Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Electromagnetic wave equation
open-in-new
<h2 id="the-origin-of-the-electromagnetic-wave-equation">The origin of the electromagnetic wave equation</h2> <p>In his 1865 paper titled <a href="/facts/A_Dynamical_Theory_of_the_Electromagnetic_Field/zbe0J5rW">A Dynamical Theory of the Electromagnetic Field</a>, <a href="/facts/James_Clerk_Maxwell/jtYyze61">James Clerk Maxwell</a> utilized the correction to Ampère's circuital law that he had made in part III of his 1861 paper <a href="/facts/On_Physical_Lines_of_Force/nQoSbLhu">On Physical Lines of Force</a>. In <i>Part VI</i> of his 1864 paper titled <i>Electromagnetic Theory of Light</i>,<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> Maxwell combined displacement current with some of the other equations of electromagnetism and he obtained a wave equation with a speed equal to the speed of light. He commented: </p> <blockquote><p>The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></p></blockquote> <p>Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics education by a much less cumbersome method involving combining the corrected version of Ampère's circuital law with <a href="/facts/Faraday%2527s_law_of_induction/PFBreB0F">Faraday's law of induction</a>. </p><p>To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. In a vacuum- and charge-free space, these equations are: </p><p> ∇ ⋅ E = 0 ∇ × E = − ∂ B ∂ t ∇ ⋅ B = 0 ∇ × B = μ 0 ε 0 ∂ E ∂ t {\displaystyle {\begin{aligned}\nabla \cdot \mathbf {E} &=0\\\nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}}\\\nabla \cdot \mathbf {B} &=0\\\nabla \times \mathbf {B} &=\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\\\end{aligned}}} </p><p>These are the general Maxwell's equations specialized to the case with charge and current both set to zero. Taking the <a href="/facts/Curl_(mathematics)/GuS8dc7o">curl</a> of the curl equations gives: </p><p> ∇ × ( ∇ × E ) = ∇ × ( − ∂ B ∂ t ) = − ∂ ∂ t ( ∇ × B ) = − μ 0 ε 0 ∂ 2 E ∂ t 2 ∇ × ( ∇ × B ) = ∇ × ( μ 0 ε 0 ∂ E ∂ t ) = μ 0 ε 0 ∂ ∂ t ( ∇ × E ) = − μ 0 ε 0 ∂ 2 B ∂ t 2 {\displaystyle {\begin{aligned}\nabla \times \left(\nabla \times \mathbf {E} \right)&=\nabla \times \left(-{\frac {\partial \mathbf {B} }{\partial t}}\right)=-{\frac {\partial }{\partial t}}\left(\nabla \times \mathbf {B} \right)=-\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}\\\nabla \times \left(\nabla \times \mathbf {B} \right)&=\nabla \times \left(\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)=\mu _{0}\varepsilon _{0}{\frac {\partial }{\partial t}}\left(\nabla \times \mathbf {E} \right)=-\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}\end{aligned}}} </p><p>We can use the <a href="/facts/Vector_calculus_identities/QWAaYA7t">vector identity</a> </p><p> ∇ × ( ∇ × V ) = ∇ ( ∇ ⋅ V ) − ∇ 2 V {\displaystyle \nabla \times \left(\nabla \times \mathbf {V} \right)=\nabla \left(\nabla \cdot \mathbf {V} \right)-\nabla ^{2}\mathbf {V} } </p><p>where V is any vector function of space. And </p><p> ∇ 2 V = ∇ ⋅ ( ∇ V ) {\displaystyle \nabla ^{2}\mathbf {V} =\nabla \cdot \left(\nabla \mathbf {V} \right)} </p><p>where ∇V is a <a href="/facts/Dyadics/FoLJnrwR">dyadic</a> which when operated on by the divergence operator ∇ ⋅ yields a vector. Since </p><p> ∇ ⋅ E = 0 ∇ ⋅ B = 0 {\displaystyle {\begin{aligned}\nabla \cdot \mathbf {E} &=0\\\nabla \cdot \mathbf {B} &=0\end{aligned}}} </p><p>then the first term on the right in the identity vanishes and we obtain the wave equations: </p><p> 1 c 0 2 ∂ 2 E ∂ t 2 − ∇ 2 E = 0 1 c 0 2 ∂ 2 B ∂ t 2 − ∇ 2 B = 0 {\displaystyle {\begin{aligned}{\frac {1}{c_{0}^{2}}}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}-\nabla ^{2}\mathbf {E} &=\mathbf {0} \\{\frac {1}{c_{0}^{2}}}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}-\nabla ^{2}\mathbf {B} &=\mathbf {0} \end{aligned}}} </p><p>where </p><p> c 0 = 1 μ 0 ε 0 = 2.99792458 × 10 8 m/s {\displaystyle c_{0}={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}=2.99792458\times 10^{8}\;{\textrm {m/s}}} </p><p>is the speed of light in free space. </p> <h2 id="covariant-form-of-the-homogeneous-wave-equation">Covariant form of the homogeneous wave equation</h2> <p>These <a href="/facts/Formulation_of_Maxwell%2527s_equations_in_special_relativity/pM1kDz2n">relativistic equations</a> can be written in <a href="/facts/Covariance_and_contravariance_of_vectors/s3HGMkhz">contravariant</a> form as </p><p> ◻ A μ = 0 {\displaystyle \Box A^{\mu }=0} </p><p>where the <a href="/facts/Electromagnetic_four-potential/S8gTFNAt">electromagnetic four-potential</a> is </p><p> A μ = ( ϕ c , A ) {\displaystyle A^{\mu }=\left({\frac {\phi }{c}},\mathbf {A} \right)} </p><p>with the <a href="/facts/Lorenz_gauge_condition/Uh8N3S1o">Lorenz gauge condition</a>: </p><p> ∂ μ A μ = 0 , {\displaystyle \partial _{\mu }A^{\mu }=0,} </p><p>and where </p><p> ◻ = ∇ 2 − 1 c 2 ∂ 2 ∂ t 2 {\displaystyle \Box =\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}} </p><p>is the <a href="/facts/D%2527Alembert_operator/hCzCvJyD">d'Alembert operator</a>. </p> <h2 id="homogeneous-wave-equation-in-curved-spacetime">Homogeneous wave equation in curved spacetime</h2> <p class="note">Main article: <a href="/facts/Maxwell%2527s_equations_in_curved_spacetime/ofggksaC">Maxwell's equations in curved spacetime</a></p> <p>The electromagnetic wave equation is modified in two ways, the derivative is replaced with the <a href="/facts/Covariant_derivative/sRGN0Xh6">covariant derivative</a> and a new term that depends on the curvature appears. </p><p> − A α ; β ; β + R α β A β = 0 {\displaystyle -{A^{\alpha ;\beta }}_{;\beta }+{R^{\alpha }}_{\beta }A^{\beta }=0} </p><p>where R α β {\displaystyle {R^{\alpha }}_{\beta }} is the <a href="/facts/Ricci_curvature_tensor/qujou7QS">Ricci curvature tensor</a> and the semicolon indicates covariant differentiation. </p><p>The generalization of the <a href="/facts/Lorenz_gauge_condition/Uh8N3S1o">Lorenz gauge condition</a> in curved spacetime is assumed: </p><p> A μ ; μ = 0. {\displaystyle {A^{\mu }}_{;\mu }=0.} </p> <h2 id="inhomogeneous-electromagnetic-wave-equation">Inhomogeneous electromagnetic wave equation</h2> <p class="note">Main article: <a href="/facts/Inhomogeneous_electromagnetic_wave_equation/CAYHP6ku">Inhomogeneous electromagnetic wave equation</a></p> <p>Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. Maxwell's equations can be written in the form of a wave equation with sources. The addition of sources to the wave equations makes the <a href="/facts/Partial_differential_equations/DyPsBlv7">partial differential equations</a> inhomogeneous. </p> <h2 id="solutions-to-the-homogeneous-electromagnetic-wave-equation">Solutions to the homogeneous electromagnetic wave equation</h2> <p class="note">Main article: <a href="/facts/Wave_equation/J2ksD2Az">Wave equation</a></p> <p>The general solution to the electromagnetic wave equation is a <a href="/facts/Superposition_principle/GKGs9C92">linear superposition</a> of waves of the form </p><p> E ( r , t ) = g ( ϕ ( r , t ) ) = g ( ω t − k ⋅ r ) B ( r , t ) = g ( ϕ ( r , t ) ) = g ( ω t − k ⋅ r ) {\displaystyle {\begin{aligned}\mathbf {E} (\mathbf {r} ,t)&=g(\phi (\mathbf {r} ,t))=g(\omega t-\mathbf {k} \cdot \mathbf {r} )\\\mathbf {B} (\mathbf {r} ,t)&=g(\phi (\mathbf {r} ,t))=g(\omega t-\mathbf {k} \cdot \mathbf {r} )\end{aligned}}} </p><p>for virtually any well-behaved function g of dimensionless argument φ, where ω is the <a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a> (in radians per second), and k = (<i>kx</i>, <i>ky</i>, <i>kz</i>) is the <a href="/facts/Wave_vector/Jk8wUEft">wave vector</a> (in radians per meter). </p><p>Although the function g can be and often is a monochromatic <a href="/facts/Sine_wave/mD1CB09M">sine wave</a>, it does not have to be sinusoidal, or even periodic. In practice, g cannot have infinite periodicity because any real electromagnetic wave must always have a finite extent in time and space. As a result, and based on the theory of <a href="/facts/Fourier_transform/XXhKJtc8">Fourier decomposition</a>, a real wave must consist of the superposition of an infinite set of sinusoidal frequencies. </p><p>In addition, for a valid solution, the wave vector and the angular frequency are not independent; they must adhere to the <a href="/facts/Dispersion_relation/oo7RM5eg">dispersion relation</a>: </p><p> k = | k | = ω c = 2 π λ {\displaystyle k=|\mathbf {k} |={\omega \over c}={2\pi \over \lambda }} </p><p>where k is the <a href="/facts/Wavenumber/cLaLAeW5">wavenumber</a> and λ is the <a href="/facts/Wavelength/kcJU0ymz">wavelength</a>. The variable c can only be used in this equation when the electromagnetic wave is in a vacuum. </p> <h3>Monochromatic, sinusoidal steady-state</h3> <p>The simplest set of solutions to the wave equation result from assuming sinusoidal waveforms of a single frequency in separable form: </p><p> E ( r , t ) = ℜ { E ( r ) e i ω t } {\displaystyle \mathbf {E} (\mathbf {r} ,t)=\Re \left\{\mathbf {E} (\mathbf {r} )e^{i\omega t}\right\}} </p><p>where </p> <ul><li>i is the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a>,</li> <li><i>ω</i> = 2<i>π</i> <i>f</i> is the <a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a> in <a href="/facts/Radians_per_second/cG0plKHD">radians per second</a>,</li> <li> <i>f</i> is the <a href="/facts/Frequency/QUSbPaW8">frequency</a> in <a href="/facts/Hertz/6Wshj5qm">hertz</a>, and</li> <li> e i ω t = cos ( ω t ) + i sin ( ω t ) {\displaystyle e^{i\omega t}=\cos(\omega t)+i\sin(\omega t)} is <a href="/facts/Euler%2527s_formula/Wq7VXIV0">Euler's formula</a>.</li></ul> <h3>Plane wave solutions</h3> <p class="note">Main article: <a href="/facts/Sinusoidal_plane-wave_solutions_of_the_electromagnetic_wave_equation/yr0KlGER">Sinusoidal plane-wave solutions of the electromagnetic wave equation</a></p> <p>Consider a plane defined by a unit normal vector </p><p> n = k k . {\displaystyle \mathbf {n} ={\mathbf {k} \over k}.} </p><p>Then planar traveling wave solutions of the wave equations are </p><p> E ( r ) = E 0 e − i k ⋅ r B ( r ) = B 0 e − i k ⋅ r {\displaystyle {\begin{aligned}\mathbf {E} (\mathbf {r} )&=\mathbf {E} _{0}e^{-i\mathbf {k} \cdot \mathbf {r} }\\\mathbf {B} (\mathbf {r} )&=\mathbf {B} _{0}e^{-i\mathbf {k} \cdot \mathbf {r} }\end{aligned}}} </p><p>where r = (<i>x</i>, <i>y</i>, <i>z</i>) is the position vector (in meters). </p><p>These solutions represent planar waves traveling in the direction of the normal vector n. If we define the z direction as the direction of n, and the x direction as the direction of E, then by Faraday's Law the magnetic field lies in the y direction and is related to the electric field by the relation </p><p> c 2 ∂ B ∂ z = ∂ E ∂ t . {\displaystyle c^{2}{\partial B \over \partial z}={\partial E \over \partial t}.} </p><p>Because the divergence of the electric and magnetic fields are zero, there are no fields in the direction of propagation. </p><p>This solution is the linearly <a href="/facts/Polarization_(waves)/fP33m0eD">polarized</a> solution of the wave equations. There are also circularly polarized solutions in which the fields rotate about the normal vector. </p> <h3>Spectral decomposition</h3> <p>Because of the linearity of Maxwell's equations in a vacuum, solutions can be decomposed into a superposition of <a href="/facts/Sine/gQ6LXK8z">sinusoids</a>. This is the basis for the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> method for the solution of differential equations. The sinusoidal solution to the electromagnetic wave equation takes the form </p><p> E ( r , t ) = E 0 cos ( ω t − k ⋅ r + ϕ 0 ) B ( r , t ) = B 0 cos ( ω t − k ⋅ r + ϕ 0 ) {\displaystyle {\begin{aligned}\mathbf {E} (\mathbf {r} ,t)&=\mathbf {E} _{0}\cos(\omega t-\mathbf {k} \cdot \mathbf {r} +\phi _{0})\\\mathbf {B} (\mathbf {r} ,t)&=\mathbf {B} _{0}\cos(\omega t-\mathbf {k} \cdot \mathbf {r} +\phi _{0})\end{aligned}}} </p><p>where </p> <ul><li>t is time (in seconds),</li> <li>ω is the <a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a> (in radians per second),</li> <li>k = (<i>kx</i>, <i>ky</i>, <i>kz</i>) is the <a href="/facts/Wave_vector/Jk8wUEft">wave vector</a> (in radians per meter), and</li> <li> ϕ 0 {\displaystyle \phi _{0}} is the <a href="/facts/Phase_(waves)/cpGtuVll">phase angle</a> (in radians).</li></ul> <p>The wave vector is related to the angular frequency by </p><p> k = | k | = ω c = 2 π λ {\displaystyle k=|\mathbf {k} |={\omega \over c}={2\pi \over \lambda }} </p><p>where k is the <a href="/facts/Wavenumber/cLaLAeW5">wavenumber</a> and λ is the <a href="/facts/Wavelength/kcJU0ymz">wavelength</a>. </p><p>The <a href="/facts/Electromagnetic_spectrum/24LARvFY">electromagnetic spectrum</a> is a plot of the field magnitudes (or energies) as a function of wavelength. </p> <h3>Multipole expansion</h3> <p>Assuming monochromatic fields varying in time as e − i ω t {\displaystyle e^{-i\omega t}} , if one uses Maxwell's Equations to eliminate B, the electromagnetic wave equation reduces to the <a href="/facts/Helmholtz_equation/SWfaXrTB">Helmholtz equation</a> for E: </p><p> ( ∇ 2 + k 2 ) E = 0 , B = − i k ∇ × E , {\displaystyle (\nabla ^{2}+k^{2})\mathbf {E} =0,\,\mathbf {B} =-{\frac {i}{k}}\nabla \times \mathbf {E} ,} </p><p>with <i>k</i> = <i>ω</i>/<i>c</i> as given above. Alternatively, one can eliminate E in favor of B to obtain: </p><p> ( ∇ 2 + k 2 ) B = 0 , E = − i k ∇ × B . {\displaystyle (\nabla ^{2}+k^{2})\mathbf {B} =0,\,\mathbf {E} =-{\frac {i}{k}}\nabla \times \mathbf {B} .} </p><p>A generic electromagnetic field with frequency ω can be written as a sum of solutions to these two equations. The <a href="/facts/Helmholtz_equation/SWfaXrTB">three-dimensional solutions of the Helmholtz Equation</a> can be expressed as expansions in <a href="/facts/Spherical_harmonics/bn98Gv2L">spherical harmonics</a> with coefficients proportional to the <a href="/facts/Spherical_Bessel_functions/hSyvFPqC">spherical Bessel functions</a>. However, applying this expansion to each vector component of E or B will give solutions that are not generically divergence-free (∇ ⋅ E = ∇ ⋅ B = 0), and therefore require additional restrictions on the coefficients. </p><p>The multipole expansion circumvents this difficulty by expanding not E or B, but r ⋅ E or r ⋅ B into spherical harmonics. These expansions still solve the original Helmholtz equations for E and B because for a divergence-free field F, ∇2 (r ⋅ F) = r ⋅ (∇2 F). The resulting expressions for a generic electromagnetic field are: </p><p> E = e − i ω t ∑ l , m l ( l + 1 ) [ a E ( l , m ) E l , m ( E ) + a M ( l , m ) E l , m ( M ) ] B = e − i ω t ∑ l , m l ( l + 1 ) [ a E ( l , m ) B l , m ( E ) + a M ( l , m ) B l , m ( M ) ] , {\displaystyle {\begin{aligned}\mathbf {E} &=e^{-i\omega t}\sum _{l,m}{\sqrt {l(l+1)}}\left[a_{E}(l,m)\mathbf {E} _{l,m}^{(E)}+a_{M}(l,m)\mathbf {E} _{l,m}^{(M)}\right]\\\mathbf {B} &=e^{-i\omega t}\sum _{l,m}{\sqrt {l(l+1)}}\left[a_{E}(l,m)\mathbf {B} _{l,m}^{(E)}+a_{M}(l,m)\mathbf {B} _{l,m}^{(M)}\right]\,,\end{aligned}}} </p><p>where E l , m ( E ) {\displaystyle \mathbf {E} _{l,m}^{(E)}} and B l , m ( E ) {\displaystyle \mathbf {B} _{l,m}^{(E)}} are the <i>electric multipole fields of order (l, m)</i>, and E l , m ( M ) {\displaystyle \mathbf {E} _{l,m}^{(M)}} and B l , m ( M ) {\displaystyle \mathbf {B} _{l,m}^{(M)}} are the corresponding <i>magnetic multipole fields</i>, and <i>aE</i>(<i>l</i>, <i>m</i>) and <i>aM</i>(<i>l</i>, <i>m</i>) are the coefficients of the expansion. The multipole fields are given by </p><p> B l , m ( E ) = l ( l + 1 ) [ B l ( 1 ) h l ( 1 ) ( k r ) + B l ( 2 ) h l ( 2 ) ( k r ) ] Φ l , m E l , m ( E ) = i k ∇ × B l , m ( E ) E l , m ( M ) = l ( l + 1 ) [ E l ( 1 ) h l ( 1 ) ( k r ) + E l ( 2 ) h l ( 2 ) ( k r ) ] Φ l , m B l , m ( M ) = − i k ∇ × E l , m ( M ) , {\displaystyle {\begin{aligned}\mathbf {B} _{l,m}^{(E)}&={\sqrt {l(l+1)}}\left[B_{l}^{(1)}h_{l}^{(1)}(kr)+B_{l}^{(2)}h_{l}^{(2)}(kr)\right]\mathbf {\Phi } _{l,m}\\\mathbf {E} _{l,m}^{(E)}&={\frac {i}{k}}\nabla \times \mathbf {B} _{l,m}^{(E)}\\\mathbf {E} _{l,m}^{(M)}&={\sqrt {l(l+1)}}\left[E_{l}^{(1)}h_{l}^{(1)}(kr)+E_{l}^{(2)}h_{l}^{(2)}(kr)\right]\mathbf {\Phi } _{l,m}\\\mathbf {B} _{l,m}^{(M)}&=-{\frac {i}{k}}\nabla \times \mathbf {E} _{l,m}^{(M)}\,,\end{aligned}}} </p><p>where <i>h</i>l(1,2)(<i>x</i>) are the <a href="/facts/Spherical_Bessel_function/hSyvFPqC">spherical Hankel functions</a>, <i>E</i>l(1,2) and <i>B</i>l(1,2) are determined by boundary conditions, and </p><p> Φ l , m = 1 l ( l + 1 ) ( r × ∇ ) Y l , m {\displaystyle \mathbf {\Phi } _{l,m}={\frac {1}{\sqrt {l(l+1)}}}(\mathbf {r} \times \nabla )Y_{l,m}} </p><p>are <a href="/facts/Vector_spherical_harmonics/GRXdSXgm">vector spherical harmonics</a> normalized so that </p><p> ∫ Φ l , m ∗ ⋅ Φ l ′ , m ′ d Ω = δ l , l ′ δ m , m ′ . {\displaystyle \int \mathbf {\Phi } _{l,m}^{*}\cdot \mathbf {\Phi } _{l',m'}d\Omega =\delta _{l,l'}\delta _{m,m'}.} </p><p>The multipole expansion of the electromagnetic field finds application in a number of problems involving spherical symmetry, for example antennae <a href="/facts/Radiation_pattern/5t0QU2mv">radiation patterns</a>, or nuclear <a href="/facts/Gamma_decay/5Drf913J">gamma decay</a>. In these applications, one is often interested in the power radiated in the <a href="/facts/Near_and_far_field/wS2Dp5lv">far-field</a>. In this regions, the E and B fields asymptotically approach </p><p> B ≈ e i ( k r − ω t ) k r ∑ l , m ( − i ) l + 1 [ a E ( l , m ) Φ l , m + a M ( l , m ) r ^ × Φ l , m ] E ≈ B × r ^ . {\displaystyle {\begin{aligned}\mathbf {B} &\approx {\frac {e^{i(kr-\omega t)}}{kr}}\sum _{l,m}(-i)^{l+1}\left[a_{E}(l,m)\mathbf {\Phi } _{l,m}+a_{M}(l,m)\mathbf {\hat {r}} \times \mathbf {\Phi } _{l,m}\right]\\\mathbf {E} &\approx \mathbf {B} \times \mathbf {\hat {r}} .\end{aligned}}} </p><p>The angular distribution of the time-averaged radiated power is then given by </p><p> d P d Ω ≈ 1 2 k 2 | ∑ l , m ( − i ) l + 1 [ a E ( l , m ) Φ l , m × r ^ + a M ( l , m ) Φ l , m ] | 2 . {\displaystyle {\frac {dP}{d\Omega }}\approx {\frac {1}{2k^{2}}}\left|\sum _{l,m}(-i)^{l+1}\left[a_{E}(l,m)\mathbf {\Phi } _{l,m}\times \mathbf {\hat {r}} +a_{M}(l,m)\mathbf {\Phi } _{l,m}\right]\right|^{2}.} </p> <h2 id="see-also">See also</h2> <h3>Theory and experiment</h3> <table><tbody><tr><td><ul><li><a href="/facts/Maxwell%2527s_equations/LfJ1drHS">Maxwell's equations</a></li><li><a href="/facts/Wave_equation/J2ksD2Az">Wave equation</a></li><li><a href="/facts/Partial_differential_equation/DyPsBlv7">Partial differential equation</a></li><li><a href="/facts/Computational_electromagnetics/uJKnwCar">Computational electromagnetics</a></li><li><a href="/facts/Electromagnetic_radiation/Ynx7ujCW">Electromagnetic radiation</a></li><li><a href="/facts/Charge_conservation/WGQXmoE8">Charge conservation</a></li><li><a href="/facts/Light/jeofpMn4">Light</a></li></ul></td><td><ul><li><a href="/facts/Electromagnetic_spectrum/24LARvFY">Electromagnetic spectrum</a></li><li><a href="/facts/Optics/p0vq4S1T">Optics</a></li><li><a href="/facts/Special_relativity/vsFFc63E">Special relativity</a></li><li><a href="/facts/General_relativity/jVsoIg8X">General relativity</a></li><li><a href="/facts/Inhomogeneous_electromagnetic_wave_equation/CAYHP6ku">Inhomogeneous electromagnetic wave equation</a></li><li><a href="/facts/Photon_polarization/fh6oqC5E">Photon polarization</a></li><li><a href="/facts/Larmor_formula/LGcfxU49">Larmor power formula</a></li></ul></td></tr></tbody></table> <h3>Phenomena and applications</h3> <table><tbody><tr><td><ul><li><a href="/facts/Rainbow/rBvLyu3u">Rainbow</a></li><li><a href="/facts/Cosmic_microwave_background/ThB2HkLS">Cosmic microwave background</a></li><li><a href="/facts/Laser/jk6PTvvT">Laser</a></li><li><a href="/facts/Laser_fusion/67Lyy7nb">Laser fusion</a></li><li><a href="/facts/Photography/ezLPT43M">Photography</a></li><li><a href="/facts/X-ray/wseP79cN">X-ray</a></li><li><a href="/facts/X-ray_crystallography/WtI0F5DN">X-ray crystallography</a></li><li><a href="/facts/Radar/jk9BTbFA">Radar</a></li></ul></td><td><ul><li><a href="/facts/Radio_wave/XmB6WPpF">Radio wave</a></li><li><a href="/facts/Optical_computing/aPl6UjC9">Optical computing</a></li><li><a href="/facts/Microwave/HmVSN31Y">Microwave</a></li><li><a href="/facts/Holography/4J1gW0M6">Holography</a></li><li><a href="/facts/Microscope/ldJp0CI3">Microscope</a></li><li><a href="/facts/Telescope/BDwYahhR">Telescope</a></li><li><a href="/facts/Gravitational_lens/LmxzvLs8">Gravitational lens</a></li><li><a href="/facts/Black-body_radiation/K4YKPKXL">Black-body radiation</a></li></ul></td></tr></tbody></table> <h3>Biographies</h3> <table><tbody><tr><td><ul><li><a href="/facts/Andr%25C3%25A9-Marie_Amp%25C3%25A8re/qk01uas1">André-Marie Ampère</a></li><li><a href="/facts/Albert_Einstein/CNTet8qu">Albert Einstein</a></li><li><a href="/facts/Michael_Faraday/NDcc3lyt">Michael Faraday</a></li><li><a href="/facts/Heinrich_Hertz/sbL2MSpM">Heinrich Hertz</a></li><li><a href="/facts/Oliver_Heaviside/ktYBpAt7">Oliver Heaviside</a></li><li><a href="/facts/James_Clerk_Maxwell/jtYyze61">James Clerk Maxwell</a></li><li><a href="/facts/Hendrik_Lorentz/JbBpKFyr">Hendrik Lorentz</a></li></ul></td></tr></tbody></table> <h2 id="notes">Notes</h2> <h2 id="further-reading">Further reading</h2> <h3>Electromagnetism</h3> <h4>Journal articles</h4> <ul><li>Maxwell, James Clerk, "<i><a href="https://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf">A Dynamical Theory of the Electromagnetic Field</a></i>", Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)</li></ul> <h4>Undergraduate-level textbooks</h4> <ul><li>Griffiths, David J. (1998). <a href="https://archive.org/details/introductiontoel00grif_0"><i>Introduction to Electrodynamics (3rd ed.)</i></a>. Prentice Hall. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-13-805326-X.</li> <li>Tipler, Paul (2004). <i>Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.)</i>. W. H. Freeman. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7167-0810-8.</li> <li>Edward M. Purcell, <i>Electricity and Magnetism</i> (McGraw-Hill, New York, 1985). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-07-004908-4.</li> <li>Hermann A. Haus and James R. Melcher, <i>Electromagnetic Fields and Energy</i> (Prentice-Hall, 1989) <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-13-249020-X.</li> <li>Banesh Hoffmann, <i>Relativity and Its Roots</i> (Freeman, New York, 1983). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7167-1478-7.</li> <li><a href="/facts/David_H._Staelin/B2AvkJth">David H. Staelin</a>, Ann W. Morgenthaler, and Jin Au Kong, <i>Electromagnetic Waves</i> (Prentice-Hall, 1994) <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-13-225871-4.</li> <li>Charles F. Stevens, <i>The Six Core Theories of Modern Physics</i>, (MIT Press, 1995) <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-262-69188-4.</li> <li>Markus Zahn, <i>Electromagnetic Field Theory: a problem solving approach</i>, (John Wiley & Sons, 1979) <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-02198-9</li></ul> <h4>Graduate-level textbooks</h4> <ul><li>Jackson, John D. (1998). <i>Classical Electrodynamics (3rd ed.)</i>. Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-30932-X.</li> <li><a href="/facts/Lev_Davidovich_Landau/lNXQTC49">Landau, L. D.</a>, <i>The Classical Theory of Fields</i> (<a href="/facts/Course_of_Theoretical_Physics/Luk16l8o">Course of Theoretical Physics</a>: Volume 2), (Butterworth-Heinemann: Oxford, 1987). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-08-018176-7.</li> <li>Maxwell, James C. (1954). <i>A Treatise on Electricity and Magnetism</i>. Dover. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-60637-6. {{cite book}}: ISBN / Date incompatibility (help)</li> <li>Charles W. Misner, <a href="/facts/Kip_Thorne/o6wkjTtP">Kip S. Thorne</a>, <a href="/facts/John_Archibald_Wheeler/I1A5t0Fd">John Archibald Wheeler</a>, <i>Gravitation</i>, (1970) W.H. Freeman, New York; <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7167-0344-0. <i>(Provides a treatment of Maxwell's equations in terms of differential forms.)</i></li></ul> <h3>Vector calculus</h3> <ul><li>P. C. Matthews <i>Vector Calculus</i>, Springer 1998, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-76180-2</li> <li>H. M. Schey, <i>Div Grad Curl and all that: An informal text on vector calculus</i>, 4th edition (W. W. Norton & Company, 2005) <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-393-92516-1.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Current practice is to use c0 to denote the speed of light in vacuum according to ISO 31. In the original Recommendation of 1983, the symbol c was used for this purpose. See NIST Special Publication 330, Appendix 2, p. 45 Archived 2016-06-03 at the Wayback Machine <a href="/wiki/ISO_31" target="_blank">/wiki/ISO_31</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Maxwell 1864, page 497. <a href="https://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf" target="_blank">https://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>See Maxwell 1864, page 499. <a href="https://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf" target="_blank">https://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>