Last modified 2017-09-25 00:25:11 CDT

“Light is an electromagnetic disturbance”

\[\\ \\\] \[{\large \text{Integral form of Maxwell-Heaviside Equations}}\] \[\begin{array}{ll} \text{Gauss's Law for Electric Fields} & \oint_S \mathbf{E} \cdot \hat{\mathbf{n}}\ da = \frac{1}{\epsilon_0} \int_V \rho\ dv \\ \text{Gauss's Law for Magnetic Fields} & \oint_S \mathbf{B} \cdot \hat{\mathbf{n}}\ da = 0 \\ \text{Faraday's Law} & \oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{\partial}{\partial t} \int_S \mathbf{B} \cdot \hat{\mathbf{n}}\ da \\ \text{Amp}\grave{\mathrm{e}}\text{re-Maxwell Law} & \oint_C \mathbf{B} \cdot d\mathbf{l} = \int_S \left( \mu_0 \mathbf{J} + \mu_0\epsilon_0 \frac{\partial}{\partial t} \mathbf{E} \right) \cdot \hat{\mathbf{n}}\ da \\ \end{array}\] \[\\ \\\] \[\text{Apply the Divergence Theorem: } \oint_S \mathbf{F} \cdot \hat{\mathbf{n}}\ da = \int_V \nabla \cdot \mathbf{F}\ dV\] \[\begin{array}{ll} \oint_S \mathbf{E} \cdot \hat{\mathbf{n}}\ da = \frac{1}{\epsilon_0} \int_V \rho\ dv \; &\Longrightarrow \; \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \\ \oint_S \mathbf{B} \cdot \hat{\mathbf{n}}\ da = 0 \; &\Longrightarrow \; \nabla \cdot \mathbf{B} = 0 \\ \end{array}\] \[\\ \\\] \[\text{Apply Stokes Theorem: } \oint_C \mathbf{F} \cdot d\mathbf{l} = \int_S (\nabla \times \mathbf{F}) \cdot \hat{\mathbf{n}}\ da\] \[\begin{array}{ll} \oint_C \mathbf{E} \cdot d\mathbf{l} = \int_S -\frac{\partial \mathbf{B}}{\partial t} \cdot \hat{\mathbf{n}}\ da \; &\Longrightarrow \; \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \\ \oint_C \mathbf{B} \cdot d\mathbf{l} = \int_S \left( \mu_0 \mathbf{J} + \mu_0\epsilon_0 \frac{\partial}{\partial t} \mathbf{E} \right) \cdot \hat{\mathbf{n}}\ da \; &\Longrightarrow \; \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \\ \end{array}\] \[\\ \\\] \[\\ \\\] \[{\large \text{Differential form of Maxwell-Heaviside Equations}}\] \[\begin{array}{ll} \text{Gauss's Law for Electric Fields} & \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \\ \text{Gauss's Law for Magnetic Fields} & \nabla \cdot \mathbf{B} = 0 \\ \text{Faraday's Law} & \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \\ \text{Amp}\grave{\mathrm{e}}\text{re-Maxwell Law} & \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \\ \end{array}\] \[\\ \\\] \[\text{Set } \rho \text{ and } \mathbf{J} \text{ to zero, as there is no free charge or current in a vacuum:}\] \[\begin{aligned} \nabla \cdot \mathbf{E} &= 0 \\ \nabla \cdot \mathbf{B} &= 0 \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \times \mathbf{B} &= \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \\ \end{aligned}\] \[\\ \\\] \[\text{Take the curl of Faraday's Law and substitute in }\text{Amp}\grave{\mathrm{e}}\text{re-Maxwell Law:}\] \[\begin{aligned} \nabla \times (\nabla \times \mathbf{E}) &= \nabla \times \left( -\frac{\partial \mathbf{B}}{\partial t} \right) \\ &= -\frac{\partial}{\partial t} (\nabla \times \mathbf{B}) \\ &= - \mu_0 \epsilon_0 \frac{{\partial}^2 \mathbf{E}}{{\partial t}^2} \\ \end{aligned}\] \[\\ \\\] \[\text{Use the vector calculus identity: } \nabla \times (\nabla \times \mathbf{F}) = \nabla (\nabla \cdot \mathbf{F}) - {\nabla}^2 \mathbf{F}\] \[\text{and substitute in Gauss's Law for Electric Fields:}\] \[\begin{aligned} \nabla \times (\nabla \times \mathbf{E}) &= \nabla (\nabla \cdot \mathbf{E}) - {\nabla}^2 \mathbf{E} \\ &= - {\nabla}^2 \mathbf{E} \\ \end{aligned}\] \[\text{Now we have the ...}\] \[\\ \\\] \[{\large \text{Electromagnetic Wave Equation}}\] \[\begin{aligned} {\nabla}^2 \mathbf{E} &= \mu_0 \epsilon_0 \frac{{\partial}^2 \mathbf{E}}{{\partial t}^2} \\ {\nabla}^2 \mathbf{B} &= \mu_0 \epsilon_0 \frac{{\partial}^2 \mathbf{B}}{{\partial t}^2} \\ \end{aligned} \\\] \[{\small \text{(Derivation of magnetic field wave equation parallels the steps above.)}}\] \[\\ \\\] \[\begin{array}{ll} \text{General Wave Equation Form: }\ c^2 {\nabla}^2 u = \frac{{\partial}^2 u}{{\partial t}^2} \\ \text{where c in the electromagnetic wave equation is:} \\ \end{array}\] \[\begin{aligned} c^2 &= \frac{1}{\mu_0 \epsilon_0} \\ c &= \frac{1}{\sqrt{\mu_0 \epsilon_0}} \\ c &= 2.99792 \times 10^8\ \text{m/s} \\ \end{aligned}\]

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and James Clerk Maxwell wrote in his A Dynamical Theory of the Electromagnetic Field paper:

At the commencement of this paper we made use of the optical hypothesis of an
elastic medium through which the vibrations of light are propagated, in order
to show that we have warrantable grounds for seeking, in the same medium, the
cause of other phenomena as well as those of light. We then examined
electromagnetic phenomena, seeking for their explanation in the properties of
the field which surrounds the electrified or magnetic bodies. In this way we
arrived at certain equations expressing certain properties of the
electromagnetic field. We now proceed to investigate whether these properties
of that which constitutes the electromagnetic field, deduced from
electromagnetic phenomena alone, are sufficient to explain the propagation of
light through the same substance.


Hence the velocity of light deduced from experiment agrees sufficiently well
with the value of v deduced from the only set of experiments we as yet possess.
The value of v was determined by measuring the electromotive force with which a
condenser of known capacity was charged, and then discharging the condenser
through a galvanometer, so as to measure the quantity of electricity in it in
electromagnetic measure. The only use made of light in the experiment was to
see the instruments. The value of V found by M. Foucault was obtained by
determining the angle through which a revolving mirror turned, while the light
reflected from it went and returned along a measured course. No use whatever
was made of electricity or magnetism.

The agreement of the results seems to show that light and magnetism are
affections of the same substance, and that light is an electromagnetic
disturbance propagated through the field according to electromagnetic laws.

-- James Clerk Maxwell, A Dynamical Theory of the Electromagnetic Field

Imagine being the first person in all of human history to comprehend that and write it down.


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