In a source-free vacuum region, write the two Maxwell curl equations that allow electromagnetic waves.

Use Maxwell's equations to state the vacuum speed of electromagnetic waves.

A plane wave in vacuum has \(E_0=180\,\mathrm{V\,m^{-1}}\). Find \(B_0\).

A wave has \(\vec E\) along \(+z\) and travels in the \(+x\) direction. Determine the direction of \(\vec B\).

Use \(\mu_0=4\pi\times10^{-7}\,\mathrm{N\,A^{-2}}\) and \(\epsilon_0=8.85\times10^{-12}\,\mathrm{F\,m^{-1}}\) to estimate \(c\), keeping the unit interpretation explicit.

A nonmagnetic dielectric has relative permittivity \(\epsilon_r=4.0\). Use Maxwell's wave-speed result to find the EM wave speed in the dielectric.

For a wave moving in \(+x\), take \(E_y=E_0\cos(kx-\omega t)\). Use Faraday's law to find the required form and sign of \(B_z\).

For the same wave \(E_y=E_0\cos(kx-\omega t)\), use the Ampere-Maxwell law to recover the speed condition.

A proposed vacuum wave has \(\vec E=E_0\cos(kx-\omega t)\hat x\). Explain why it cannot be a source-free plane electromagnetic wave traveling in \(x\).

A plane wave in vacuum is claimed to have \(E/B=2.4\times10^8\,\mathrm{m\,s^{-1}}\). Give two possible interpretations and decide whether it is a valid vacuum plane wave.

Derive the one-dimensional wave equation for \(E_y(x,t)\) from \(\partial E_y/\partial x=-\partial B_z/\partial t\) and \(-\partial B_z/\partial x=\mu_0\epsilon_0\partial E_y/\partial t\).

A medium has \(\epsilon=9\epsilon_0\) and \(\mu=4\mu_0\). Determine the wave speed and the field ratio \(E/B\).

Explain why displacement current is not just a mathematical patch if electromagnetic waves exist in vacuum.

A field pair has \(E_y=E_0\cos(kx-\omega t)\) and \(B_z=-B_0\cos(kx-\omega t)\), with positive \(E_0,B_0\). Determine the propagation direction and whether the pair can represent a wave moving in \(+x\).

Show that the electromagnetic wave speed decreases if either \(\epsilon\) or \(\mu\) increases, and interpret the result physically.

A source-free plane wave travels in direction \(\hat n\). Use \(\nabla\cdot\vec E=0\) to argue that \(\vec E\perp\hat n\).

A vacuum plane wave has \(\omega=7.5\times10^{14}\,\mathrm{rad\,s^{-1}}\). Find \(k\), then state what extra condition fixes the direction of \(\vec B\).

A wave in a medium has \(E_0=90\,\mathrm{V\,m^{-1}}\), \(B_0=6.0\times10^{-7}\,\mathrm{T}\), and frequency \(1.0\times10^{14}\,\mathrm{Hz}\). Find its wavelength in the medium.

Construct a consistent set of field directions for a vacuum plane wave traveling in \(-z\) with \(\vec E\) along \(+x\), and justify the magnetic-field direction.

Prove, for the 1D plane-wave ansatz, that Maxwell's equations require \(E\), \(B\), and propagation direction to form a right-handed triad rather than allowing either magnetic-field sign.