Statistics-lab™可以为您提供stanford.edu PHYSICS220 electrodynamics电动力学的代写代考和辅导服务！

PHYSICS220 electrodynamics课程简介

This course provides a deeper look into the theory of electricity and magnetism.

We shall study:

– magnetostatics

– magnetic fields in matter

– Maxwell’s equations

– electromagnetic waves

PREREQUISITES 

Upon successful completion of this course you will be able to:

• Understand the implications of Ampère’s and Biot-Savart’s laws
• Understand the nature of the vector potential
• Understand the behaviour of magnetic fields in medium
• Be perfectly familiar with Maxwell’s equations
• Know the nature of momentum and angular moment in Electrodynamics
• Understand the details of propagation of electromagnetic waves

PHYSICS220 electrodynamics HELP（EXAM HELP， ONLINE TUTOR）

The Lienard-Wiechert potentials for a charge moving in hyperbolic motion can be obtained using the following equations:
$$
\begin{gathered}
\mathbf{r}=\mathbf{r}^{\prime}-\mathbf{v} t+\frac{1}{c^2} \mathbf{a} t^2 \
\phi(\mathbf{r}, t)=\frac{1}{4 \pi \epsilon_0} \frac{q}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} \
\mathbf{A}(\mathbf{r}, t)=\frac{\mu_0}{4 \pi} \frac{q \mathbf{v}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}
\end{gathered}
$$
where $\$ \backslash m a t h b f{r}$ ‘ $\$$ is the position of the charge at some earlier time, $\$ \backslash m a t h b f{v} \$$ and $\$ \backslash m a t h b f{a} \$$ are the velocity and acceleration of the charge at the retarded time $\$ \mathrm{t} r=\mathrm{t}$ are the scalar and vector potentials, respectively.
For the given hyperbolic motion $\$ w(t)=\mid s q r t\left{b^{\wedge} 2+\right.$ $\left.(c t)^{\wedge} 2\right}, \backslash \operatorname{bar}{x} \$$, we can choose the reference position $\$ \backslash$ mathbf ${r} ‘ \$$ to be the point where $\$ w\left(t^{\prime}\right)=0 \$$ for some earlier time $\$ t^{\prime} \$$. Since $\$ w(t) \$$ is symmetric about the $\$ y \$$ and $\$ z \$$ axes, we can assume that $\$ \backslash$ mathbf ${r}$ ‘ lies on the $\$ x \$$ axis. Let $\$ r^{\prime}=\mid \backslash$ mathbf $\left.{r}\right}^{\prime} \mid \$$ be the distance of this point from the origin. Then we have
$$
r^{\prime}=\sqrt{b^2+\left(c t^{\prime}\right)^2}
$$
and$$
\mathbf{r}^{\prime}=r^{\prime} \bar{x}
$$
The velocity and acceleration of the charge at time $\$ t_{-} r=t$ | $\backslash$ mathbf ${r}-\backslash$ mathbf{r}’ $\mid / c \$$ can be obtained as follows:
$$
\begin{gathered}
\mathbf{v}=\frac{d \mathbf{w}}{d t}=\frac{c^2 t}{\sqrt{b^2+(c t)^2}} \bar{x} \
\mathbf{a}=\frac{d \mathbf{v}}{d t}=\frac{c^2}{\sqrt{b^2+(c t)^2}}\left(\frac{b^2}{b^2+(c t)^2}-\frac{c t}{\sqrt{b^2+(c t)^2}}\right) \bar{x} .
\end{gathered}
$$
Using these expressions, we can write the Lienard-Wiechert potentials as
$$
\begin{aligned}
& \phi(\mathbf{r}, t)=\frac{1}{4 \pi \epsilon_0} \frac{q}{\sqrt{(r-c t)^2+r^{\prime 2}}} \
& \mathbf{A}(\mathbf{r}, t)=\frac{\mu_0}{4 \pi} \frac{q \mathbf{v}}{\sqrt{(r-c t)^2+r^{\prime 2}}},
\end{aligned}
$$
where $\$ r=\mid \backslash$ mathbf ${r} \mid \$$ is the distance from the charge to the observation point $\$ \backslash$ mathbf{r}$\$$ on the $\$ x \$$ axis. Note that $\$ r$-ct\$ is the distance between the charge and the point on the $\$ \times \$$ axis that is reached by the charge at the retarded time $\$$ t_r\$. Substituting the expressions for $\$ \backslash$ mathbf ${\mathrm{v}$

(a) To find the scalar and vector potentials, we need to express the two dipoles as current densities $\$ \backslash$ mathbf ${J}_{-} 1 \$$ and $\$ \backslash$ mathbf ${J}_{-} 2 \$$, and then integrate the expressions for the scalar and vector potentials over these current densities. Since we have two oppositely oriented electric dipoles, we can model each dipole as a pair of equal and opposite point charges of magnitude $\$ q \$$ separated by a distance $\$ 1 \$$, with $\$ \mathrm{I}<\mathrm{d} \$$. The charge separation is in the $\$ \times \$$ direction, and the dipoles oscillate in the $\$ z \$$ direction with a frequency \$\omega $\$$. Let $\$ \backslash$ mathbf ${r} _1 \$$ and $\$ \backslash$ mathbf ${r} 2 \$$ be the positions of the two charges in each dipole, as shown in the figure. We can then express the current densities as
$$
\begin{gathered}
\mathbf{J}_1=q \omega\left(\delta\left(\mathbf{r}-\mathbf{r}_1\right)-\delta\left(\mathbf{r}-\mathbf{r}_2\right)\right) \hat{z} \
\mathbf{J}_2=q \omega\left(\delta\left(\mathbf{r}-\mathbf{r}_1\right)-\delta\left(\mathbf{r}-\mathbf{r}_2\right)\right)-\hat{z}
\end{gathered}
$$
where $\$ \backslash$ delta(\mathbf ${r}) \$$ is the Dirac delta function, and we have assumed that the charges have equal and opposite charges of magnitude $\$ q \$$.

Using these current densities, we can then write the scalar and vector potentials as
$$
\phi(\mathbf{r}, t)=\frac{1}{4 \pi \epsilon_0} \int \frac{\rho\left(\mathbf{r}^{\prime}, t_r\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} d^3 \mathbf{r}^{\prime}
$$$$
\mathbf{A}(\mathbf{r}, t)=\frac{\mu_0}{4 \pi} \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}, t_r\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} d^3 \mathbf{r}^{\prime},
$$
where $\$ \backslash r h o(\backslash$ mathbf ${r}, t) \$$ is the charge density corresponding to the current density $\$ \backslash$ mathbf ${J}$ $(\backslash$ mathbf ${r}, t)=\backslash f r a c{$ partial $} \backslash \backslash$ partial $t} \backslash$ mathbf ${P}$ $(\backslash$ mathbf ${r}, t) \$$, with $\$ \backslash$ mathbf ${P}(\backslash$ mathbf ${r}, t) \$$ being the polarization density. The retarded time $\$$ \$_ $_{-} r$ is given by $\$ t_{-} r=t-\mid \backslash$ mathbf ${r}-\backslash$ mathbf ${r}^{\prime} \mid / c \$$.
Assuming that the charges in each dipole oscillate harmonically with a frequency \$\omega\$, we can write the polarization densities as
$$
\begin{gathered}
\mathbf{P}_1(\mathbf{r}, t)=q \omega l \cos (\omega t)\left(\delta\left(\mathbf{r}-\mathbf{r}_1\right)-\delta\left(\mathbf{r}-\mathbf{r}_2\right)\right) \hat{z} \
\mathbf{P}_2(\mathbf{r}, t)=q \omega l \cos (\omega t)\left(\delta\left(\mathbf{r}-\mathbf{r}_1\right)-\delta\left(\mathbf{r}-\mathbf{r}_2\right)\right)-\hat{z}
\end{gathered}
$$
Using these polarization densities and integrating over

Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

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