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Figure shows a rectangular box of dimensions l by l by delta x. The top and bottom sides, parallel to the xz plane are labeled side 2 and side 1 respectively. The front and back sides, parallel to the xy plane are labeled side 3 and side 4 respectively. Three arrows originate from a point on the left side. These are along the x, y and z axis and are respectively labeled E subscript x parentheses x, t parentheses, E subscript y parentheses x, t parentheses and E subscript z parentheses x, t parentheses. Three more arrows originate from the point where the x axis intersects the right side of the box. These, too, are along the the x, y and z axis and are respectively labeled E subscript x parentheses x plus delta x, t parentheses, E subscript y parentheses x plus delta x, t parentheses and E subscript z parentheses x plus delta x, t parentheses.
The surface of a rectangular box of dimensions l × l × Δ x is our Gaussian surface. The electric field shown is from an electromagnetic wave propagating along the x -axis.

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.

The speed of propagation of electromagnetic waves

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

E · d s = E y ( x , t ) l + E y ( x + Δ x , t ) l .

Assuming Δ x is small and approximating E y ( x + Δ x , t ) by

E y ( x + Δ x , t ) = E y ( x , t ) + E y ( x , t ) x Δ x ,

we obtain

E · d s = E y ( x , t ) x ( l Δ x ) .
Figure shows a rectangular box of dimensions l by l by delta x. The top and bottom sides, parallel to the xz plane are labeled side 2 and side 1 respectively. The front and back sides, parallel to the xy plane are labeled side 3 and side 4 respectively. The boundary of side 3 is labeled integration path. Two arrows along side 3, pointing upwards in the positive y direction are labeled E subscript y parentheses x,t parentheses and E subscript y parentheses x plus delta x,t parentheses. An arrow on the left side of the box, pointing in the positive z direction is labeled B subscript z parentheses x plus delta x by 2,t parentheses.
We apply Faraday’s law to the front of the rectangle by evaluating E · d s along the rectangular edge of Side 3 in the direction indicated, taking the B field crossing the face to be approximately its value in the middle of the area traversed.

Because Δ 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 ( x + Δ x 2 , t ) . The flux of the B field through Face 3 is then the B field times the area,

S B · n d A = B z ( x + Δ x 2 , t ) ( l Δ x ) .

From Faraday’s law,

E · d s = d d t S B · n d A .

Therefore, from [link] and [link] ,

E y ( x , t ) x ( l Δ x ) = t [ B z ( x + Δ x 2 , t ) ] ( l Δ x ) .

Canceling l Δ x and taking the limit as Δ x = 0 , we are left with

E y ( x , t ) x = B z ( x , t ) t .

We could have applied Faraday’s law instead to the top surface (numbered 2) in [link] , to obtain the resulting equation

E z ( x , t ) x = B y ( x , t ) t .

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] ,

B · d s = μ 0 ε 0 ( d / d t ) S E · n d a

to Surface 3, and then to Surface 2, yields the two equations

B y ( x , t ) x = ε 0 μ 0 E z ( x , t ) t , and
B z ( x , t ) x = ε 0 μ 0 E y ( x , t ) t .

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

2 E y x 2 = x ( E y x ) = x ( B z t ) = t ( B z x ) = t ( ε 0 μ 0 E y t ) or
2 E y x 2 = ε 0 μ 0 2 E y t 2 .

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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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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