Final answer:
The energy density of an electromagnetic wave with a given electric field can be calculated using the equation u = (\epsilon_0 E^2 + B^2 / \mu_0) / 2, where E is the electric field and B is the magnetic field. In free space, we can simplify this to u = \epsilon_0 E^2 / 2, assuming E is given in standard units and that E/B = c, the speed of light.
Step-by-step explanation:
The question is asking to find the energy density of an electromagnetic wave that propagates along the z-direction with a peak electric field component of 4 units (since the precise unit wasn't provided, we will assume standard SI units).
The formula to find the energy density (u), which is the energy per unit volume, for an electromagnetic wave in free space involves both the electric field (E) and the magnetic field (B). Using the equation u = (\epsilon_0 E^2 + B^2 / \mu_0) / 2, where \(\epsilon_0\) is the vacuum permittivity and \(\mu_0\) is the vacuum permeability, you can calculate this value assuming that we know the relationship between the electric and magnetic fields in free space as E/B = c, where c is the speed of light. Without the explicit peak magnetic field from the question, we assume that it is E/c in free space based on the given information.
Assuming the value of the peak electric field (E) is given in volts per meter (V/m) and considering the wave is in a vacuum, you have:
u = \epsilon_0 E^2 / 2.
Inserting the given peak electric field value and the known value of \(\epsilon_0\) (approximately 8.85 x 10^-12 F/m) would yield the energy density. However, since the unit for the given electric field is not specified, we cannot calculate a numerical answer.