Plane Electromagnetic Waves

In reality, the E and B fields (electric and magnetic fields) in space must be in closed loops. So when we write an equation like E = E_msin(kx-w t), it cannot be exactly correct. Nevertheless, it can be very close, and this is a useful approach to the theory.

Before you read this, you should know a little calculus and know about the energy per volume in E and B fields, item #7 in my thing on fields.

Consider a wave going in the x direction and E in the y direction. Later we will find that E and B are in phase, and that light travels in the direction of ExB, so B is in the z direction. We look at the equation, ò E× ds = -¶ f _B /¶ t, where the integral is around a closed loop in the counterclockwise direction, and the minus sign is irrelevant to our derivation. On the left we integrate around a tiny rectangle of height y and width ¶ x (using partial derivative terminology because it is a function of x only). Start at the lower left. The first part of the integral is zero, because E is in the y direction and ds is in the x direction. Then we have (E + ¶ E)y, then zero, then Ey, so the total is (¶ E)y. On the right-hand side we have area y¶ x times -¶ B/¶ t, with the minus because if the fields are positive and increasing with x, they must be decreasing in time (picture the wave moving to the right and you will see). So

¶ E/¶ x = - ¶ B/¶ t (equation 1)

Note that if we wanted a wave moving to the left we would have a positive sign in the equation above. Another equation to work with is ò B× ds = m _o e _o ¶ f _E/¶ t, and with identical reasoning we obtain

¶ B/¶ x = - m _o e _o ¶ E/¶ t (equation 2)

(And again it would be + if moving to the left.) Now take the partial derivative of equation 1 with respect to t and 2 w.r.t. x, equate ¶ ²E/¶ t¶ x and ¶ ²E/¶ x¶ t and get

¶ ²B/¶ x² = m _o e _o ¶ ²B/¶ t² (equation 3)

Similarly, take the partial of 1 wrt x and 2 wrt t (wrt=with respect to) and show

¶ ²E/¶ x² = m _o e _o ¶ ²E/¶ t² (equation 4)

(A math note: mixed partials like ¶ ²E/¶ t¶ x and ¶ ²E/¶ x¶ t are equal where the function is continuous, which is the case for these fields.)

Equations 3 and 4 show that the wave is a combination of E and B, both moving together with speed (m _o e _o )^-1/2, and if you don't know why, go back at the speed of light to my main page on waves. If you do know, go back there anyway.

Or go to one of the other main physics pages:
Mechanics
Fluids, heat, electricity and magnetism
Quantum

Or my alphabetized junk at Index.

 or send comments & questions to fredrick.gram