Electromagnetic wave equation

The electromagnetic wave equation is a second-order partial differential equation that governs the propagation of electromagnetic waves through free space. The equation, written in terms of either the electric field E or the magnetic field H, takes the form:

${\displaystyle {\partial ^{2}\mathbf {E} \over \partial t^{2}}\ -\ c^{2}\cdot \nabla ^{2}\mathbf {E} \ \ =\ \ 0}$

${\displaystyle {\partial ^{2}\mathbf {H} \over \partial t^{2}}\ -\ c^{2}\cdot \nabla ^{2}\mathbf {H} \ \ =\ \ 0}$

where

${\displaystyle c={1 \over {\sqrt {\mu _{o}\varepsilon _{o}}}}=2.998\times 10^{8}}$ meters per second

is the speed of light in free space. The magnetic permeability ${\displaystyle \ \mu _{o}}$ and the electric permittivity ${\displaystyle \ \varepsilon _{o}}$ are important physical constants that play a key role in electromagnetic theory.

These equations derive from Maxwell's equations and describe the propagation of electromagnetic waves in a vacuum.

Conservation of charge

Conservation of charge requires that the time rate of change of the total charge density within an enclosed volume V must equal the net current density J flowing into the surface S enclosing the volume:

${\displaystyle \int _{S}\mathbf {J} \cdot d\mathbf {a} =-{d \over dt}\int _{V}\rho \cdot dV}$ 

From the divergence theorem, we can convert this relationship from integral form to differential form:

${\displaystyle \nabla \cdot \mathbf {J} =-{d\rho \over dt}}$ 

Ampere's Law prior to Maxwell's correction

In its original form, Ampere's Law relates the magnetic field H to its source, the current density J:

${\displaystyle \oint _{C}\mathbf {H} \cdot d\mathbf {l} =\int _{S}\mathbf {J} \cdot d\mathbf {a} }$ 

Again, we can convert to differential form, this time using Stoke's theorem:

${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} }$ 

Inconsistency between Ampere's Law and Conservation of Charge

Maxwell discovered an important inconsistency between Ampere's Law and the Conservation of Charge.

If we take the divergence of both sides of Ampere's Law, we find

${\displaystyle \nabla \cdot (\nabla \times \mathbf {H} )=\nabla \cdot \mathbf {J} }$ 

The divergence of the curl of any vector field – in this case, the magnetic field H – is always equal to zero:

${\displaystyle \nabla \cdot (\nabla \times \mathbf {H} )=0}$ 

Combining these two equations implies that

${\displaystyle \nabla \cdot \mathbf {J} =0}$ 

From the conservation of charge, we know that

${\displaystyle \nabla \cdot \mathbf {J} =-{d\rho \over dt}}$ 

Ampere's Law, then, implies that

${\displaystyle {d\rho \over dt}=0}$ 

This last result suggests that the net charge density at any point in space is a fixed constant that cannot ever change, which is of course absurd. Not only is this outcome contrary to all physical intuition, it also directly contradicts the empirical results of thousands of laboratory experiments. It requires not only that electrical charge is conserved, but that it cannot be re-distributed from one place to another. But we know that electrical currents can and do re-distribute electrical charge. As long as the total amount of charge remains constant, conservation of charge allows for the movement of charge from one place to another. So this last result is incorrect.

Something was clearly missing from Ampere's Law, and Maxwell figured out what it was.

Maxwell's correction to Ampere's Law

To understand Maxwell's correction to Ampere's Law, we need to look at another of Maxwell's Equations, namely, Gauss's Law in integral form:

${\displaystyle \oint _{S}\varepsilon _{o}\mathbf {E} \cdot d\mathbf {a} =\int _{V}\rho \cdot dV}$ 

Again, using the divergence theorem, we can convert this equation to differential form:

${\displaystyle \nabla \cdot \varepsilon _{o}\mathbf {E} =\rho }$ 

Taking the derivative with respect to time of both sides, we find:

${\displaystyle {\partial \over \partial t}(\nabla \cdot \varepsilon _{o}\mathbf {E} )={\partial \rho \over \partial t}}$ 

Reversing the order of differentiation on the left-hand side, we obtain

${\displaystyle \nabla \cdot \varepsilon _{o}{\partial \mathbf {E} \over \partial t}={\partial \rho \over \partial t}}$ 

This last result, along with Ampere's Law and the conservation of charge equation, suggests that there are actually two sources of the magnetic field: the current density J, as Ampere had already established, and the so-called displacement current:

${\displaystyle {\partial \mathbf {D} \over \partial t}=\varepsilon _{o}{\partial \mathbf {E} \over \partial t}}$ 

So the corrected form of Ampere's Law, which Maxwell discovered, becomes:

${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} +\varepsilon _{o}{\partial \mathbf {E} \over \partial t}}$ 

Maxwell's discovery that light is an electromagnetic wave

Maxwell's correction of Ampere's Law set the stage for an even more important and, at the time, startling discovery. Maxwell realized that the equations of electromagnetism suggest that electric and magnetic fields can propagate through free space – in other words, in the absence of matter – as electromagnetic waves, and further, that the speed of propagation of these waves is exactly the same as the speed of light. Upon making this discovery in 1865, Maxwel wrote:

This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.

To obtain electromagnetic waves in a vacuum note that Maxwell's equations in a vacuum are

${\displaystyle \nabla \cdot \mathbf {E} =0}$ 

${\displaystyle \nabla \times \mathbf {E} =-\mu _{0}{\frac {\partial \mathbf {H} }{\partial t}}}$ 

${\displaystyle \nabla \cdot \mathbf {H} =0}$ 

${\displaystyle \nabla \times \mathbf {H} =\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}$ 

If we take the curl of these equations and note the vector identity

${\displaystyle \nabla \times \left(\nabla \times \mathbf {A} \right)=\nabla \left(\nabla \cdot \mathbf {A} \right)-\nabla ^{2}\mathbf {A} }$ 

we recover the wave equations, which describe wave propagation at speed c.

See also

• Maxwell's equations
• Electromagnetic waves
• Charge conservation
• Light
• Albert Einstein
• James Clerk Maxwell
• Michael Faraday
• Heinrich Hertz