(a) Evaluate the energy density \(W_{1}\) and the Poynting vector \(s\) for the simple plane wave of Sec. \(8.3\) to show that the electric and magnetic energy densities are equal and that $$ |\mathbf{s}|=c\left(\boldsymbol{W}_{1}\right)_{\text {field }} $$ (b) Show that the time-average Poynting vector is $$ \mathrm{s}=\frac{E_{0}{ }^{2}}{2 Z_{0}} \hat{\mathrm{k}}=\frac{1}{2} H_{0}{ }^{2} Z_{0} \hat{\mathrm{k}}, $$ where \(E_{0}\) and \(H_{0}=B_{0} / \mu_{0}\) are the (peak) field amplitudes, \(Z_{0}\) is the wave impedance given by \((8.3 .10)\), and \(\hat{k}\) is a unit vector in the direction of propagation \((c)\) Evaluate the field-dependent terms in (8.4.15) for a plane wave to show that the time-average force density is $$ \vec{F}_{1}=-\nabla\left(\bar{W}_{1}\right) \text { field } $$ which is a simple generalization of the result quoted in (8.4.20).

Understanding the concept of energy density is crucial when studying electromagnetic waves, like plane waves. Energy density refers to the amount of energy stored in a given volume of space. For electromagnetic fields, we have two types of energy densities: electric energy density, represented as \( W_E \) which involves the electric field \( \mathbf{E} \), and magnetic energy density, denoted as \( W_H \) which involves the magnetic field \( \mathbf{H} \).

The formula for electric energy density is \( W_E = \frac{1}{2} \epsilon_0 E^2 \) and for magnetic energy density \( W_H = \frac{1}{2 \mu_0} H^2 \). In the context of a simple plane wave, these quantities become particularly significant when we consider that, due to the wave impedance relation, the electric and magnetic energy densities are equal. This equilibrium plays a pivotal role in the stable propagation of plane waves through space.

For a student working through an exercise involving energy densities, it's helpful to remember that for a plane wave, the magnetic field \( \mathbf{H} \) and the electric field \( \mathbf{E} \) are related by the medium's intrinsic impedance, leading to equal energy densities. This harmonious relationship not only simplifies calculations but also reveals a deeper symmetry in electromagnetic wave phenomena.

A plane wave in electromagnetism is an idealized wave that travels through space with uniform amplitude and phase fronts that are infinitely extended, flat planes. For students tackling problems involving plane waves, visualize these waves as infinite sheets of energy moving in a single direction, characterized by a wave vector \( \hat{k} \) that indicates their direction of propagation.

In mathematical terms, a plane wave can be expressed as \( \mathbf{E}(\mathbf{r},t) = \mathbf{E}_0 \sin(kz - \omega t) \) for the electric field and analogously for the magnetic field. Here, \( \mathbf{E}_0 \) and \( \mathbf{H}_0 \) represent the maximum amplitudes of the fields, while the term \( kz - \omega t \) ensures that the wave's shape is maintained over both space and time.

It's this simplicity that makes plane waves such a fundamental model; they provide a clear and manageable way of understanding wave behavior without the complexity found in real-world scenarios. For exercises involving plane waves, focusing on their defining characteristics – uniform amplitude and phase, as well as the wave impedance relationship – can greatly facilitate comprehension.

Force density is a concept that describes the amount of force exerted per unit volume within a field. In electromagnetic theory, this provides insight into how the electromagnetic fields exert forces on charged particles or on distributions of charge. The term \( \vec{F}_{1} \) represents force density and can be seen in the context of the Poynting vector and energy density as a way to quantify the momentum change within an electromagnetic field.

In the exercise provided, the Poynting vector represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. By considering the plane wave solution, we find that the time-average force density is linked to the gradient of the averaged energy density \( \bar{W}_1 \) within the field. The equation \( \bar{\mathbf{F}}_1 = -abla(\bar{W}_{1})_{\text {field}} \) indicates that force density is directly related to the spatial variations of the energy density.

For a student, it is essential to grasp that force density is an abstract concept manifesting how electromagnetic fields influence their surroundings. It's not only about how strong the fields are but also about how those fields change in space. Making the connection between force density, energy density, and the Poynting vector can help students develop an integrated understanding of the dynamics within electromagnetic fields.