Maxwell's Equation

Maxwell's Equation

The Basic Equations of Electromagnetisms

In vacuum, we have

\[ \begin{cases} \oiint \vec{E} \bullet d\vec{A} = \dfrac{q_0}{\epsilon_0} \\ \oiint \vec{B} \bullet d\vec{A} = 0 \\ \oint \vec{E} \bullet d\vec{l} = - \dfrac{d\Phi_B}{dt} = - \iint \dfrac{\partial \vec{B}}{\partial t} \bullet d\vec{A} \\ \oint \vec{B} \bullet d\vec{l} = \mu_0 i \end{cases} \]

In the dielectric and magnetic materials,

\[ \begin{gathered} \begin{cases} \oiint \vec{D} \bullet d\vec{A} = q_0 \\ \oiint \vec{B} \bullet d\vec{A} = 0 \\ \oint \vec{E} \bullet d\vec{l} = - \iint \dfrac{\partial \vec{B}}{\partial t} \bullet d\vec{A} \\ \oint \vec{H} \bullet d\vec{l} = i_0 = \iint \vec{j}_0 \bullet d\vec{A} \end{cases} \Rightarrow \begin{cases} \nabla \bullet \vec{D} = \rho_0 \\ \nabla \bullet \vec{B} = 0 \\ \nabla \times \vec{E} = - \dfrac{\partial \vec{B}}{\partial t} \\ \nabla \times \vec{H} = \vec{j}_0 \end{cases} \end{gathered} \]

where \(\vec{D} = \epsilon_0 \vec{E} + \vec{P}\), \(\vec{H} = \dfrac{\vec{B}}{\mu_0} - \vec{M}\), \(\epsilon_0 = 8.85\times 10^{-12} F/m\), \(\mu_0 = 4\pi\times 10^{-7} N/A^2\).

Induced Magnetic Field and the Displacement current

To make \(\oint \vec{H} \bullet d\vec{l}\) same for different surfaces, we need to introduce displacement current \(i_D\) in the region between the plates, equal to the current in the wire. Then we have \(\(\oint \vec{H} \bullet d\vec{l} = i_0 + i_D\)\)

Because

\[ \begin{gathered} \Phi_D = \iint \vec{D} \bullet d\vec{A} \\\\ i_D = \frac{d\Phi_D}{dt} = \iint \frac{\partial \vec{D}}{\partial t} \bullet d\vec{A} \\\\ \vec{j}_D = \frac{\partial \vec{D}}{\partial t} \end{gathered} \]

Now we have the New Ampere’s Loop Law \(\(\oint \vec{H} \bullet d\vec{l} = \iint (\vec{j}_0 + \frac{\partial \vec{D}}{\partial t}) \bullet d\vec{A} = i_0 + \iint \frac{\partial \vec{D}}{\partial t} \bullet d\vec{A}\)\)

So the changing electric field will generate induced magnetic field.

For a enclosed surface, we have

\[ \begin{gathered} \begin{cases} \oiint \vec{j}_0 \bullet d\vec{A} = - \dfrac{dq_0}{dt} \\ \oiint \vec{D} \bullet d\vec{A} = q_0 \end{cases} \\\\ \Rightarrow \oiint \vec{j}_0 \bullet d\vec{A} = - \oiint \frac{\partial \vec{D}}{\partial t} \bullet d\vec{A} \\\\ \Rightarrow \oiint (\vec{j}_0 + \frac{\partial \vec{D}}{\partial t}) \bullet d\vec{A} = 0 \end{gathered} \]

Maxwell’s Equations

\[ \begin{gathered} \begin{cases} \oiint \vec{D} \bullet d\vec{A} = q_0 \\ \oiint \vec{B} \bullet d\vec{A} = 0 \\ \oint \vec{E} \bullet d\vec{l} = - \iint \dfrac{\partial \vec{B}}{\partial t} \bullet d\vec{A} \\ \oint \vec{H} \bullet d\vec{l} = i_0 + \iint \dfrac{\partial \vec{D}}{\partial t} \bullet d\vec{A} \end{cases} \Rightarrow \begin{cases} \nabla \bullet \vec{D} = \rho_{e_0} \\ \nabla \bullet \vec{B} = 0 \\ \nabla \times \vec{E} = - \dfrac{\partial \vec{B}}{\partial t} \\ \nabla \times \vec{H} = \vec{j}_0 + \dfrac{\partial \vec{D}}{\partial t} \end{cases} \end{gathered} \]