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物理代写|电动力学作业代写electrodynamics代考|Radiation from Localized Charge and Current Distributions

如果你也在 怎样代写电动力学electrodynamics这个学科遇到相关的难题,请随时右上角联系我们的24/7代写客服。电动力学electrodynamics是物理学的一个分支,涉及到对电磁力的研究,这是一种发生在带电粒子之间的物理作用。电磁力是由电场和磁场组成的电磁场所承载的,它是诸如光这样的电磁辐射的原因。它与强相互作用、弱相互作用和引力一起,是自然界的四种基本相互作用(通常称为力)之一。

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物理代写|电动力学作业代写electrodynamics代考|Radiation from Localized Charge and Current Distributions

物理代写|电动力学作业代electrodynamics代考|Green Function for Inhomogeneous Maxwell’s Equation in Lorenz Gauge

If we look at the inhomogeneous Maxwell equations in the Lorentz gauge, as discussed in the first chapter, we see that their general structure is like
$$
\nabla^{2} \Psi-\frac{1}{c^{2}} \frac{\partial^{2} \Psi}{\partial t^{2}}=-4 \pi f(\mathbf{x}, t)
$$

Here, $\Psi$ stands for the scalar electrodynamic potential in Lorenz gauge. The source function $f(\mathbf{x}, t)$ may be related to charge density or components of the total current. To proceed we Fourier expand $\Psi(\mathbf{x}, t)$ and $f(\mathbf{x}, t)$ as
$$
\begin{aligned}
&\Psi(\mathbf{x}, t)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \Psi(\mathbf{x}, \omega) e^{-i \omega t} d \omega \
&f(\mathbf{x}, t)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(\mathbf{x}, \omega) e^{-i \omega t} d \omega
\end{aligned}
$$
with the inverse Fourier transforms as
$$
\begin{aligned}
\Psi(\mathbf{x}, \omega) &=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \Psi(\mathbf{x}, t) e^{i \omega t} d \omega \
f(\mathbf{x}, t) &=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(\mathbf{x}, \omega) e^{i \omega t} d \omega
\end{aligned}
$$
From these equations, we see that
$$
\frac{\partial^{2} \Psi}{\partial t^{2}}=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \Psi(\mathbf{x}, \omega) \frac{d^{2}\left(e^{-i \omega t}\right)}{d t^{2}} d \omega=-\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \Psi(\mathbf{x}, \omega) \omega^{2} e^{-i \omega t} d \omega
$$
As a result, the equation for $\Psi$ becomes
$$
\begin{aligned}
\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\left(\nabla^{2} \Psi(\mathbf{x}, \omega)\right) e^{-i \omega t} d \omega &+\frac{1}{\sqrt{2 \pi} c^{2}} \int_{-\infty}^{\infty} \Psi(\mathbf{x}, \omega) \omega^{2} e^{-i \omega t} d \omega \
&=-4 \pi\left[\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(\mathbf{x}, \omega) e^{-i \omega t} d \omega\right]
\end{aligned}
$$
which can also be written as
$$
\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\left[\nabla^{2} \Psi(\mathbf{x}, \omega)+k^{2} \Psi(\mathbf{x}, \omega)+4 \pi f(\mathbf{x}, \omega)\right] e^{-i \omega t} d \omega=0,
$$
where
$$
k=\frac{\omega}{c} .
$$
If the integral vanish, then generally
$$
\left(\nabla^{2}+k^{2}\right) \Psi(\mathbf{x}, \omega)=-4 \pi f(\mathbf{x}, \omega),
$$

物理代写|电动力学作业代electrodynamics代考|Radiation from Localized Charges

Now we know how to handle time-dependent situations in electrodynamics giving rise to radiation. We will apply our knowledge in an important problem. First of all, we recall that the Maxwell equations in the Lorenz gauge were
$$
\nabla^{2} \Phi-\frac{1}{c^{2}} \frac{\partial^{2} \Phi}{\partial t^{2}}=-4 \pi \rho, \quad \nabla^{2} \mathbf{A}-\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{A}}{\partial t^{2}}=-\frac{4 \pi}{c} \mathbf{J}
$$
For time-varying charges and currents, we can use their Fourier expansions as
$$
\mathbf{J}(\mathbf{x}, t)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \mathbf{J}(\mathbf{x}, \omega) e^{-i \omega t} d \omega, \quad \rho(\mathbf{x}, t)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \rho(\mathbf{x}, \omega) e^{-i \omega t} d \omega
$$
Similar Fourier expansion can also be done for $\Phi$ and $\mathbf{A}$. As the differential equations are linear in nature, we can actually work with one Fourier component alone. Henceforth, we work with the $\omega>0$ component and write the source functions as
$$
\rho(\mathbf{x}, t)=\rho(\mathbf{x}) e^{-i \omega t}, \quad \mathbf{J}(\mathbf{x}, t)=\mathbf{J}(\mathbf{x}) e^{-i \omega t}
$$

In our present case, all the relevant time varying fields can be written as
$$
\mathbf{A}(\mathbf{x}, t)=\mathbf{A}(\mathbf{x}) e^{-i \omega t}, \quad \mathbf{E}(\mathbf{x}, t)=\mathbf{E}(\mathbf{x}) e^{-i \omega t} \quad \mathbf{B}(\mathbf{x}, t)=\mathbf{B}(\mathbf{x}) e^{-i \omega t}
$$
This amounts to saying that the whole system is oscillating with time at the same frequency and the only thing to know is the spatial dependence of the fields. In general, all the fields and source functions which are independent of time can be complex. Time dependence introduces another complex phase. To connect with reality, we must remember that we have to take the real part of the complex solutions as physically relevant.

物理代写|电动力学作业代ELECTRODYNAMICS代考|The Vector Potential in the Electric Dipole Approximation

We have seen that in the region where $r \gg d$ the vector potential is given by
$$
\mathbf{A}(\mathbf{x}, t)=\frac{e^{i(k r-\omega t)}}{r} \mathbf{f}(\theta, \phi)
$$
where
$$
\mathbf{f}(\theta, \phi)=\frac{1}{c} \int_{-\infty}^{\infty} \mathbf{J}\left(\mathbf{x}^{\prime}\right) e^{-i k \hat{\mathbf{n}} \cdot \mathbf{x}^{\prime}} d^{3} x^{\prime}
$$
As the source dimension is much small compared to the wavelength of radiation we can actually expand $e^{-i k \hat{\mathbf{n}} \cdot \mathbf{x}^{\prime}}$ in a power series and write
$$
\mathbf{f}(\theta, \phi)=\frac{1}{c} \int_{-\infty}^{\infty} \mathbf{J}\left(\mathbf{x}^{\prime}\right)\left[1-i k\left(\mathbf{n} \cdot \mathbf{x}^{\prime}\right)-\frac{1}{2} k^{2}\left(\mathbf{n} \cdot \mathbf{x}^{\prime}\right)^{2}+\cdots\right] d^{3} x^{\prime} .
$$
The successive terms in the bracket inside the integral contribute decreasingly. If we take the first term of the above series, we get
$$
\mathbf{A}(\mathbf{x}, t)=\frac{e^{i(k r-\omega t)}}{c r} \int_{-\infty}^{\infty} \mathbf{J}\left(\mathbf{x}^{\prime}\right) d^{3} x^{\prime}
$$
Working with the first term of the series gives rise to the electric dipole approximation. Why we call it the dipole approximation will become clear later. This approximation plays an important part in physics related to radiation. The reader must note that one does not get other relevant radiation patterns, as the magnetic dipole radiation or electric quadrupole radiation, by simply taking higher order terms inside the bracket on the right-hand side of the above integral. To obtain the other radiation patterns, one has to be a bit more careful. We will show how one obtains the other radiation patterns later in this chapter.

物理代写|电动力学作业代写electrodynamics代考|Radiation from Localized Charge and Current Distributions

电动力学代写

物理代写|电动力学作业代ELECTRODYNAMICS代考|GREEN FUNCTION FOR INHOMOGENEOUS MAXWELL’S EQUATION IN LORENZ GAUGE

如果我们看一下洛伦兹规范中的非齐次麦克斯韦方程,如第一章所讨论的,我们会看到它们的一般结构是
∇2Ψ−1C2∂2Ψ∂吨2=−4圆周率F(X,吨)

这里,Ψ代表洛伦兹规范中的标量电动势。源函数F(X,吨)可能与电荷密度或总电流的分量有关。为了继续我们的傅立叶展开Ψ(X,吨)和F(X,吨)作为
Ψ(X,吨)=12圆周率∫−∞∞Ψ(X,ω)和−一世ω吨dω F(X,吨)=12圆周率∫−∞∞F(X,ω)和−一世ω吨dω
傅里叶逆变换为
Ψ(X,ω)=12圆周率∫−∞∞Ψ(X,吨)和一世ω吨dω F(X,吨)=12圆周率∫−∞∞F(X,ω)和一世ω吨dω
从这些方程中,我们看到
∂2Ψ∂吨2=12圆周率∫−∞∞Ψ(X,ω)d2(和−一世ω吨)d吨2dω=−12圆周率∫−∞∞Ψ(X,ω)ω2和−一世ω吨dω
结果,方程为Ψ变成
12圆周率∫−∞∞(∇2Ψ(X,ω))和−一世ω吨dω+12圆周率C2∫−∞∞Ψ(X,ω)ω2和−一世ω吨dω =−4圆周率[12圆周率∫−∞∞F(X,ω)和−一世ω吨dω]
也可以写成
12圆周率∫−∞∞[∇2Ψ(X,ω)+到2Ψ(X,ω)+4圆周率F(X,ω)]和−一世ω吨dω=0,
在哪里
到=ωC.
如果积分消失,那么通常
(∇2+到2)Ψ(X,ω)=−4圆周率F(X,ω),

物理代写|电动力学作业代ELECTRODYNAMICS代考|RADIATION FROM LOCALIZED CHARGES

现在我们知道如何处理引起辐射的电动力学中与时间相关的情况。我们将把我们的知识应用到一个重要的问题上。首先,我们记得洛伦兹规范中的麦克斯韦方程是
∇2披−1C2∂2披∂吨2=−4圆周率ρ,∇2一种−1C2∂2一种∂吨2=−4圆周率CĴ
对于随时间变化的电荷和电流,我们可以使用它们的傅里叶展开式:
Ĵ(X,吨)=12圆周率∫−∞∞Ĵ(X,ω)和−一世ω吨dω,ρ(X,吨)=12圆周率∫−∞∞ρ(X,ω)和−一世ω吨dω
也可以进行类似的傅里叶展开披和一种. 由于微分方程本质上是线性的,我们实际上可以单独使用一个傅立叶分量。今后,我们与ω>0组件并将源函数编写为
ρ(X,吨)=ρ(X)和−一世ω吨,Ĵ(X,吨)=Ĵ(X)和−一世ω吨

在我们目前的情况下,所有相关的时变场都可以写成
一种(X,吨)=一种(X)和−一世ω吨,和(X,吨)=和(X)和−一世ω吨乙(X,吨)=乙(X)和−一世ω吨
这等于说整个系统以相同的频率随时间振荡,唯一要知道的是场的空间依赖性。一般来说,所有与时间无关的场和源函数都可以是复杂的。时间依赖性引入了另一个复杂的阶段。为了与现实联系起来,我们必须记住,我们必须将复杂解决方案的真实部分与物理相关。

物理代写|电动力学作业代ELECTRODYNAMICS代考|THE VECTOR POTENTIAL IN THE ELECTRIC DIPOLE APPROXIMATION

我们已经看到,在该地区r≫d矢量势由下式给出
一种(X,吨)=和一世(到r−ω吨)rF(θ,φ)
在哪里
F(θ,φ)=1C∫−∞∞Ĵ(X′)和−一世到n^⋅X′d3X′
由于源尺寸与辐射波长相比要小得多,我们实际上可以扩展和−一世到n^⋅X′在幂级数中并写
F(θ,φ)=1C∫−∞∞Ĵ(X′)[1−一世到(n⋅X′)−12到2(n⋅X′)2+⋯]d3X′.
积分内括号中的连续项贡献递减。如果我们取上述系列的第一项,我们得到
一种(X,吨)=和一世(到r−ω吨)Cr∫−∞∞Ĵ(X′)d3X′
使用该系列的第一项会产生电偶极子近似。为什么我们称其为偶极子近似,稍后将变得清楚。这种近似在与辐射有关的物理学中起着重要的作用。读者必须注意,通过简单地在上述积分右侧的括号内取高阶项,不会得到其他相关的辐射模式,如磁偶极子辐射或电四极子辐射。要获得其他辐射模式,必须更加小心。我们将在本章后面展示如何获得其他辐射图。

物理代写|电动力学作业代写electrodynamics代考

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