Q.Discuss diffraction at a grating and obtain the condition for the $m$th maximum.

A diffraction grating is a plane transparent sheet ruled with a very large number of closely, equally spaced opaque lines (typically thousands per centimetre), so that the unruled gaps between the lines act as a huge number of identical, evenly-spaced parallel slits. If $a$ is the width of each transparent slit and $b$ the width of each opaque ruling, the grating element is $e=a+b$ (the spacing between corresponding points on adjacent slits). Light diffracted from all the slits at angle $\theta$ to the grating normal interferes; the path difference between corresponding points of adjacent slits is $(a+b)\sin\theta$, and a bright maximum (of order $m$) forms wherever this equals a whole number of wavelengths: $(a+b)\sin\theta=m\lambda$, $m=0,1,2,\ldots$, with $m=0$ giving the central (zero-order) maximum straight ahead. Writing $N=1/(a+b)$ for the number of grating elements (lines) per unit width -- the value usually printed on a commercial grating -- this becomes $\sin\theta=Nm\lambda$; despite its superficial resemblance to the single-slit minimum formula $a\sin\theta=n\lambda$, the grating formula is a condition for a maximum with order $m$, not a minimum with order $n$, and the two must not be confused. This grating equation, together with a spectrometer, is used experimentally to measure an unknown wavelength: the collimator produces a parallel beam through the gr …