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  "body": "\n## Example\n\nLet $S$ be the smooth closed surface of a perfectly conducting body in $\\R^3$, with outward unit normal $n$.\nLet a time-harmonic incident field $E^{\\mathrm{inc}}$ with time factor $e^{-i\\omega t}$ illuminate the body in a medium with permittivity $\\varepsilon$ and permeability $\\mu$, and let $k = \\omega \\sqrt{\\varepsilon \\mu}$.\nLet $G(x, y) = \\frac{e^{ik\\abs{x - y}}}{4\\pi \\abs{x - y}}$.\nThe unknown is the tangential surface current density $J$ on $S$.\nThe electric field integral equation is\n$$n(x) \\times E^{\\mathrm{inc}}(x) = -i\\omega\\mu \\, n(x) \\times \\left[\\int_S G(x, y) J(y) \\, dS(y) + \\frac{1}{k^2} \\nabla_x \\int_S G(x, y) \\, (\\nabla_S \\cdot J)(y) \\, dS(y)\\right], \\quad x \\in S.$$\nThe unknown $J$ appears only under the integrals, so it is an [[integral equation of the first kind|ZDS77Q0F06]].\n\nCheck: the current $J$ radiates the scattered field $E^{\\mathrm{s}} = i\\omega A - \\nabla \\Phi$.\nHere $A = \\mu \\int_S G J \\, dS$ is the vector potential and $\\Phi = \\frac{1}{\\varepsilon} \\int_S G \\sigma \\, dS$ the scalar potential.\nThe charge density is $\\sigma = \\frac{1}{i\\omega} \\nabla_S \\cdot J$ by the continuity equation.\nSince $\\frac{1}{i\\omega\\varepsilon} = \\frac{-i\\omega\\mu}{k^2}$, this gives $E^{\\mathrm{s}} = i\\omega\\mu \\left[\\int_S G J \\, dS + \\frac{1}{k^2} \\nabla \\int_S G \\, \\nabla_S \\cdot J \\, dS\\right]$.\nOn a perfect conductor the tangential total field vanishes, $n \\times (E^{\\mathrm{inc}} + E^{\\mathrm{s}}) = 0$ on $S$, which is the equation above.\n\n## Remarks\n\nThe equation is uniquely solvable except at the countably many wavenumbers $k$ at which the interior of $S$ has a nonzero cavity resonance field.\nNear these wavenumbers numerical solutions become unstable; combining it with the magnetic field integral equation removes the problem.\nDiscretized with the method of moments, it is a standard tool for computing radar scattering and antenna fields.\n\n## Explanations\n\n### L1 human\n\nWhen a radio wave hits a metal object, it drives electric currents on the surface, and these currents send out the reflected wave.\nOn metal the total electric field along the surface must be zero.\nThis condition gives an equation for the unknown surface currents, in which each current element contributes through an integral over the whole surface.\n",
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