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In some cases permittivity of substance turns out to be a complex or a negative quantity, and refractive index, respectively, a complex (n' = n+ ix) or an imaginary (n' = ix) quantity. Write the equation of a plane wave for both of these cases and find out the physical meaning of such refractive indices. |
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Answer» Solution :Let us write the solutions of the wave equation in the form `A = A_(0) e^(i(omegat - kx))` where `k = (2pi)/(lambda)` and `lambda` is the WAVELENGTH in the medium. If `n' = n+ ichi`, then `k = (2pi)/(lambda_(0))n'` `(lambda_(0)` is the wavelength in vaccume) and the equation BECOMES `A = A_(0) e^(chi'x) `EXP `(i(omegat_(1) - k' x))` where `chi' = (2pi)/(lambda_(0)) chi` and `k' = (2pi)/(lambda_(0)) n`. In real form, `A = A_(0) e^(chi'x) cos (omegat - k' x)` This REPRESENTS a plane wave whose amplitude diminished as it propagates to the right (privided `chi' lt 0)`. when `n' = i chi`, the similarly `A = A_(0)e^(chi' x) cos omega t` (on putting `n = 0` in the above equation). This represents a STANDING wave whose amplitude diminishes as one goes to the right (if `chi' lt 0`). The wavelength of the wave is infinite `(k' = 0`). Waves of the former type are realized inside metals as well as inside dielectrics when there is total reflection. (pentration of wave). |
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