Write a sinusoidal electric field for a wave moving in \(+x\) with \(\vec E\) along \(+y\).

A sinusoidal wave has \(E_0=75\,\mathrm{V\,m^{-1}}\). Find \(B_0\) in vacuum.

A sinusoidal wave has \(\lambda=600\,\mathrm{nm}\). Find \(f\), \(k\), and \(\omega\) in vacuum.

For \(E_y=120\cos(5x-1.5\times10^9t)\), find the speed and magnetic-field amplitude.

For \(E_y=E_0\cos(kx-\omega t)\), find the phase change when \(x\) increases by \(\lambda/4\) at fixed time.

For \(E_y=E_0\cos(kx-\omega t)\), find the phase change after time \(T/4\) at fixed position.

A wave is \(E_y=80\cos(4x-1.2\times10^9t+\pi/3)\). Find \(E_y\) at \(x=0\), \(t=0\).

A wave has \(E_y=E_0\cos(kx+\omega t)\). If \(\vec E\) points \(+y\) when the cosine is positive, what direction must \(\vec B\) point?

A proposed wave has \(E_y=E_0\cos(kx-\omega t)\) and \(B_z=B_0\sin(kx-\omega t)\). Explain whether this is a standard traveling vacuum plane wave.

Given \(E_y=E_0\cos(kx-\omega t)\), write a consistent \(B_z\) and state the required amplitude relation.

Derive the relation \(E_0/B_0=\omega/k\) for \(E_y=E_0\cos(kx-\omega t)\) and \(B_z=B_0\cos(kx-\omega t)\).

A sinusoidal wave in a medium is \(E_y=60\cos(2.0\times10^7x-4.0\times10^{15}t)\). Find the refractive index of the medium.

A detector at fixed \(x\) sees adjacent electric-field maxima separated by \(2.0\,\mathrm{fs}\). Find the wavelength in vacuum.

At one instant, adjacent zero crossings of \(E_y\) along \(x\) are separated by \(0.25\,\mathrm{m}\). Find \(\lambda\), \(k\), and \(f\) in vacuum.

A wave has \(E_y=E_0\cos(kx-\omega t)\). Find the velocity of the point where the phase equals \(\pi/3\).

A student writes \(E_y=E_0\cos(kx-\omega t)\) and \(B_z=(E_0/c)\cos(kx+\omega t)\). Identify the inconsistency.

Show that the time-average of \(E_y\) over one cycle is zero but the time-average of \(E_y^2\) is not.

A sinusoidal wave has measured magnetic field \(B_z=B_0\cos(kx-\omega t+\pi)\) while \(E_y=E_0\cos(kx-\omega t)\). Could it still travel in \(+x\)? Explain with the Poynting direction.

Design a sinusoidal vacuum plane wave traveling in \(-y\) with electric field along \(+z\). Give \(\vec E\), \(\vec B\), and the amplitude relation.

Starting from a sinusoidal ansatz, prove that an arbitrary phase offset between \(\vec E\) and \(\vec B\) is not allowed for a source-free vacuum traveling plane wave.