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# 物理代写|电磁学代写Electromagnetism代考|Continuity Equation and Relaxation Time

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## 物理代写|电磁学代写Electromagnetism代考|Continuity Equation and Relaxation Time

Consider a region with volume $v$ bounded by its surface $s$. Let the total current coming out of this surface be $I$. Outflow of this current causes a progressive reduction of any charge $Q$, that may be present in the volume, provided that the electric charge is neither created nor annihilated. Therefore,
$$I=-\frac{d Q}{d t}$$
This is called the continuity equation. Since the current $I$ is given by integrating the current density over the closed surface $s$, and the charge $Q$ is obtained by integrating the charge density in the volume $v$, the integral form of the continuity equation can be given as
$$\oiint_s J \cdot d s=-\frac{d}{d t} \iiint_v \rho d v$$
To obtain the point form of the continuity equation, apply Stokes’ theorem to the LHS of this equation. Noting that the time and space coordinates are independent of each other, the sequence of operations on the RHS of Equation 2.64 can be interchanged. Since the resulting equation is valid for any arbitrary volume $v$, integrands on the two sides can be equated. Therefore, we get
$$\nabla \cdot J=-\frac{\partial \rho}{\partial t}$$

## 物理代写|电磁学代写Electromagnetism代考|A Rear Window View

In our journey through the kingdom of Maxwell’s equations, we have now covered some distance. It is now time to look back. Maxwell’s equations are the experimental results described in mathematical language. These equations are based on two axioms, namely,

1. Nonexistence of magnetic monopoles
2. Conservation of electric charges
Consider Maxwell’s third equation, that is, $\nabla \times E=-(\partial B / \partial t)$. If we take divergence on both sides of this equation, and interchange the order of differentiations with respect to time and space coordinates, we get
$$\frac{\partial}{\partial t}(\nabla \cdot \boldsymbol{B})=0$$
It follows from the above equation that at every point in the field the divergence of $B$ is time invariant. Thus, if $\nabla \cdot B$ is ever zero, it remains at the zero value. Therefore, Maxwell’s first equation, that is, $\nabla \cdot \boldsymbol{B}=0$, is included in (consistent with) Maxwell’s third equation.

Next, consider Maxwell’s fourth equation, that is, $\nabla \times H=J+(\partial D / \partial t)$. Again, if we take divergence on both sides of this equation, and interchange the order of differentiations with respect to time and space coordinates, we get
$$\nabla \cdot J=-\frac{\partial}{\partial t}(\nabla \cdot D)$$

## 物理代写|电磁学代写Electromagnetism代考|Continuity Equation and Relaxation Time

$$I=-\frac{d Q}{d t}$$

$$\oiint_s J \cdot d s=-\frac{d}{d t} \iiint_v \rho d v$$

$$\nabla \cdot J=-\frac{\partial \rho}{\partial t}$$

## 物理代写|电磁学代写Electromagnetism代考|A Rear Window View

$$\frac{\partial}{\partial t}(\nabla \cdot \boldsymbol{B})=0$$

## MATLAB代写

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