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Interdependence of Electric and Magnetic Fields

It has been known for a long time that a changing magnetic field induces and electric field and vice versa. An electromagnetic waves consists of electric fields and magnetic fields constantly changing and in a sense giving rise to each other. This means that electic and magnetic fields are interdependent.

Suppose we have a magnetic field which varies in time:

then

and similarly for an electric field

We can obtain an expression for

Similarly,

Maxwell's equationbecomes

Substitutingandgives

A second relationship can be similarly be found to write the equationasor

Take the dot product ofandto give

The left hand side can be written

sobut the dot product is even soandis perpendicular tofor an electromagnetic wave.