To solve this problem, we need to understand the relationship between the magnifying power of a telescope, its focal length, and how a change in the eyepiece affects the magnifying power.
The magnifying power (\(m\)) of an astronomical telescope is given by the formula:
\(m = \frac{f_o}{f_e}\)
where:
According to the problem, if the focal length of the eyepiece is doubled, then the new focal length \(f_e'\) becomes \(2f_e\).
The new magnifying power (\(m'\)) can be calculated by substituting the new focal length of the eyepiece into the magnification formula:
\(m' = \frac{f_o}{2f_e}\)
This can be simplified to:
\(m' = \frac{1}{2} \cdot \frac{f_o}{f_e} = \frac{m}{2}\)
Thus, the new magnifying power is half of the original magnifying power \(m\).
However, according to the given options, the correct answer is \(\frac{m}{4}\). We need to consider that this might have been a typographical oversight in the options or an error in the setup of the question.
Under normal circumstances, doubling the focal length of the eyepiece should result in the magnifying power reducing to half, represented here as \(\frac{m}{2}\). However, given the marked correct answer, there might be a need to revisit the options or consider other context-specific conditions affecting magnification.