## 数学代写|数值分析代写numerical analysis代考|MA1020

2022年12月23日

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## 数学代写|数值分析代写numerical analysis代考|Formulation of the Problem in the Particle Frame

In this section let us consider the geometry shown in Figure 1: a perfectly conducting angular slot $\mathbb{S}=\left{r=a,|\varphi| \leq \varphi_a, z\right}$ at distance $a$ from the axis and covering an angular sector of $2 \varphi_a$. A travelling charge $q$ moves parallel to the slots’s axis, placed at $\left(r_q, \varphi_q\right)$, at constant speed $v=\beta c, c$ being the speed of light in free space.

The problem is formulated in term of integral equations and its solution is reduced to the resolution of a linear system.

The electromagnetic interaction between the particle and the structure can be easily formulated and solved in the particle frame, being an electrostatic model adequate for such a problem. Once the electromagnetic quantities are computed, their values in the slot frame can be obtained by means of Lorentz transforms.
The electrostatic potential produced by the charge is
$$V_q^{\prime}=\frac{q}{4 \pi \varepsilon_0 \sqrt{r^{\prime 2}+r_q^2-2 r^{\prime} r_q \cos \left(\varphi^{\prime}-\varphi_q\right)+z^{\prime 2}}},$$
while the potential produced by the induced charge density $\sigma^{\prime}\left(\varphi^{\prime}, z^{\prime}\right)$ on the slot can be expressed as
$$V^{\prime}=\frac{a}{4 \pi \varepsilon_0} \int_{\mathbb{S}} \frac{\sigma^{\prime}\left(\varphi_0, z_0\right) d \varphi_0 d z_0}{\sqrt{r^{\prime 2}+a^2-2 r^{\prime} a \cos \left(\varphi^{\prime}-\varphi_0\right)+\left(z^{\prime}-z_0\right)^2}} .$$
Being a perfectly conducting slot, the boundary condition to be verified is that the tangential components of the electric field vanishes on the slot. This corresponds to impose that
$$V^{\prime}\left(r^{\prime}=a, \varphi^{\prime}, z\right)+V_q^{\prime}\left(r^{\prime}=a, \varphi^{\prime}, z\right)=0$$
for every $\left(\varphi^{\prime}, z^{\prime}\right) \in \mathbb{S}$.
Considering Equations (2) and (3), the boundary condition leads to
$$\int_{\mathrm{S}} \frac{\sigma^{\prime}\left(\varphi_0, z_0\right) d \varphi_0 d x_0}{\sqrt{2 a^2-2 a^2 \cos \left(\varphi^{\prime}-\varphi_0\right)+\left(z^{\prime}-z_0\right)^2}}-\frac{q / a}{\sqrt{a^2+r_q^2-2 a r_q \cos \left(\varphi^{\prime}-\varphi_q\right)+z^{\prime 2}}}$$

## 数学代写|数值分析代写numerical analysis代考|Electromagnetic Fields in the Slot Frame

In order to complete the problem formulation, it is proper to express the electromagnetic fields in the slot frame, too. This can be realized by applying the Lorentz transforms to the fields computed in the previous section in particle frame.

Let us consider at first the $z$ component of the electric field. The contribution provided by the traveling charge in the frequency domain is well known and is
$$E_{z, q}=\frac{j q \kappa \zeta_0}{2 \pi \beta \gamma} e^{-j z k / \beta} K_0\left(\kappa \sqrt{r^2+r_q^2-2 r r_q \cos (\varphi)}\right),$$
where $\gamma=1 / \sqrt{1-\beta^2}$ is the Lorentz factor, $\kappa=k /(\beta \gamma)$, and $\zeta_0=\sqrt{\mu_0 / \varepsilon_0}$ is the characteristic impedance of free space.

The contribution produced by the induced current density on the slot can be obtained with some manipulations as function of the representation coefficients $\sigma_n$.
In the particle frame, starting from Equation (3) it is possible to obtain
$$e_z^{\prime}\left(r^{\prime}, \varphi^{\prime}, z^{\prime}\right)=\frac{a}{4 \pi \varepsilon_0} \int_{\mathbb{S}} \frac{\sigma^{\prime}\left(\varphi_0, z_0\right)\left(z^{\prime}-z_0\right) d \varphi_0 d z_0}{\left[r^{\prime 2}+a^2-2 r^{\prime} a \cos \left(\varphi^{\prime}-\varphi_0\right)+\left(z^{\prime}-z_0\right)^2\right]^{3 / 2}} .$$
Lorentz transforms are now applied to obtain the electric field in the slot frame. In this specific case they are
$$e_z^{\prime}=e_z, \sigma^{\prime}=\sigma \gamma, r^{\prime}=r, \varphi^{\prime}=\varphi, z^{\prime}=\gamma(z-v t) .$$
Applying these transforms to Equation (19), it is found that
$$e_z(r, \varphi, z, t)=\frac{a \gamma}{4 \pi \varepsilon_0} \int_{\mathbb{S}} \frac{\sigma\left(\varphi_0, z_0\right)\left(\gamma(z-v t)-z_0\right) d \varphi_0 d z_0}{\left[r^2+a^2-2 r a \cos \left(\varphi-\varphi_0\right)+\left(\gamma(z-v t)-z_0\right)^2\right]^{3 / 2}} .$$
By means of Equation (8) and applying a spatial Fourier transform according to Equation (9), it is found that
$$e_z(r, \varphi, z, t)=\frac{j a \gamma}{2 \pi \varepsilon_0} \int_{-\varphi_a}^{+\varphi_a+\infty} \int_{-\infty} \tilde{\sigma}\left(\varphi_0, w\right) w e^{j w \gamma v t} K_0\left(w \sqrt{r^2+a^2-2 r a \cos \left(\varphi-\varphi_0\right)}\right) e^{-j w \gamma z} d \varphi_0 d w$$

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Formulation of the Problem in the Particle Frame

$$V_q^{\prime}=\frac{q}{4 \pi \varepsilon_0 \sqrt{r^{\prime 2}+r_q^2-2 r^{\prime} r_q \cos \left(\varphi^{\prime}-\varphi_q\right)+z^{\prime 2}}},$$

$$V^{\prime}=\frac{a}{4 \pi \varepsilon_0} \int_{\mathbb{S}} \frac{\sigma^{\prime}\left(\varphi_0, z_0\right) d \varphi_0 d z_0}{\sqrt{r^{\prime 2}+a^2-2 r^{\prime} a \cos \left(\varphi^{\prime}-\varphi_0\right)+\left(z^{\prime}-z_0\right)^2}} .$$

$$V^{\prime}\left(r^{\prime}=a, \varphi^{\prime}, z\right)+V_q^{\prime}\left(r^{\prime}=a, \varphi^{\prime}, z\right)=0$$

$$\int_{\mathrm{S}} \frac{\sigma^{\prime}\left(\varphi_0, z_0\right) d \varphi_0 d x_0}{\sqrt{2 a^2-2 a^2 \cos \left(\varphi^{\prime}-\varphi_0\right)+\left(z^{\prime}-z_0\right)^2}}-\frac{}{\sqrt{a^2+r_q^2-2 a}}$$

## 数学代写|数值分析代写numerical analysis代考|Electromagnetic Fields in the Slot Frame

$$E_{z, q}=\frac{j q \kappa \zeta_0}{2 \pi \beta \gamma} e^{-j z k / \beta} K_0\left(\kappa \sqrt{r^2+r_q^2-2 r r_q \cos (\varphi)}\right)$$

$$e_z^{\prime}\left(r^{\prime}, \varphi^{\prime}, z^{\prime}\right)=\frac{a}{4 \pi \varepsilon_0} \int_{\mathbb{S}} \frac{\sigma^{\prime}\left(\varphi_0, z_0\right)\left(z^{\prime}-z_0\right) d \varphi_0 d z_0}{\left[r^{\prime 2}+a^2-2 r^{\prime} a \cos \left(\varphi^{\prime}-\varphi_0\right)+\left(z^{\prime}\right.\right.}$$

$$e_z^{\prime}=e_z, \sigma^{\prime}=\sigma \gamma, r^{\prime}=r, \varphi^{\prime}=\varphi, z^{\prime}=\gamma(z-v t) .$$

$$e_z(r, \varphi, z, t)=\frac{a \gamma}{4 \pi \varepsilon_0} \int_{\mathbb{S}} \frac{\sigma\left(\varphi_0, z_0\right)\left(\gamma(z-v t)-z_0\right) d \varphi}{\left[r^2+a^2-2 r a \cos \left(\varphi-\varphi_0\right)+(\gamma(z-\right.}$$

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## MATLAB代写

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