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## 物理代写|电磁学代写electromagnetism代考|Magnetic Induction and Time-Varying Fields

Topics. Magnetic induction. Faraday’s law. Electromotive force. The slowly varying current approximation. Mutual inductance and self-inductance. Energy stored in an inductor. Magnetically coupled circuits. Magnetic energy. Displacement current and the complete Maxwell’s equations.

Basic equations In the presence of a time-varying magnetic field, Equation (1.5) is modified into the exact equation
$$\boldsymbol{\nabla} \times \mathbf{E}=-b_{\mathrm{m}} \partial_{t} \mathbf{B}$$
so that the line integral of $\boldsymbol{\nabla} \times \mathbf{E}$ around a closed path $C$ is
$$\oint_{C} \mathbf{E} \cdot \mathrm{d} \ell=-b_{\mathrm{m}} \int_{S} \partial_{t} \mathbf{B} \cdot \mathrm{d} \mathbf{S}$$
Thus, for a fixed path, the line integral of $\mathbf{E}$ equals the time derivative of the flux of the time-varying field $\mathbf{B}$ through a surface delimited by the contour $C$.

The electromotive force (emf) $\mathcal{E}$ in a real circuit having moving parts is the work done by the Lorentz force on a unit charge over the circuit path,
$$\mathcal{E}=\oint_{\text {circ. }}\left(\mathbf{E}+b_{\mathrm{m}} \mathbf{V} \times \mathbf{B}\right) \cdot \mathrm{d} \boldsymbol{\ell} \equiv-b_{\mathrm{m}} \frac{\mathrm{d}}{\mathrm{d} t} \Phi_{\text {circ }}(\mathbf{B})$$
where $\mathbf{V}$ is the velocity of the circuit element; now in $(6.3)$ the flux $\Phi_{\text {circ }}(\mathbf{B})$ of $\mathbf{B}$ through the circuit may vary because of both the temporal variation of $\mathbf{B}$ and of the circuit geometry. Equation (6.3) is the general Faraday’s law of induction.

For a system of two electric circuits, the magnetic flux through each circuit can be written as a function of the currents flowing in each circuit,
$$\Phi_{2}=L_{1} I_{1}+M_{21} I_{2}, \quad \Phi_{1}=L_{2} I_{2}+M_{12} I_{1}$$

where the terms containing the (self-)inductance coefficients $L_{i}$ are the contribution to flux generated by the circuit itself, and the terms containing the mutual inductance coefficients $M_{21}=M_{12}$ give the flux generated by one circuit over the other.

Finally, for time-varying fields the complete Maxwell’s equation replacing (5.3) is
$$\boldsymbol{\nabla} \times \mathbf{B}=4 \pi k_{\mathrm{m}} \mathbf{J}+\frac{k_{\mathrm{m}}}{k_{\mathrm{e}}} \partial_{t} \mathbf{E}=\left{\begin{array}{l} \frac{4 \pi}{c} \mathbf{J}+\frac{1}{c} \partial_{t} \mathbf{E} \quad(\text { Gaussian }) \ \mu_{0} \mathbf{J}+\mu_{0} \varepsilon_{0} \partial_{t} \mathbf{E}(\text { SI. }) \end{array}\right.$$

## 物理代写|电磁学代写electromagnetism代考|A Circuit with “Free-Falling” Parts

In the presence of the Earth’s gravitational field $\mathbf{g}$, two high-conducting bars are located vertically, at a distance $a$ from each other. A uniform, horizontal magnetic field $\mathbf{B}$ is perpendicular to the plane defined by the vertical bars. Two horizontal bars, both of mass $m$, resistance $R / 2$ and length $a$, are constrained to move, without friction, with their ends steadily in contact with the two vertical bars. The resistance of the two fixed vertical bars is assumed to be much smaller than $R / 2$, so that the net resistance of the resulting rectangular circuit is, with very good approximation, always $R$, independently of the positions of the two horizontal bars.

First, assume that the upper horizontal bar is fixed, while the lower bar starts a “free” fall at $t=0$. Let’s denote by $v=$ $v(t)$ the velocity of the falling bar at time $t$, with $v(0)=0$.
a) Write the equation of motion for the falling bar, find the solution for $v(t)$ and show that, asymptotically, the bar approaches a terminal velocity $v_{\mathrm{t}}$.
b) Evaluate the power dissipated in the circuit by Joule heating when $v(t)=v_{\mathrm{t}}$, and the mechanical work done per unit time by gravity in these conditions.

Now consider the case in which, at $t=0$, the upper bar already has a velocity $v_{0} \neq 0$ directed downwards, while the lower bar starts a “free” fall.
c) Write the equations of motion for both falling bars, and discuss the asymptotic behavior of their velocities $v_{1}(t)$ and $v_{2}(t)$, and of the current in the circuit $I(t)$.

## 物理代写|电磁学代写electromagnetism代考|The Tethered Satellite

The Earth’s magnetic field at the Earth’s surface roughly approximates the field of a magnetic dipole placed at the Earth’s center. Its magnitude ranges from $2.5 \times 10^{-5}$ to $6.5 \times$ $10^{-5} \mathrm{~T}$ ( $0.25$ to $0.65 \mathrm{G}$ in Gaussian units), with a value $B_{\mathrm{eq}} \simeq 3.2 \times 10^{-5} \mathrm{~T}$ at the equator. A satellite moves on the magnetic equatorial plane with a velocity $v \simeq 8 \mathrm{~km} / \mathrm{s}$ at a constant height $h \simeq 100 \mathrm{~km}$ over the Earth’s surface, as shown in the figure (not to scale!). A tether (leash, or lead line), consisting in a metal cable of length $\ell=1 \mathrm{~km}$, hangs from the satellite, pointing to the Earth’s center.
a) Find the electromotive force on the wire.
b) The satellite is traveling through the ionosphere, where charge carriers in outer space are available to close the circuit, thus a current can flow along the wire. Assume that the ionospheere is rigidly rotating at the same angularr vèlocity as the Earth. Find the power dissipated by Joule heating in the wire and the mechanical force on the wire as a function of its resistance $R$.

## 物理代写|电磁学代写electromagnetism代考|Magnetic Induction and Time-Varying Fields

∇×和=−b米∂吨乙

∮C和⋅dℓ=−b米∫小号∂吨乙⋅d小号

$$\boldsymbol{\nabla} \times \mathbf{B}=4 \pi k_{\mathrm{m}} \mathbf{J}+\ frac{k_{\mathrm{m}}}{k_{\mathrm{e}}} \partial_{t} \mathbf{E}=\left{4圆周率CĴ+1C∂吨和( 高斯 ) μ0Ĵ+μ0e0∂吨和( 和。 )\正确的。$$

## 物理代写|电磁学代写electromagnetism代考|A Circuit with “Free-Falling” Parts

a) 写出落杆的运动方程，求解在(吨)并表明，该条渐近地接近最终速度在吨.
b) 评估通过焦耳加热在电路中消耗的功率，当在(吨)=在吨，以及在这些条件下单位时间内通过重力所做的机械功。

c) 写出两个落杆的运动方程，并讨论它们速度的渐近行为在1(吨)和在2(吨)，以及电路中的电流我(吨).

## 物理代写|电磁学代写electromagnetism代考|The Tethered Satellite

a) 求导线上的电动势。
b) 卫星正在穿过电离层，外层空间的电荷载流子可用于闭合电路，因此电流可以沿导线流动。假设电离层以与地球相同的角速度刚性旋转。求导线中焦耳热耗散的功率和导线上的机械力与其电阻的函数关系R.

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## MATLAB代写

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