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Wave equation from Maxwell equations

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Maxwell PDEs

The "differential forms" of the Maxwell equations:

$\nabla \times \mathbf{E} = -\frac {\partial \mathbf{B}}{\partial t}$ -- Faraday's Law

$\nabla \times \mathbf{H} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{D}}{\partial t}$ -- Ampere's Law

$\nabla \cdot \mathbf{D} = \frac{\rho}{\varepsilon_0}$ -- Gauss' Law

$\nabla \cdot \mathbf{B} = 0$ -- Gauss's Law for Magnetism


These letters stand for the following vector fields:

  • $E$: electric field, also called • the electric field intensity;
  • $B$: magnetic field, also called • the magnetic induction• the magnetic field density, • the magnetic flux density;
  • $D$: electric displacement field, also called • the electric induction, • the electric flux density;
  • $H$: magnetizing field, also called • auxiliary magnetic field, • magnetic field intensity, • magnetic field;
  • $J$: total current density (including both free and bound current);

as well as these constants:

  • $ε_0$: permittivity of free space, also called the electric constant, a universal constant;
  • $μ_0$: permeability of free space, also called the magnetic constant, a universal constant.

Maxwell equations of differential forms

As differential forms:

  • $E$: 1-form on $TM$ or the primal complex in the discrete case,
  • $B$: 2-form on $TM$ or on the primal complex in the discrete case,
  • $D$: 2-form on $T^*M$ or the the dual complex in the discrete case,
  • $H$: 1-form on $T^*M$ or the the dual complex in the discrete case,
  • $J$: 2-form on $T^*M$ or the the dual complex in the discrete case.

They are also $0$-forms with respect to time.

Faraday’s Law: $$d E dt= -d_t B,$$ Ampere’s Law: $$d^* H dt = \mu_0\varepsilon_0 d_t D + \mu_0 J dt.$$ Here:

  • $d$ and $d^*$ are the exterior derivative of the primal and the dual complex respectively,
  • $d_t$ is the exterior derivative with respect to time == the forward difference == difference of values of the function,
  • $dt$ is the basic 1-form with respect to time,
  • the product of forms is, in fact, the exterior product.

The rest of the laws are: $$d^* D = \frac{\rho}{\varepsilon_0},$$ $$d B = 0.$$

In terms of the Hodge star operator, the constitutive relations are: $$D=\varepsilon_0 \star E,$$ $$B=\mu_0 \star H.$$


Special case of Maxwell equations

Let's consider the special case when $J=0$. Then $$d_t B = -dEdt = - d \star Ddt.$$

Consider $$d^2_t B := d_t^* \star _t d_t B, $$ the "second order" exterior derivative with respect to time == the second order central difference. It produces a 1-form.

Substitute: $$d^2_t B = - d_t^* \star _t d \star D dt.$$ Substitute more: $$d^2_t B = -d \star d_t^* \star _t Ddt = -d \star d_t D$$ $$= -d \star d^*Hdt = -d \star d^* \star B dt.$$

The result $$d^2_t B = -d \star d^* \star B dt$$ is the wave equation!


Continuous model for electromagnetic waves

Same goal for the PDEs.

The equation of conservation of electric charge: \[ \nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0. \]

Now consider Faraday's Law in differential form: \[ \nabla \times \mathbf{E} = -\frac{ \partial \mathbf{B}}{\partial t}. \] Taking the curl of both sides: \[ \nabla \times (\nabla \times \mathbf{E}) = \nabla \times (- \frac{ \partial \mathbf{B}}{\partial t}). \]

The right-hand side may be simplified by noting that \[ \nabla \times (\frac{ \partial \mathbf{B}}{\partial t}) = - \frac{ \partial}{\partial t} (\nabla \times \mathbf{B}). \] Recalling Ampere's Law, \[ - \frac{ \partial}{\partial t} (\nabla \times \mathbf{B}) = -\mu_0 \epsilon_0 \frac{ \partial^2 \mathbf{E}}{\partial t^2}. \] Therefore \[ \nabla \times (\nabla \times \mathbf{E}) = -\mu_0 \epsilon_0 \frac{ \partial^2 \mathbf{E}}{\partial t^2}. \] Hence \[ \nabla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{ \partial^2 \mathbf{E}}{\partial t^2} \] Applying the same analysis to Ampere's Law then substituting in Faraday's Law leads to the wave equation: \[ \nabla^2 \mathbf{B} = \mu_0 \epsilon_0 \frac{ \partial^2 \mathbf{E}}{\partial t^2}. \]