Concept explainers
The focal length of the eyepiece required to construct a telescope which can resolve features 6.5 km across the Moon and the resolution limit set by the size of the objective lens due to diffraction.
Answer to Problem 53P
Solution:
The focal length of the eyepiece is found to be 8.5 cm and the angular resolution limit of the objective is found to be with the spatial resolution limit as 2.4 km.:
Explanation of Solution
In a telescope, the resolving powers of the objective and the eyepiece must be equal and it should be equal to the resolving power of the eye.
The objective resolves two points at a distance located on Moon, which is at a distance R from the Earth. Then,
If the angular resolution of the objective is , then, the resolving power of the objective is given by,
Similarly, the angular resolution and the resolving power of the eye, which can resolve two objects separated by a distance and located at a distance can be written as,
and
Since , therefore,
Using the expressions for and in the expression,the focal length of the eyepiece can be calculated.
The angular resolution limit set by the objective lens is given by,
The spatial resolution is given by,
Given:
The distance that can be resolved by the telescope on the Moon
The distance between Earth and Moon
Focal length of the objective
Diameter of the objective’s aperture
The distance that can be resolved by the eye
The distance at which the eye resolves the objects
Wavelength of light used
Formula:
The focal length of the eyepiece is calculated using the expression,
The angular resolution limit of the objective is calculated using the expression,
The spatial resolution is given by,
Calculation:
Using the given quantities in the expression for focal length of the eyepiece, the following calculation is done.
The resolution limit set by the objective is calculated as follows:
The spatial resolution is calculated as follows:
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