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A similar argument holds by substituting E for B and using Gauss’s law for magnetism instead of Gauss’s law for electric fields. This shows that the B field is also perpendicular to the direction of propagation of the wave. The electromagnetic wave is therefore a transverse wave, with its oscillating electric and magnetic fields perpendicular to its direction of propagation.
We can next apply Maxwell’s equations to the description given in connection with [link] in the previous section to obtain an equation for the E field from the changing B field, and for the B field from a changing E field. We then combine the two equations to show how the changing E and B fields propagate through space at a speed precisely equal to the speed of light.
First, we apply Faraday’s law over Side 3 of the Gaussian surface, using the path shown in [link] . Because ${E}_{x}(x,t)=0,$ we have
Assuming $\text{\Delta}x$ is small and approximating ${E}_{y}\left(x+\text{\Delta}x,t\right)$ by
we obtain
Because $\text{\Delta}x$ is small, the magnetic flux through the face can be approximated by its value in the center of the area traversed, namely ${B}_{z}\left(x+\frac{\text{\Delta}x}{2},t\right)$ . The flux of the B field through Face 3 is then the B field times the area,
From Faraday’s law,
Therefore, from [link] and [link] ,
Canceling $l\text{\Delta}x$ and taking the limit as $\text{\Delta}x=0$ , we are left with
We could have applied Faraday’s law instead to the top surface (numbered 2) in [link] , to obtain the resulting equation
This is the equation describing the spatially dependent E field produced by the time-dependent B field.
Next we apply the Ampère-Maxwell law (with $I=0$ ) over the same two faces (Surface 3 and then Surface 2) of the rectangular box of [link] . Applying [link] ,
to Surface 3, and then to Surface 2, yields the two equations
These equations describe the spatially dependent B field produced by the time-dependent E field.
We next combine the equations showing the changing B field producing an E field with the equation showing the changing E field producing a B field. Taking the derivative of [link] with respect to x and using [link] gives
This is the form taken by the general wave equation for our plane wave. Because the equations describe a wave traveling at some as-yet-unspecified speed c , we can assume the field components are each functions of x – ct for the wave traveling in the + x -direction, that is,
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