Physics Answer the following in Detail

(a) $\lambda_1=500 \mathrm{~nm}=5 \times 10^{-7} \mathrm{~m}$

(b) $\lambda_2=50 \mu \mathrm{m}=5 \times 10^{-5} \mathrm{~m}$

(c) $\lambda_3=0.500 \mathrm{~nm}=5 \times 10^{-10} \mathrm{~m}$

Let a be the slit width.

(a) $\begin{aligned} W & =\frac{y \lambda_1}{a} \\ \therefore a & =\frac{y \lambda_1}{W}=\frac{(2.5)\left(5 \times 10^{-7}\right)}{3 \times 10^{-3}} \\ & =4.167 \times 10^{-4} m \\ & =0.4167 mm \end{aligned}$

(b) $\begin{aligned} W & =\frac{y \lambda_2}{a} \\ \therefore a & =\frac{y \lambda_2}{W}=\frac{(2.5)\left(5 \times 10^{-5}\right)}{3 \times 10^{-3}} \\ & =4.167 \times 10^{-2} m \\ & =41.67 mm \end{aligned}$

(c) $\begin{aligned} W & =\frac{y \lambda_3}{a} \\ \therefore a & =\frac{y \lambda_3}{W}=\frac{(2.5)\left(5 \times 10^{-10}\right)}{3 \times 10^{-3}} \\ & =4.167 \times 10^{-7} m \\ & =4.167 \times 10^{-4} mm \end{aligned}$

The ‘sharpness’ of the central maximum of a diffraction pattern is measured by the angular separation between the centre of the peak and the first minimum. It gives the limit of resolution.

Two overlapping diffraction patterns due to two point sources are not resolved if the angular separation between the central peaks is less than the limit of resolution. They are said to be just separate, or resolved, if the angular separation between the central peaks is equal to the limit of resolution. They are said to be well resolved if the angular separation between the central peaks is more than the limit of resolution.

Resolving power of an optical instrument:
The primary aim of using an optical instrument is to see fine details, whether observing a star system through a telescope or a living cell through a microscope. After passing through an optical system, light from two adjacent parts of the object should produce sharp, distinct (separate) images of those parts. The objective lens or mirror of a telescope or microscope acts like a circular aperture. The diffraction pattern of a circular aperture consists of a central bright spot (called the Airy disc and corresponds to the central maximum) and concentric dark and bright rings.

Light from two close objects or parts of an object after passing through the aperture of an optical system produces overlapping diffraction patterns that tend to obscure the image. If these diffraction patterns are so broad that their central maxima overlap substantially, it is difficult to decide if the intensity distribution is produced by two separate objects or by one.

The resolving power of an optical instrument, e.g., a telescope or microscope, is a measure of its ability to produce detectably separate images of objects that are close together.

Definition : The smallest linear or angular separation between two point objects which appear just resolved when viewed through an optical instrument is called the limit of resolution of the instrument and its reciprocal is called the resolving power of the instrument.

Rayleigh’s criterion for minimum resolution : Two overlapping diffraction patterns due to two point sources are acceptably or just resolved if the centre of the central peak of one diffraction pattern is as far as the first minimum of the other pattern.

The ‘sharpness’ of the central maximum of a diffraction pattern is measured by the angular separation between the centre of the peak and the first minimum. It gives the limit of resolution.

The resolving power of a telescope is defined as the reciprocal of the angular limit of resolution between two closely-spaced distant objects so that they are just resolved when seen through the telescope.

Consider two stars seen through a telescope. The diameter (D) of the objective lens or mirror corresponds to the diffracting aperture. For a distant point source, the first diffraction minimum is at an angle θ away from the centre such that

$D \sin \theta=1.22 \lambda$
where $\lambda$ is the wavelength of light. The angle $\theta$ is usually so small that we can substitute $\sin \theta \approx \theta(\theta$ in radian). Thus, the Airy disc for each star will be spread out over an angular half-width $\theta=1.22 \lambda / D$ about its geometrical image point. The radius of the Airy disc at the focal plane of the objective lens is $r=f \theta=1.22 f \lambda / D$, where / is the focal length of the objective.
When observing two closely-spaced stars, the Rayleigh criterion for just resolving the images as that of two point sources (instead of one) is met when the centre of one Airy disc falls on the first minimum of the other pattern. Thus, the angular limit (or angular separation) of resolution is
$
\theta=\frac{1.22 \lambda}{D} \ldots(1)
$
and the linear separation between the images at the focal plane of the objective lens is $y=f \theta$
$\therefore$ Resolving power of a telescope,
$
R =\frac{1}{\theta}=\frac{D}{1.22 \lambda} \ldots \text { (3) }
$
It depends
1. directly on the diameter of the objective lens or mirror,
2. inversely on the wavelength of the radiation.

1. $S_1$ and $S_2$ act as secondary sources. The crests/-troughs of the secondary wavelets superpose and interfere constructively along straight lines joining the black dots shown in above figure. The point where these lines meet the screen have high intensity and are bright.
2. Similarly, there are points shown with red dots where the crest of one wave coincides with the trough of the other. The corresponding points on the screen are dark due to destructive interference.
3. These dark and bright regions are called fringes or bands and the whole pattern is called interference pattern.