A single slit of width \(b\) is illuminated by a coherent monochromatic light of wavelength \(\lambda \). If the second and fourth minima in the diffraction pattern at a distance \(1\,m\) from the slit are at \(3\, cm\) and \(6\, cm\) respectively from the central maximum, what is the width of the central maximum ?......\(cm\) (i.e. distance between first minimum on either side of the central maximum)

\(b \sin \theta=n \lambda \Rightarrow \sin \theta=\frac{n \lambda}{b}\)

Distance of \(\mathrm{n}^{\text {th }}\) secondary minima \(\mathrm{x}=\mathrm{D} \sin\, \theta\)

or \(\sin \theta_{1}=\frac{x_{1}}{D}\)

\(\sin \theta_{1}=\frac{2 \lambda}{b}\)

\(n=4\)

\(\sin \theta_{2}=\frac{4 \lambda}{b}=\frac{x_{2}}{D}\)

\(x_{2}-x_{1}=\frac{4 \lambda}{b}-\frac{2 \lambda}{b}=\frac{2 \lambda}{b}\)

\(3=\frac{2 \lambda}{b} \Rightarrow b=\frac{2 \lambda}{3}\)  ...... \((i)\)

Width of central maxima \(=\frac{2 \lambda}{b}\)

\(=\frac{2 \lambda}{\frac{2 \lambda}{3}}=3 \mathrm{cm}\)    ...  from eq. \((i)\)