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Electromagnetic Waves

Field: Electrodynamics

Waves that are created as a result of vibrations between an electric field and a magnetic field.

Sequence of Expressions

The Electromagnetic Field is represented by the four-potential Aμ=(ϕ/c,A⃗)A^\mu = (\phi/c, \vec{A}). The field strength tensor FμνF_{\mu\nu} is defined as the exterior derivative of this potential:\n\nFμν=∂μAν−∂νAμF_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu\n\nThe physical fields are extracted from the components of this tensor:\n\nE⃗=−∇ϕ−∂A⃗∂t\vec{E} = -\nabla \phi - \frac{\partial \vec{A}}{\partial t} \nB⃗=∇×A⃗\vec{B} = \nabla \times \vec{A} \n\nIn a vacuum, these fields must satisfy the wave equation, ensuring that changes in the distribution of charge or current propagate at the speed of light cc: \n\n(∇2−1c2∂2∂t2)E⃗=0\left( \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \right) \vec{E} = 0
In the vacuum (ρ=0,J⃗=0\rho=0, \vec{J}=0), the propagation of the electric field E⃗(r⃗,t)\vec{E}(\vec{r}, t) and the magnetic field B⃗(r⃗,t)\vec{B}(\vec{r}, t) is governed by the homogeneous Maxwell's equations. The resulting wave equation for both fields is given by the d'Alembertian operator □2\Box^2: \n\n□2E⃗=(∇2−1c2∂2∂t2)E⃗=0\Box^2 \vec{E} = \left( \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \right) \vec{E} = 0 \n\n□2B⃗=(∇2−1c2∂2∂t2)B⃗=0\Box^2 \vec{B} = \left( \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \right) \vec{B} = 0 \n\nWhere c=1/ϵ0μ0c = 1/\sqrt{\epsilon_0 \mu_0} is the speed of light in vacuum, and ∇2\nabla^2 is the Laplacian operator. These equations imply that the fields satisfy the general form of a hyperbolic partial differential equation, characterizing wave propagation in a source-free medium.
In the context of classical electrodynamics, the speed of light cc in a vacuum is derived from the constitutive relations governing the electromagnetic field tensor FμνF^{\mu \nu}. Specifically, cc is defined by the relationship between the vacuum permittivity ϵ0\epsilon_0 and the vacuum permeability μ0\mu_0, which are fundamental constants derived from the vacuum Maxwell equations. The magnitude of the propagation speed of any electromagnetic wave is given by:\n\nc=1ϵ0μ0c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}\n\nFurthermore, this speed dictates the dispersion relation for the wave vector k⃗\vec{k} and angular frequency ω\omega of a plane wave propagating in vacuum, satisfying:\n\nω2=c2k⃗2\omega^2 = c^2 \vec{k}^2\n\nThis relationship confirms that cc represents the maximum signal velocity in the vacuum spacetime manifold, consistent with the structure of the Minkowski metric.
The relationship between the electric field E⃗\vec{E} and the magnetic field B⃗\vec{B} in a vacuum is defined by their mutual coupling in Maxwell's equations. They propagate as transverse waves where:\n\n1. The fields are mutually orthogonal: E⃗⋅B⃗=0\vec{E} \cdot \vec{B} = 0\n2. The ratio of their magnitudes is constant: c=∣E⃗∣∣B⃗∣c = \frac{|\vec{E}|}{|\vec{B}|}\n3. They are in phase and perpendicular to the direction of propagation k⃗\vec{k}.\n\nThe instantaneous energy flux density is given by the Poynting vector:\n\nS⃗=1μ0(E⃗×B⃗)\vec{S} = \frac{1}{\mu_0} (\vec{E} \times \vec{B})
Maxwell's Equations constitute a system of coupled partial differential equations governing the behavior of the electric field E⃗\vec{E} and the magnetic field B⃗\vec{B} in a vacuum. In SI units:\n\n**1. Gauss's Law for Electricity:**\n∇⋅E⃗=ρϵ0\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}\n\n**2. Gauss's Law for Magnetism:**\n∇⋅B⃗=0\nabla \cdot \vec{B} = 0\n\n**3. Faraday's Law of Induction:**\n∇×E⃗=−∂B⃗∂t\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}\n\n**4. Ampère-Maxwell Law:**\n∇×B⃗=μ0J⃗+μ0ϵ0∂E⃗∂t\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}
The differential form of Faraday's Law, derived from Maxwell's equations, relates the curl of the electric field E⃗\vec{E} to the time rate of change of the magnetic field B⃗\vec{B}. For any point in space, the following vector identity holds:\n\n∇×E⃗=−∂B⃗∂t\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}\n\nWhere:\n* E⃗(r⃗,t)\vec{E}(\vec{r}, t) is the electric field vector, measured in Volts per meter (V/m).\n* B⃗(r⃗,t)\vec{B}(\vec{r}, t) is the magnetic field vector, measured in Tesla (T).\n* ∇×E⃗\nabla \times \vec{E} is the curl operator applied to E⃗\vec{E}, representing the rotational component of the electric field.\n* ∂B⃗∂t\frac{\partial \vec{B}}{\partial t} is the partial time derivative of the magnetic field, quantifying the rate of change of the magnetic flux density at a fixed spatial point.
In the context of classical electrodynamics, Ampère-Maxwell's Law is stated in differential form as:\n\n∇×B⃗=μ0J⃗+μ0ϵ0∂E⃗∂t\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t} \n\nWhere:\n* B⃗\vec{B} is the magnetic field vector (T\text{T}).\n* ∇×\nabla \times is the curl operator.\n* μ0\mu_0 is the permeability of free space (H/m\text{H/m}).\n* J⃗\vec{J} is the conduction current density vector (A/m2\text{A/m}^2).\n* ϵ0\epsilon_0 is the permittivity of free space (F/m\text{F/m}). \n* ∂E⃗∂t\frac{\partial \vec{E}}{\partial t} is the time rate of change of the electric field vector (V/m/s\text{V/m/s}). \n\nThis equation mathematically asserts that the circulation of the magnetic field around an infinitesimal loop is proportional to the sum of the conduction current passing through the loop and the displacement current generated by the changing electric field.
The Principle of Least Time asserts that the path r⃗(s)\vec{r}(s) taken by an electromagnetic wave propagating from point AA to point BB in a medium characterized by a spatially dependent refractive index n(r⃗)n(\vec{r}) is the path that extremizes the travel time functional TT. This minimization is formulated via the calculus of variations:\n\nT[r⃗(s)]=∫ABn(r⃗)c(dxds)2+(dyds)2+(dzds)2dsT[\vec{r}(s)] = \int_{A}^{B} \frac{n(\vec{r})}{c} \sqrt{\left(\frac{dx}{ds}\right)^2 + \left(\frac{dy}{ds}\right)^2 + \left(\frac{dz}{ds}\right)^2} ds\n\nThe path r⃗(s)=(x(s),y(s),z(s))\vec{r}(s) = (x(s), y(s), z(s)) must satisfy the condition that the variation of the time functional vanishes: \n\nδT=δ∫ABn(r⃗)cds=0\delta T = \delta \int_{A}^{B} \frac{n(\vec{r})}{c} ds = 0\n\nThis leads to the Euler-Lagrange equations for the components of the path, where the Lagrangian density is L=n(r⃗)cx˙2+y˙2+z˙2L = \frac{n(\vec{r})}{c} \sqrt{\dot{x}^2 + \dot{y}^2 + \dot{z}^2} (using x˙=dx/ds\dot{x} = dx/ds for simplicity in the functional form). The resulting differential equations define the trajectory of the wave front, which are equivalent to the eikonal equation derived from the wave equation in the paraxial approximation.