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Unformatted text preview: UAHuntsville Spring, 2010 PH 113: Homework Solutions No. 6 Solution 35: The dispersion of a grating is given by D = d / d , where is the angular position of a line associated with wavelength . The angular position and wavelength are related by d sin = m , where d is the slit separation (which we made boldfaced in order not to confuse it with the d used in the derivative, below) and m is an integer. We differentiate this expression with respect to to obtain d d m d cos , = or D d d m = = d cos . Now m = ( d / ) sin , so D = = d d sin tan . cos Solution 36: (a) From the expression for the half-width hw (given by hw = N d cos ( ) ) and that for the resolving power R (given by R = N m ), we find the product of hw and R to be hw R = N d cos N m = m d cos = d sin d cos = tan where we used m = d sin . (b) For first order m = 1, so the corresponding angle 1 satisfies d sin 1 = m = . Thus the product in question is given by ( ) ( ) ( ) 1 1 1 2 2 2 1 1 1 2 sin sin 1 1 tan cos 1 sin 1/ sin 1 / 1 1 0.89....
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