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Differential Form and Integral Form of Maxwell Equation

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Differential Form of Maxwell Equation

Gauss’s law

$$div {\bf D}({\bf r},t)=ρ({\bf r},t)$$

Law of the magnetic flux conservation

$$div {\bf B}({\bf r},t)=0$$

Maxwell-Ampere law

$$rot {\bf H}({\bf r},t)={\bf i}({\bf r},t)+\frac{∂{\bf D}({\bf r},t)}{∂t}$$

Faraday’s law of induction

$$rot {\bf E}({\bf r},t)=-\frac{∂{\bf B}({\bf r},t)}{∂t}$$

Relation between electric field and electric flux density, magnetic field and magnetic flux density

$${\bf D}({\bf r},t)=ε{\bf E}({\bf r},t)$$
$${\bf B}({\bf r},t)=μ{\bf H}({\bf r},t)$$

\({\bf E}({\bf r},t)\):Electric field   \({\bf D}({\bf r},t)\):Electric flux density   \({\bf H}({\bf r},t)\):Magnetic field   \({\bf B}({\bf r},t)\):Magnetic flux density

\(ρ({\bf r},t)\):Charge density   \({\bf i}({\bf r},t)\):Current density   \(ε\):Dielectric constant (Dielectric)   \(μ\):Magnetic permeability (Permeability)

The first four equations are collectively called the Maxwell equation. The form is called the differential form of the Maxwell equation. The latter two supplements wrote the electric field and electric flux density, and the relational expressions of the magnetic field and the magnetic flux density, respectively.

Divergence Theorem of Gauss and Stokes’ Theorem

To convert from a differential form to an integral form, divergence theorem of Gauss and Stokes’ theorem are necessary.

Divergence Theorem of Gauss

$$\int_{V} div {\bf A} dV=\int_{S} {\bf A} ･d{\bf S}$$

Here, V on the left side represents a part surrounded by the closed surface S. Therefore, the integral by \(dV\) on the left side represents the volume of the part surrounded by the closed surface S. S on the right side represents the area of this closed surface S.

Stokes’ Theorem

$$\int_{S} rot{\bf A} ･d{\bf S}=\int_{C} {\bf A} ･d{\bf r}$$

Here, \(S\) on the left side is the area of this closed surface \(S\). \(C\) on the right side represents a curve surrounding the closed surface \(S\).

The Integral Forms of Maxwell Equations

Gauss’s law

$$div {\bf D}({\bf r},t)=ρ$$

First, calculate the volume integral of both sides.

$$\int_{V} div {\bf D}({\bf r},t) dV=\int_{V} ρ({\bf r},t)  dV$$

When the left side of this equation is transformed by the divergence theorem, the following equation is obtained.

$$\int_{S} {\bf E}({\bf r},t) ･d{\bf S}=\frac{1}{ε} \int_{V} ρ({\bf r},t) dV$$

This equation is the integral form of Gauss’s law. It means the number of lines of electric force passing vertically through the closed surface \(S\) is equal to the sum of the charges existing in the region \(V\) divided by the permittivity \(ε\).

Law of the magnetic flux conservation

In the same way, we also try to integrate the formula of the law of the magnetic flux conservation.

$$\int_{S} {\bf B}({\bf r},t) ･d{\bf S}=0$$

This law comes up as a proof that there is no “single magnetic charge” that creates a magnetic field by itself.

Maxwell-Ampere law

$$rot {\bf H}({\bf r},t)={\bf i}({\bf r},t)+\frac{∂{\bf D}({\bf r},t)}{∂t}$$

First, calculate the surface integral of both sides.

$$\int_{S} rot {\bf H}({\bf r},t) ･d{\bf S}=\int_{S} {\bf i}({\bf r},t) ･d{\bf S}+\int_{S} \frac{∂{\bf D}({\bf r},t)}{∂t} ･d{\bf S}$$

The left side of this equation can be expressed as follows with Stokes’ theorem.

$$\int_{C} {\bf H}({\bf r},t)･d{\bf r}=\int_{S} {\bf i}({\bf r},t) ･d{\bf S}+\int_{S} \frac{∂{\bf D}({\bf r},t)}{∂t} ･d{\bf S}$$

\({\bf i}({\bf r}, t)\) in the first term on the right-hand side is the current density. If you calculate the surface integral of \(i\), the current \({\bf I }({\bf r}, t)\) passing through the surface \(S\) can be obtained.

$$\int_{S} {\bf i}({\bf r},t) ･d{\bf S}={\bf I}({\bf r},t)$$

From the above, the integral form of Maxwell-Ampere law appears.

$$\int_{C} {\bf H}({\bf r},t)･d{\bf r}={\bf I}({\bf r},t)+\int_{S} \frac{∂{\bf D}({\bf r},t)}{∂t} ･d{\bf S}$$

Here, if the electric field does not change with time, that is, \(\frac{∂{\bf D}}{∂t}=0\), the law becomes as follows.

$$\int_{C} {\bf H}({\bf r},t)･d{\bf r}={\bf I}({\bf r},t)$$

From this equation, we see that when the current passes through the surface \ (S \), it creates a magnetic field.

Faraday’s law of induction

In the same way, we also try to integrate the formula of the Faraday’s law of induction.

$$\int_{C} {\bf E}({\bf r},t)･d{\bf r}=-\int_{S}\frac{∂{\bf B}({\bf r},t)}{∂t}･d{\bf S}$$

From this equation, we see that the temporal change in magnetic flux density through the surface \(S\) creates an electric field.

Summary of the Integral Form

Gauss’s law

$$\int_{S} {\bf E}({\bf r},t) ･d{\bf S}=\frac{1}{ε} \int_{V} ρ({\bf r},t) dV$$

Law of the magnetic flux conservation

$$\int_{S} {\bf B}({\bf r},t) ･d{\bf S}=0$$

Maxwell-Ampere law

$$\int_{C} {\bf H}({\bf r},t)･d{\bf r}={\bf I}({\bf r},t)+\int_{S} \frac{∂{\bf D}({\bf r},t)}{∂t} ･d{\bf S}$$

Faraday’s law of induction

$$\int_{C} {\bf E}({\bf r},t)･d{\bf r}=-\int_{S}\frac{∂{\bf B}({\bf r},t)}{∂t}･d{\bf S}$$

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