A slit of width \(a\) is illuminated by light of wavelength \(\lambda\). The linear separation between \(1^{st}\) and \(3^{rd}\) minima in the diffraction pattern produced on a screen placed at a distance \(D\) from the slit system is _____.

The position of the \(n^{th}\) minimum in a single slit diffraction pattern on a screen at distance \(D\) is given by \(y_n = \dfrac{n D \lambda}{a}\).

For the \(1^{st}\) minimum, \(n = 1\), so \(y_1 = \dfrac{D \lambda}{a}\).

For the \(3^{rd}\) minimum, \(n = 3\), so \(y_3 = \dfrac{3 D \lambda}{a}\).

The linear separation between the \(1^{st}\) and \(3^{rd}\) minima on the same side of the central maximum is \(\Delta y = y_3 - y_1\).

\(\Delta y = \dfrac{3 D \lambda}{a} - \dfrac{D \lambda}{a} = 2\dfrac{D \lambda}{a}\).

Answer: \(2\dfrac{D\lambda}{a}\)