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物理代考|电磁学代考Electromagnetism代考|PHYS2213 Complex Field

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物理代考|电磁学代考ELECTROMAGNETISM代考|Complex Field

The use of a complex exponential time dependence for the fields is a convenience that may allow us to express Maxwell equations in simple forms. The physically meaningful fields are the real parts of the complex quantities. The notation
$$\mathbf{E}=\mathbf{E}{0} e^{i \omega t}$$ implies that the field is actually given by $$\mathbf{E}=\mathbf{E}{0} \cos \omega t$$

The representation of physical variables by means of complex quantities is based on the understanding that, at the end of the calculations involving these variables, only their real parts will be used. This prescription applies also to the square of a physical variable, for which the meaningful quantity is the square of the real part (which is different from the real part of the square). Therefore, if a field $\mathbf{A}$ is expressed in complex form, in order to obtain the time average of its square value, we have first to find the real part of $\mathbf{A}$, square it, and then average the result over one period. In many cases, such as that of a plane, monochromatic wave, such a calculation is simple. In other circumstances the complex functions are more complicated than $e^{i \omega t}$ and the calculations require more elaborate efforts. However, some useful relations may be found that simplify the task of obtaining the time average of squared monochromatic quantities.

物理代考|电磁学代考ELECTROMAGNETISM代考|Electromagnetic Waves in Vacuum and in Continuous Media

The Maxwell equations in the absence of charges and currents $(\rho=j=0)$ become homogeneous equations that may have solutions different from zero. We will not be concerned at this point with the production of electromagnetic waves, but rather with waves that can exist independently from the sources.

The macroscopic Maxwell equations, in the absence of true charges and true currents $\left(\rho_{\text {true }}=\mathbf{j}_{\text {true }}=0\right)$ can be written as follows:
$$\begin{array}{r} \nabla \cdot \mathbf{D}=0 \ \nabla \times \mathbf{E}+\frac{1}{c} \frac{\partial \mathbf{B}}{\partial t}=0 \ \nabla \cdot \mathbf{B}=0 \ \boldsymbol{\nabla} \times \mathbf{H}-\frac{1}{c} \frac{\partial \mathbf{D}}{\partial t}=0 \end{array}$$

物理代考|电磁学代考ELECTROMAGNETISM代考|Complex Field

$$\mathbf{E}=\mathbf{E} 0 e^{i \omega t}$$

$$\mathbf{E}=\mathbf{E} 0 \cos \omega t$$

物理代考|电磁学代考ELECTROMAGNETISM代 考|Electromagnetic Waves in Vacuum and in Continuous Media

$$\nabla \cdot \mathbf{D}=0 \nabla \times \mathbf{E}+\frac{1}{c} \frac{\partial \mathbf{B}}{\partial t}=0 \nabla \cdot \mathbf{B}=0 \boldsymbol{\nabla} \times \mathbf{H}-\frac{1}{c} \frac{\partial \mathbf{D}}{\partial t}=0$$

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