Verify by substitution that the following equations are...

Verify by substitution that the following equations are solutions to Equations 24.16 and 24.17 , respectively:
$$\begin{array}{l}E=E_{\max } \cos (k x-\omega t) \\B=B_{\max } \cos (k x-\omega t)\end{array}$$

Key Concepts

Verification by Substitution

This process involves taking a proposed solution to a differential equation and inserting it back into the equation to see if it holds true. It is a direct method to confirm that the function satisfies the required relationships of the equation, demonstrating that the solution is valid.

Electromagnetic Waves

Electromagnetic waves are solutions to Maxwell’s equations that describe how electric (E) and magnetic (B) fields propagate through space. Their wave-like behavior is characterized by sinusoidal variations in space and time, and these variations satisfy specific differential equations derived from Maxwell’s framework.

Maxwell’s Equations

Maxwell’s equations are the fundamental laws governing electricity and magnetism. They relate the time-varying electric and magnetic fields to each other and to their sources. In free space, these equations can be combined to yield wave equations for the fields.

Plane Wave Solutions

Plane wave solutions, such as the cosine functions in the given equations, represent waves that propagate in a specific direction with uniform amplitude and phase fronts. They are ideal representations used to model electromagnetic waves in free space or homogeneous media.

Dispersion Relation

The dispersion relation is a fundamental aspect of wave phenomena that relates the angular frequency (?) to the wave number (k). This relation is obtained when the trial solution is inserted into the wave equation and is essential for ensuring consistency between the mathematical description of the wave and its physical propagation properties.

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