Derivation of wave equation

• Starting from the inhomogeneous Maxwell equations

  $$ \vec E=-\vec \nabla V-\frac{\partial \vec A}{\partial t} \qquad \text{and} \qquad \vec B =\vec \nabla \times \vec A $$

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Lorenz gauge:

$$ \boxed{\mu_0 \epsilon_0 \frac{\partial V}{\partial t} + \vec \nabla \cdot \vec A =0} $$

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⚡ Inhomogeneous wave equation for $V$

• Start by taking the div

$$ \vec \nabla \cdot \vec E=- \nabla^2 V - \frac{\partial }{\partial t} ( \vec \nabla \cdot \vec A)=\frac{\rho}{\epsilon_0} $$

🧲 Inhomogeneous wave equation for $\vec A$

• Start by taking the curl

$$ \vec \nabla \times \vec B = \mu_0 \vec j + \mu_0 \epsilon_0 \frac{\partial \vec E}{\partial t} $$

• Subbing in the value for $\vec E$ we get

$$ \begin{aligned} \vec \nabla \times \vec B = \mu_0 \vec j + \mu_0 \epsilon_0 \left [ -\vec \nabla\frac{\partial V}{\partial t}-\frac{\partial ^2 \vec A}{\partial t^2} \right ] \end{aligned} $$

• And using a vector identity we can show

$$ \begin{aligned} \vec \nabla \times \vec B &=\vec \nabla \times (\vec \nabla \times \vec A) \\ &=\vec \nabla ( \vec \nabla \cdot \vec A)-\nabla^2 \vec A \end{aligned} $$

• Combining both gives us

$$ \begin{aligned} \footnotesize{\mu_0\epsilon_0 \frac{\partial ^2 \vec A}{\partial t^2} - \nabla^2 \vec A + \nabla (\mu_0 \epsilon_0 \frac{\partial V}{\partial t} + \vec \nabla \cdot \vec A)=\mu_0 j} \end{aligned} $$

• Applying the Lorenz gauge to both sides we get

  $$ \boxed{\mu_0 \epsilon_0 \frac{\partial^2 V}{\partial t^2}-\nabla^2 V=\frac{\rho}{\epsilon_0}} \qquad \qquad \boxed{\mu_0 \epsilon_0 \frac{\partial^2 \vec A}{\partial t^2}-\nabla^2 \vec A=\mu_0 \vec j} $$

🗒️ note: since $\epsilon_0 \mu_0 = \frac{1}{c^2}$ the potentials (and fields) propagate at the speed of light

For convenience we define the D’Alambertian

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D’Alambertian:

$$ \square ^2 \equiv \frac{1}{c^2} \frac{\partial ^2 }{\partial t^2} -\nabla^2 $$

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• We can now re-write the wave equations as

  $$ \square ^2 V=\frac{\rho}{\epsilon_0} \qquad \qquad \square ^2 \vec A= \mu_0 \vec j $$

🗒️ Notes:

• Symmetry between the wave equation for $V$ and $\vec A$
• We can think of these as $4$-dimensional versions of Poisson’s equation
• We can think of the equations for $\vec A$ as 3 equations relating the components
• Equations contain the same information as maxwell equations for $\vec E$ and $\vec B$ but in $4$-component form rather than the $6$-components we had before
• If $\rho$ and $\vec j$ are we obtain the homogeneous wave equations
• If there is no time-dependence we recover Poisson’s equation for electro- and magneto-statics

Solutions to the wave equations for potentials

💼 Case: Similar setup to before except this time we need to define the time at which we are determining the field

• We take into account time-varying $\rho$, $\vec j$

• Thus we take into consideration the time delay as the EM waves travel from $\delta t'$ to $P(\vec r,t)$

  $$ t_\text{ret}=t-\frac{r}{c} $$

  $t_\text{ret}$ is the retarded time