Plane Electromagnetic Waves II

Faraday's law is applied to a rectangular wavefront traveling in the x-direction to derive the relationship between electric and magnetic fields. The electric field contributes a non-zero value to the line integral along one side of the rectangle, while the magnetic field's time-varying flux through the rectangle satisfies the law. This relationship shows that electric and magnetic field magnitudes are directly connected through the propagation speed of electromagnetic waves.

When applying Faraday's law to a rectangular wavefront, the electric field integral produces a non-zero value along one side. To satisfy Faraday's law, a corresponding non-zero time derivative of magnetic flux must exist. This requires a magnetic field component in the z-direction perpendicular to both the electric field and the wave's propagation direction, maintaining the wave's consistency with electromagnetic theory.

As the plane wavefront travels a distance c dt in time dt, it sweeps an area equal to ac dt across the rectangle. During this motion, the magnetic flux through the rectangle increases because the wavefront carries the magnetic field into the region previously outside it. This increasing magnetic flux creates the time derivative required by Faraday's law to establish the E-B relationship.

The relationship between electric field E, magnetic field B, and wave speed c emerges directly from applying Faraday's law to the moving wavefront. The substitution of the magnetic flux change rate into Faraday's law yields an expression showing that the wave is consistent with the law only if wave speed and the perpendicular field magnitudes are related through a specific mathematical equation derived from the geometry and physics of the wavefront.

The electric field is zero along the side of the rectangle lying outside the wavefront, contributing nothing to the integral. Along the two perpendicular sides, the electric field is perpendicular to the length elements, so their dot product is zero. Only along one side inside the wavefront is the electric field parallel to the length, producing a non-zero contribution to the counterclockwise line integral around the rectangle.

The area vector points in the positive z-direction, perpendicular to the xy-plane containing the rectangle. This orientation determines the direction of the magnetic flux through the rectangle and ensures consistency with the right-hand rule for Faraday's law. The z-directed area vector aligns with the required magnetic field component, allowing the time-varying flux to produce the non-zero right-hand side of Faraday's law equation.

The rectangle's geometry isolates the relevant field components: one side captures the electric field contribution, while the area swept by the moving wavefront quantifies the magnetic flux change. By choosing height a and tracking the wavefront's motion over time dt, the geometry creates a clear mathematical framework connecting the electric field integral to the magnetic flux derivative, enabling direct derivation of the E-B relationship.