In figure \(A\) and \(B\) represent the edges of the slit \(AB\) of width \(a\) and \(C\) represents the midpoint of the slit. For the first minimum at \(P\),

\(a \sin \theta=\lambda\)   ...... \((i)\)

where \(\lambda\) is the wavelength of light. The path difference between the wavelets from \(A\) to \(C\) is

\(\Delta x=\frac{a}{2} \sin \theta=\frac{1}{2}(a \sin \theta)=\frac{\lambda}{2}\)   (using \((i)\))

The corresponding phase difference \(\Delta \phi\) is

\(\Delta \phi=\frac{2 \pi}{\lambda} \Delta x=\frac{2 \pi}{\lambda} \times \frac{\lambda}{2}=\pi\)