Did god say “Let there be light!” using these equations?

Maxwell’s equations (and their intuitive meaning):

\begin{equation}\label{eqn:1} \nabla . \vec{E} = \frac{\rho}{\epsilon_0} \end{equation} The equation above (Gauss’s law for electric field) tells that the electric field at any point is directly proportional to the free charge nearby.

\begin{equation}\label{eqn:2} \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} \end{equation} Equation (\ref{eqn:2}) (Faraday’s (+Lenz’s) law) says that if you change the magnetic field (let’s say, by moving a magnet), then it produces an electric filed.

\begin{equation}\label{eqn:3} \nabla.\vec{B} = 0 \end{equation} Equation (\ref{eqn:3}) (Gauss’s law for magnetic field) says that magnetic fields don’t have any origin, but loop onto itself. This is not the case with electric field as electric fields can originate from point charges.

\begin{equation}\label{eqn:4} \nabla \times \vec{B} = \mu_0 \left( \vec{J} + \epsilon_0 \frac{\partial \vec{E}}{\partial t} \right) \end{equation} This equation (Ampere-Maxwell’s law) says that if you have a current carrying wire or a changing electric field, then it produces a magnetic field around.

Also, an identity from vector calculus,

\begin{equation} \nabla \times \nabla \times \vec{A} = \nabla ( \nabla. \vec{A}) - \nabla^2 \vec{A} \end{equation}

So,

\[\nabla \times \nabla \times \vec{E} = \nabla ( \nabla. \vec{E} ) - \nabla^2 \vec{E}\]

Substituting value of $\nabla \times \vec{E}$ from equation (\ref{eqn:2}) and that of $\nabla. \vec{E}$ from equation (\ref{eqn:1}) respectively, we get,

\[-\nabla \times \frac{\partial \vec{B}}{\partial t} = \nabla ( \frac{\rho}{\epsilon_0} ) - \nabla^2 \vec{E}\]

In vacuum, the free charge density at any point is zero. So, the quantity first term on the RHS above becomes zero.

Hence,

\[\nabla \times \frac{\partial \vec{B}}{\partial t} = \nabla^2 \vec{E}\]

Or,

\[\frac{\partial }{\partial t} \left( \nabla \times \vec{B} \right) = \nabla^2 \vec{E}\]

Substituting equation (\ref{eqn:4}) in above expression, we get,

\[\frac{\partial }{\partial t} \left( \mu_0 \left( \vec{J} + \epsilon_0 \frac{\partial \vec{E}}{\partial t} \right) \right) = \nabla^2 \vec{E}\]

Again, in vacuum $\vec{J}$ (current density) is zero.

So,

\[\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} = \nabla^2 \vec{E}\]

Hence, \begin{equation} \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} \end{equation}

which represents an wave equation, whose standard form is: \begin{equation} \nabla^2 \vec{\psi} = \frac{1}{v^2} \frac{\partial^2 \vec{\psi}}{\partial t^2} \end{equation}

So, the velocity of the wave is: \(v = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8 m/s (approx.)\)

So the point is:

• Electric and magnetic fields aren’t entirely different things but manifestation of same thing. (?? Deep!!)
• The change in electric / magnetic field travels in space as a wave (known as electromagnetic wave).
• The wave has a velocity of $3\times10^8 m/s$ in vacuum.
• By experimental measurement, it is confirmed that light (which is also electromagnetic wave) travels at a speed of $3\times10^8 m/s$ in vacuum.
• All radio-tv signals, wireless telecommunication signals, wifi signals etc. are electromagnetic field and are produced in accordance with the above laws!
• From engineering perspective, this is probably the most important equation for late 20th and early 21st century.

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Last Updated: Wednesday, 30 Jan, 2020, 11:54 NPT
Author: Madhav Humagain (scimad)