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# 物理代写|电磁学代写Electromagnetism代考|Approximations and Similarity Theorems

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## 物理代写|电磁学代写Electromagnetism代考|Approximations and Similarity Theorems

Before delving into that subject, here we show how a rough, still useful approximation of diffusion problems can be obtained with hardly any calculation at all. For instance, insert a conductor suddenly into a magnetic field. Its inside shall initially be without a field. Once inside the field, it gradually penetrates, diffuses into the conductor. One would like to approximate how long it will take for the field to penetrate the conductor (Fig. 6.14).

The typical length of the conductor (which may be cube shaped) shall be of the order of magnitude $l$. The diffusion time shall be approximately $t_0$. This allows for a rough approximation of eq. (6.29) in the form
$$\frac{\mathbf{B}}{l^2} \approx \mu \kappa \frac{\mathbf{B}}{t_0} .$$
This is justified because it is approximately
$$\nabla^2 \mathbf{B} \approx \frac{\mathbf{B}}{l^2} .$$
and
$$\frac{\partial \mathbf{B}}{\partial t} \approx \frac{\mathbf{B}}{t_0}$$
Consequently
$$t_0 \approx \mu \kappa l^2 \text {. }$$

## 物理代写|电磁学代写Electromagnetism代考|Laplace Transform

The Laplace transform is oftentimes a very helpful tool to solve initial value problems. In the theory of circuits, this approach is used to reduce ordinary differential equations, whose independent variable is the time, to algebraic equations. In field theory, the equations are partial differential equations, for example in $x, y, z$, and $t$. Applying the Laplace transform results in a new partial differential equation in $x, y$, and $z$, where the initial values are automatically considered. This reduces the problem to a spatial boundary value problem, which can be solved using the methods introduced in Chapter 3 . In the simplest case, the spatial problem is one dimensional, for example, $x$-dependent (planar) or $r$ dependent (cylindrical). In this case, the Laplace transform reduced the original partial differential equation to an ordinary differential equation in $x$ or $r$.
Now, we will compile a list of important relations about the Laplace transform, whereby no explanation or derivation is provided. More over, no attempt for completeness will be made.

The (one sided) Laplace transform of $\tilde{f}(p)$ for a function $f(t)$ is defined by $\tilde{f}(p)=\int_0^{\infty} f(t) \exp (-p t) d t$.

$\mathcal{L}$ shall serve as the symbol to represent the Laplace transform of a time dependent function and $\mathcal{L}^{-1}$ shall represent the inverse Laplace transformation. Thus, we will frequently write
$$\mathcal{L}{f(t)}=\tilde{f}(p)$$
or
$$\mathcal{L}^{-1}{\tilde{f}(p)}=f(t) .$$

## 物理代写|电磁学代写Electromagnetism代考|Approximations and Similarity Theorems

$$\frac{\mathbf{B}}{l^2} \approx \mu \kappa \frac{\mathbf{B}}{t_0}$$

$$\nabla^2 \mathbf{B} \approx \frac{\mathbf{B}}{l^2}$$

$$\frac{\partial \mathbf{B}}{\partial t} \approx \frac{\mathbf{B}}{t_0}$$

$$t_0 \approx \mu \kappa l^2$$

## 物理代写|电磁学代写Electromagnetism代考|Laplace Transform

$\mathcal{L}$ 应作为表示时间相关函数的拉普拉斯变换的符号，并且 $\mathcal{L}^{-1}$ 应代表逆拉普拉斯变换。因此，我们会经常写
$$\mathcal{L} f(t)=\tilde{f}(p)$$

$$\mathcal{L}^{-1} \tilde{f}(p)=f(t)$$

## MATLAB代写

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