Explanation:

When the focal length of the eyepiece of a telescope is doubled, the magnifying power (m) of the telescope changes. Let's understand why the correct answer is option 'C' - m/2.

Magnifying Power:
The magnifying power (m) of a telescope is defined as the ratio of the angle subtended by the image formed by the telescope to the angle subtended by the object when viewed with the naked eye.

Magnifying Power Formula:
m = θi/θo

Where:
- m is the magnifying power
- θi is the angle subtended by the image formed by the telescope
- θo is the angle subtended by the object when viewed with the naked eye

Effect of Doubling Eyepiece Focal Length:
When the focal length of the eyepiece is doubled, it means the new focal length is twice the original focal length. Let's consider the original focal length of the eyepiece as f and the new focal length as 2f.

Effect on Image Distance:
The image distance formed by the eyepiece is given by the formula:
1/f = 1/v - 1/u

Where:
- f is the focal length of the eyepiece
- v is the image distance
- u is the object distance

When the focal length is doubled, the new image distance (v2) can be calculated as:
1/(2f) = 1/v2 - 1/u

Simplifying the equation, we find:
v2 = 2v

This means that the new image distance is twice the original image distance.

Effect on Angle Subtended by Image:
The angle subtended by the image formed by the telescope is given by the formula:
θi = h/v

Where:
- θi is the angle subtended by the image
- h is the height of the object being observed
- v is the image distance

Since the new image distance (v2) is twice the original image distance, the new angle subtended by the image (θi2) can be calculated as:
θi2 = h/(2v)

Simplifying the equation, we find:
θi2 = θi/2

This means that the new angle subtended by the image is half the original angle subtended by the image.

Effect on Magnifying Power:
Using the magnifying power formula, we can substitute the new values of θi2 and θo (which remains the same) into the equation:
m2 = θi2/θo
m2 = (θi/2)/θo
m2 = θi/2θo
m2 = m/2

Therefore, when the focal length of the eyepiece of a telescope is doubled, the magnifying power (m) of the telescope becomes m/2. Hence, the correct answer is option 'C'.