What is the wave equation derived from Maxwell

The wave equation derived from Maxwell’s equations describes the propagation of electromagnetic waves (such as light) in a vacuum or other mediums. The derivation involves Maxwell’s four equations, which are:

1. Gauss’s Law for Electricity:

   $$nabla cdot mathbf{E} = frac{rho}{epsilon_0}$$

   where $mathbf{E}$ is the electric field, $rho$ is the charge density, and $epsilon_0$ is the permittivity of free space.

2. Gauss’s Law for Magnetism:

   $$nabla cdot mathbf{B} = 0$$

   where $mathbf{B}$ is the magnetic field. This equation states that there are no “magnetic charges” and magnetic field lines are always closed loops.

3. Faraday’s Law of Induction:

   $$nabla times mathbf{E} = – frac{partial mathbf{B}}{partial t}$$

   which describes how a changing magnetic field can induce an electric field.

4. Ampère’s Law (with Maxwell’s correction):

   $$nabla times mathbf{B} = mu_0 mathbf{J} + mu_0 epsilon_0 frac{partial mathbf{E}}{partial t}$$

   where $mathbf{J}$ is the current density, $mu_0$ is the permeability of free space, and $epsilon_0$ is the permittivity of free space. This equation shows how a changing electric field and electric current can generate a magnetic field.

Derivation of the Wave Equation

To derive the wave equation, we’ll consider the behavior of the electric and magnetic fields in free space (where $rho = 0$ and $mathbf{J} = 0$):

1. Taking the curl of Faraday’s Law:

   $$nabla times (nabla times mathbf{E}) = – nabla times frac{partial mathbf{B}}{partial t}$$

2. Using the vector identity $nabla times (nabla times mathbf{A}) = nabla (nabla cdot mathbf{A}) – nabla^2 mathbf{A}$, the left-hand side becomes:

   $$nabla (nabla cdot mathbf{E}) – nabla^2 mathbf{E} = – frac{partial}{partial t} (nabla times mathbf{B})$$

3. Substitute Ampère’s Law ($nabla times mathbf{B} = mu_0 epsilon_0 frac{partial mathbf{E}}{partial t}$) into the above equation:

   $$nabla (nabla cdot mathbf{E}) – nabla^2 mathbf{E} = – mu_0 epsilon_0 frac{partial^2 mathbf{E}}{partial t^2}$$

4. In free space, $nabla cdot mathbf{E} = 0$ (from Gauss’s Law for Electricity), so the equation simplifies to:

   $$– nabla^2 mathbf{E} = – mu_0 epsilon_0 frac{partial^2 mathbf{E}}{partial t^2}$$

   Therefore, the wave equation for the electric field is:

   $$nabla^2 mathbf{E} = mu_0 epsilon_0 frac{partial^2 mathbf{E}}{partial t^2}$$

   This is a standard wave equation with a wave speed $c = frac{1}{sqrt{mu_0 epsilon_0}}$.

5. Similarly, we can derive the wave equation for the magnetic field $mathbf{B}$ using the curl of Ampère’s Law (with Maxwell’s correction), yielding:

   $$nabla^2 mathbf{B} = mu_0 epsilon_0 frac{partial^2 mathbf{B}}{partial t^2}$$

Thus, both the electric and magnetic fields in free space satisfy the wave equation with the same wave speed $c = frac{1}{sqrt{mu_0 epsilon_0}}$, which is the speed of light in a vacuum. This shows that electromagnetic waves propagate with the speed of light.