The first dark ring of the diffraction pattern produced by a circular aperture…

PHYSICS · Multi-part · CBSE · Physics · Ray Optics and Optical Instruments · Ray Optics and Optical Instruments

Problem

The first dark ring of the diffraction pattern produced by a circular aperture defines the angular limit beyond which two point sources can be distinguished as separate. (i) Starting from the condition for the first minimum in the Fraunhofer diffraction pattern of a circular aperture, derive an expression for the minimum angular separation

\theta_{\min}

that can be resolved by an optical telescope in terms of the wavelength

\lambda

of light used and the diameter

D

of its objective. (ii) A distant planet is effectively a luminous disc of diameter

3.0\times10^{3}\,\text{km}

situated

5.0\times10^{11}\,\text{m}

from Earth. Calculate the minimum objective diameter required for a ground–based optical telescope, working with light of wavelength

550\,\text{nm}

, to just resolve the planet's disc as a non–point object. (iii) An objective of the diameter found in part (ii) is to be paired with an eyepiece of focal length

2.5\,\text{cm}

. If the objective focal length is

1.0\,\text{m}

, compute the angular magnification of the telescope and state whether this magnification is sufficient to utilise the full resolving capability calculated in part (ii).

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