To solve this problem, we need to understand the diffraction pattern produced by a single slit.$\\$When light passes through a single slit of width $a,$ it creates a diffraction pattern on a screen placed at a distance $D$ from the slit.$\\$The central maximum is the brightest and widest part of the diffraction pattern.$\\$The angular width of the central maximum is given by the condition for the first minimum on either side of the central maximum.$\\$The condition for the first minimum in single-slit diffraction is:$\\ a \sin \theta = \lambda \\$For small angles, $\sin \theta \approx \theta,$ so we have:$\\ a \theta = \lambda \\$Therefore, $\theta = \frac{\lambda}{a} \\$The width of the central maximum on the screen is given by the angle $2\theta$ (since it extends equally on both sides of the central axis).$\\$Thus, the linear width of the central maximum on the screen is:$\\$Width$= 2D \theta = 2D \left(\frac{\lambda}{a}\right) \\$Simplifying, we get:$\\$Width$= \frac{2D\lambda}{a} \\$Therefore, the width of the central maximum is $\frac{2D\lambda}{a},$ which corresponds to Option 4.