Question

Discuss the conservation of energy and linear momentum for a macroscopic system of sources and electromagnetic fields in a uniform, isotropic medium described by a permittivity \(\epsilon\) and a permeability \(\mu\). Show that in a straightforward calculation the energy density, Poynting vector, field- momentum density, and Maxwell stress tensor are given by the Minkowski expressions, $$ \begin{aligned} u &=\frac{1}{2}\left(\epsilon E^{2}+\mu H^{2}\right) \\ \mathbf{S} &=\mathbf{E} \times \mathbf{H} \\ \mathbf{g} &=\mu \in \mathbf{E} \times \mathbf{H} \\ T_{d} &=\left[\epsilon E_{i} E_{j}+\mu H_{i} H_{j}-\frac{1}{2} \delta_{i j}\left(\epsilon E^{2}+\mu H^{2}\right)\right] \end{aligned} $$ What modifications arise if \(\epsilon\) and \(\mu\) are functions of position?

Short answer

Energy and momentum expressions are constant for uniform media; variability adds gradient terms.

Step-by-step solution

Define Energy Conservation

Energy conservation in an electromagnetic system involves the use of the energy density \( u \) and the Poynting vector \( \mathbf{S} \). The energy density \( u \) for an electromagnetic field is given by \( u = \frac{1}{2}(\epsilon E^2 + \mu H^2) \), representing the energy stored per unit volume. The Poynting vector \( \mathbf{S} = \mathbf{E} \times \mathbf{H} \) represents the rate of energy flow per unit area.

Define Momentum Conservation

Momentum conservation in the system is described using the field-momentum density \( \mathbf{g} = \mu \epsilon \mathbf{E} \times \mathbf{H} \). This provides the amount of momentum stored in the fields. Additionally, the Maxwell stress tensor \( T_{d} \) describes the force exerted by the electromagnetic fields and is expressed as \( T_{d} = \epsilon E_i E_j + \mu H_i H_j - \frac{1}{2} \delta_{ij}(\epsilon E^2 + \mu H^2) \).

Interpret the Minkowski Expressions

The expressions provided (Minkowski's formulation) are specifically attributed to a medium characterized by constant \( \epsilon \) and \( \mu \). Each term accounts for how energy and momentum are stored and transferred in the medium, with energy density and momentum density tied through electromagnetic fields, and the Poynting vector indicating directional energy flow.

Consider Variable Permittivity and Permeability

When \( \epsilon \) and \( \mu \) are functions of position, additional terms involving the gradients of \( \epsilon \) and \( \mu \) appear in the conservation equations. This position-dependence affects how energy and momentum are distributed and transferred within the medium, leading to within the expressions, gradient \\( abla \epsilon \times E \) and \( abla \mu \times H \) potentially appearing in the modified equations.

Key concepts

Energy Conservation

Energy conservation in the realm of electromagnetic fields is about managing and accounting for where energy exists and moves. Imagine energy as a flow, like water in a stream. With electromagnetic fields, this stream is described using energy density and the Poynting vector.
Energy density \(u\) captures how much energy the electromagnetic fields store in a given space. This is important because fields can store energy just as a battery does, owing to their electric \(E\) and magnetic \(H\) components. The equation \(u = \frac{1}{2}(\epsilon E^2 + \mu H^2)\) articulates this storage per unit volume, depending on the medium's permittivity \(\epsilon\) and permeability \(\mu\).

On the flip side, the energy doesn't merely stay put. The Poynting vector \(\mathbf{S} = \mathbf{E} \times \mathbf{H}\) tells us the rate at which energy flows through space, like how fast the water flows down the stream. This vector is crucial because it describes the directional flow of energy, providing insights into how electromagnetic waves propagate through space.

• Energy Density \(u\): Where energy is stored.
• Poynting Vector \(\mathbf{S}\): How and where energy flows.

Understanding these concepts helps us visualize and calculate how electromagnetism manages energy, critical for everything from understanding light spread to managing wireless signals.

Momentum Conservation

Momentum conservation in electromagnetic fields works by analyzing how momentum is contained and transferred by the fields. This is similar to how physical objects store and transfer momentum when they move or collide.
The field-momentum density \(\mathbf{g} = \mu \epsilon \mathbf{E} \times \mathbf{H}\) indicates how much momentum is packed into a given space by the electromagnetic fields. It gives a grasp of how electromagnetic forces can push or pull on charges and currents within a material.

Moreover, the Maxwell stress tensor is like a detailed map indicating how forces are distributed in fields. Expressed as \(T_{d} = \epsilon E_i E_j + \mu H_i H_j - \frac{1}{2} \delta_{ij}(\epsilon E^2 + \mu H^2)\), it shows the stress (or force) that comes into play in the system. This tensor impacts how electromagnetic forces structure objects at their core, even affecting their physical forms in processes like magnetostriction.

• Field-Momentum Density \(\mathbf{g}\): Momentum stored in the fields.
• Maxwell Stress Tensor \(T_{d}\): Distribution of force within the fields.

These tools allow us to decipher how electromagnetic fields exert forces and propel movements, enriching our grasp of interactions in both natural and engineered systems.

Poynting Vector

The Poynting vector is a cornerstone of electromagnetic theory, named after British physicist John Henry Poynting. It represents the directional energy flux, or power flow, of an electromagnetic field, and is a vector quantity.

Mathematically defined as \(\mathbf{S} = \mathbf{E} \times \mathbf{H}\), this vector tells us about the rate at which energy is transported through a unit area. In practical terms, the Poynting vector conveys how energy flows in a medium, such as inside transmission lines, or spread across space, like in the case of radio waves.
For engineers and physicists, understanding this flow is key to designing systems that efficiently transmit and receive energy, like antennas and wireless networks. It also helps in placing equipment safely, as it directs where the energy or signal strength will be most concentrated.

• Poynting Vector \(\mathbf{S}\): Describes power flow direction and magnitude.

This vector is not just theoretical but foundational to innovations in technology, guiding how waves should be harnessed or avoided.

Minkowski Expressions

The Minkowski expressions provide a framework for understanding the dynamics of electromagnetic fields in media that are homogeneous and isotropic, meaning they have uniform properties in all directions. This theoretical approach was named after Hermann Minkowski, who contributed significantly to the formulation of electromagnetic theory.
Minkowski's description handles how energy and momentum are deployed and accounted for in such mediums, with expressions for energy density, momentum density, the Poynting vector, and the Maxwell stress tensor fitted into this perspective. When \(\epsilon\) and \(\mu\) (permittivity and permeability) remain constant, computations are straightforward. However, if these properties change across different positions in the material, additional complexities arise.

This position-dependence might lead to terms like \( abla \epsilon \times E \) and \( abla \mu \times H \) cropping up, reflecting the influence of spatial variation on energy and momentum flow. Such modifications complicate the original equations, showing how field characteristics and interactions shift in a non-homogeneous medium.

• Minkowski Framework: A model for consistent media interactions.
• Position-Dependent Variations: Additional complexities in non-uniform settings.

By mastering these expressions, we gain a rigorous method for assessing how fields behave in various materials, aiding in predicting interactions within complex environments.