Editing Fresnel equations (section)

=== Equal permittivities ===
Although it is not encountered in practice, the equations can also apply to the case of two media with a common permittivity but different refractive indices due to different permeabilities. From equations ({{EquationNote|4}}) and ({{EquationNote|5}}), if {{math|''ϵ''}} is fixed instead of {{math|''μ''}}, then {{mvar|Y}} becomes ''inversely'' proportional to {{mvar|n}}, with the result that the subscripts 1 and 2 in equations ({{EquationNote|29}}) to ({{EquationNote|38}}) are interchanged (due to the additional step of multiplying the numerator and denominator by {{math|''n''<sub>1</sub>''n''<sub>2</sub>}}). Hence, in ({{EquationNote|29}}) and ({{EquationNote|31}}), the expressions for {{math|''r''<sub>s</sub>}} and {{math|''r''<sub>p</sub>}} in terms of refractive indices will be interchanged, so that Brewster's angle ({{EquationNote|39}}) will give {{math|''r''<sub>s</sub> {{=}} 0}} instead of {{math|''r''<sub>p</sub> {{=}} 0}}, and any beam reflected at that angle will be p-polarized instead of s-polarized.<ref>More general Brewster angles, for which the angles of incidence and refraction are not necessarily complementary, are discussed in C.L.&nbsp;Giles and W.J.&nbsp;Wild, [http://clgiles.ist.psu.edu/pubs/brewster-magnetic.pdf "Brewster angles for magnetic media"], ''International Journal of Infrared and Millimeter Waves'', vol.{{tsp}}6, no.{{tsp}}3 (March 1985), pp.{{tsp}}187–97.</ref> Similarly, Fresnel's sine law will apply to the p polarization instead of the s polarization, and his tangent law to the s polarization instead of the p polarization.

This switch of polarizations has an analog in the old mechanical theory of light waves (see ''[[#History|§{{nnbsp}}History]]'', above). One could predict reflection coefficients that agreed with observation by supposing (like Fresnel) that different refractive indices were due to different ''densities'' and that the vibrations were ''normal'' to what was then called the [[plane of polarization]], or by supposing (like [[James MacCullagh|MacCullagh]] and [[Franz Ernst Neumann|Neumann]]) that different refractive indices were due to different ''elasticities'' and that the vibrations were ''parallel'' to that plane.<ref>Whittaker, 1910, pp. 133, 148–149; Darrigol, 2012, pp. 212, 229–231.</ref> Thus the condition of equal permittivities and unequal permeabilities, although not realistic, is of some historical interest.