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Chapter 3 Electromagnetic Fields

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Chapter 3 Electromagnetic Fields 3 Electromagnetic Fields 3.1. Maxwell's equations Electromagnetism Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and 50 Chapter 3 Electromagnetic Fields communications technologies. Maxwell's equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents. They are named after the physicist and mathematician James Clerk Maxwell, who published an early form of those equations between 1861 and 1862. The equations have two major variants. The "microscopic" set of Maxwell's equations uses total charge and total current, including the complicated charges and currents in materials at the atomic scale; it has universal applicability but may be infeasible to calculate. The "macroscopic" set of Maxwell's equations defines two new auxiliary fields that describe large-scale behaviour without having to consider these atomic scale details, but it requires the use of parameters characterizing the electromagnetic properties of the relevant materials. The term "Maxwell's equations" is often used for other forms of Maxwell's equations. For example, space-time formulations are commonly used in high energy and gravitational physics. These formulations, defined on space-time rather than space and time separately, are manifestly [44] compatible with special and general relativity. In quantum mechanics and analytical mechanics, versions of Maxwell's equations based on the electric and magnetic potentials are preferred. Since the mid-20th century, it has been understood that Maxwell's equations are not exact but are a classical field theory approximation to the more accurate and 51 Chapter 3 Electromagnetic Fields fundamental theory of quantum electrodynamics. In many situations, though, deviations from Maxwell's equations are immeasurably small. Exceptions include nonclassical light, photon-photon scattering, quantum optics, and many other phenomena related to photons or virtual photons[44]. 52 Chapter 3 Electromagnetic Fields 3.2. Formulation in terms of electric and magnetic fields The powerful and most widely familiar form of Maxwell's equations, whose formulation is due to Oliver Heaviside, in the vector calculus formalism, is used throughout unless otherwise explicitly stated. Symbols in bold represent vector quantities, and symbols in italics represent scalar quantities, unless otherwise indicated. The equations introduce the electric field E, a vector field, and the magnetic field B, a pseudovector field, where each generally have time-dependence. The sources of these fields are electric charges and electric currents, which can be expressed as local densities namely charge density ρ and current density J. A separate law of nature, the Lorentz force law, describes how the electric and magnetic field act on charged particles and currents. A version of this law was included in the original equations by Maxwell but, by convention, is no longer. In the electric and magnetic field formulation there are four equations. Two of them describe how the fields vary in space due to sources, if any; electric fields emanating from electric charges in Gauss's law, and magnetic fields as closed field lines not due to magnetic monopoles in Gauss's law for magnetism. The other two describe how the fields "circulate" around their respective sources; the magnetic 53 Chapter 3 Electromagnetic Fields field "circulates" around electric currents and time varying electric fields in Ampère's law with Maxwell's addition, while the electric field "circulates" around time varying magnetic fields in Faraday's law. The precise formulation of Maxwell's equations depends on the precise definition of the quantities involved. Conventions differ with the unit systems, because various definitions and dimensions are changed by absorbing dimensionful factors like the speed of light c. This makes constants come out differently. 3.3. Conventional formulation in SI units The equations in this section are given in the convention used with SI units. Other units commonly used are Gaussian units based on the cgs system, [44] Lorentz– Heaviside units (used mainly in particle physics), and Planck units (used in theoretical physics). See below for the formulation with Gaussian units. Name Integral equations Differential equations Meaning The electric field Gauss's leaving a volume is ∯ ∰ proportional to the law charge inside. There are no magnetic magnetis monopoles; the total Gauss's ∯ magnetic flux law piercing a closed surface is zero. Maxwell– The voltage Faraday accumulated around equation a closed circuit is ∮ ∯ (Faraday' proportional to the s law of time rate of change induction) of the magnetic flux 54 Chapter 3 Electromagnetic Fields it encloses. Electric currents and Ampère's changes in electric circuital fields are law (with proportional to the Maxwell's magnetic field addition) circulating about the area they pierce. where the universal constants appearing in the equations are the permittivity of free space ε0 and the permeability of free space μ0. In the differential equations, a local description of the fields, the nabla symbol ∇ denotes the three-dimensional gradient operator, and from it the divergence operator is ∇· the curl operator is ∇×. The sources are taken to be the electric charge density (charge per unit volume) ρ and the electric current density (current per unit area) J. In the integral equations, a description of the fields within a region of space, Ω is any fixed volume with boundary surface ∂Ω, and 55 Chapter 3 Electromagnetic Fields Σ is any fixed open surface with boundary curve ∂Σ, is a surface integral over the surface ∂Ω (the oval indicates the surface is closed and not open), is a volume integral over the volume Ω, is a surface integral over the surface Σ, is a line integral around the curve ∂Σ (the circle indicates the curve is closed). Here "fixed" means the volume or surface do not change in time. Although it is possible to formulate Maxwell's equations with time-dependent surfaces and volumes, this is not actually necessary: the equations are correct and complete with time-independent surfaces. The sources are correspondingly the total amounts of charge and current within these volumes and surfaces, found by integration. The volume integral of the total charge density ρ over any fixed volume Ω is the total electric charge contained in Ω: where dV is the differential volume element, and 56 Chapter 3 Electromagnetic Fields the net electric current is the surface integral of the electric current density J, passing through any open fixed surface Σ: where dS denotes the differential vector element of surface area S normal to surface Σ. (Vector area is also denoted by A rather than S, but this conflicts with the magnetic potential, a separate vector field). The "total charge or current" refers to including free and bound charges, or free and bound currents. These are used in the macroscopic formulation below[2]. 3.4. Relationship between differential and integral formulations The differential and integral formulations of the equations are mathematically equivalent, by the divergence theorem in the case of Gauss's law and Gauss's law for magnetism, and by the Kelvin–Stokes theorem in the case of Faraday's law and Ampère's law. Both the differential and integral formulations are useful. The integral formulation can often be used to simplify and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential formulation is a more natural starting point for calculating the fields in more complicated (less symmetric) situations, for example using finite element analysis[45]. 57 Chapter 3 Electromagnetic Fields 3.5. Flux and divergence Closed volume Ω and its boundary ∂Ω, enclosing a source (+) and sink (−) of a vector field F. Here, F could be the E field with source electric charges, but not the B field which has no magnetic charges as shown. The outward unit normal is n. The "fields emanating from the sources" can be inferred from the surface integrals of the fields through the closed surface ∂Ω, defined as the electric flux and magnetic flux , as well as their respective divergences ∇ · E and ∇ · B. These surface integrals and divergences are connected by the divergence theorem. 58 Chapter 3 Electromagnetic Fields 3.6. Circulation and curl Open surface Σ and boundary ∂Σ. F could be the E or B fields. Again, n is the unit normal. (The curl of a vector field doesn't literally look like the "circulations", this is a heuristic depiction). The "circulation of the fields" can be interpreted from the line integrals of the fields around the closed curve ∂Σ: where dℓ is the differential vector element of path length tangential to the path/curve, as well as their curls: These line integrals and curls are connected by Stokes' theorem, and are analogous to quantities in classical fluid dynamics: the circulation of a fluid is the line 59 Chapter 3 Electromagnetic Fields integral of the fluid's flow velocity field around a closed loop, and the vorticity of the fluid is the curl of the velocity field. 3.7. Time evolution The "dynamics" or "time evolution of the fields" is due to the partial derivatives of the fields with respect to time: These derivatives are crucial for the prediction of field propagation in the form of electromagnetic waves. Since the surface is taken to be time-independent, we can make the following transition in Faraday's law: 3.8. Conceptual descriptions Gauss's law Gauss's law describes the relationship between a static electric field and the electric charges that cause it: The static electric field points away from positive charges and towards negative charges. In the field line description, electric field lines begin only at positive electric charges and end only at negative electric charges.
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