{array}$$

$$\begin{array}{rl}\frac{?}{?t}{\sigma }_S& =\frac{?}{?t}?{?}_0{E}_0{J}_0(w_0{r}_{||}/c)\sin (w_{0}t)\\ & =?{?}_0{E}_0{J}_0(w_0{r}_{||}/c)\cos (w_{0}t)\end{array}$$

Above,
$$??{\text{J}}_S+\frac{?}{?t}{\sigma }_S=0$$

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