Show that the magnetic field Bz(x, t) in aplane...

Show that the magnetic field $B_{z}(x, t)$ in aplane electromagnetic wave propagating in the $+x$ -direction must satisfy Eq. $(32.15) .$ (Hint . Take the partial derivative of Eq. $(32.12)$ with respect to $t$ and the partial derivative of Eq. $(32.14)$ with respect to $x .$ Then combine the results.)

Key Concepts

Use of Partial Differentiation

The technique of taking partial derivatives is crucial when working with partial differential equations like the wave equation. In this derivation, differentiating with respect to time and space allows one to combine and eliminate certain field components, thereby isolating the behavior of the magnetic field component and demonstrating that it satisfies the wave equation.

Plane Electromagnetic Waves

These are solutions to Maxwell's equations in which the electric and magnetic fields propagate as waves in a specific direction, with their magnitudes and phases varying sinusoidally in space and time. In a plane wave propagating in the +x-direction, for example, the fields depend only on x and t, and satisfy the derived wave equations.

Derivation of the Wave Equation

By taking appropriate spatial and temporal derivatives of Maxwell's equations, one can combine them to yield a second-order partial differential equation known as the wave equation. This process involves differentiating one field component with respect to time and another with respect to space and then eliminating intermediate variables, thereby revealing the wave nature of the fields.

Maxwell's Equations

These are the fundamental laws governing electromagnetism, relating electric and magnetic fields to their sources and to one another. In free space, they simplify to forms that directly relate the spatial and temporal derivatives of the fields, allowing the derivation of wave equations for each field component.

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