The angular magnification of a refracting telescope is 40 x.When the object and final image are both at infinity,the distance between the eyepiece and the objective is 143.5 cm.The telescope is used to view a distant radio tower.The real image of the tower,formed by the objective,is 6.0 mm in height.The focal point of the eyepiece is positioned at the real image.What is the angle subtended by the final image of the tower.
A) 0.15 rad
B) 0.17 rad
C) 0.19 rad
D) 0.21 rad
E) 0.23 rad

Question

Quizplus · Accepted Answer

The angular magnification (M) of a telescope is given by M = f_o / f_e, where f_o is the focal length of the objective lens and f_e is the focal length of the eyepiece. Given M = 40x and the distance between the eyepiece and the objective (which is approximately the sum of their focal lengths for distant objects) is 143.5 cm, we can find f_o and f_e. Since M = 40 = f_o / f_e and f_o + f_e ≈ 143.5 cm, solving these equations gives us f_o ≈ 40f_e and f_o + 40f_e ≈ 143.5 cm. This implies f_o is significantly larger than f_e, but exact values of f_o and f_e are not needed to find the angle.The height of the real image formed by the objective is 6.0 mm, and this image acts as the object for the eyepiece. The angular size of the image as seen through the eyepiece can be found using the formula for angular magnification, M = h'/h, where h is the height of the object (real image height = 6.0 mm) and h' is the height of the final image. Since the magnification is 40x, the height of the final image is not directly needed to find the angle subtended by it.The angle subtended by the final image (θ) can be approximated by θ = h' / f_e, where h' is the height of the final image. However, since we know the magnification and the height of the real image, we can also express this as θ = M * h / f_o. Given that the magnification is 40 and the height of the real image is 6.0 mm = 0.006 m, and without needing the exact focal lengths, we can find the angle by understanding that the angular size is proportional to the height of the image and the magnification.Given the information provided, the exact focal lengths of the lenses are not needed to calculate the angle directly from the options provided. The correct approach involves understanding the relationship between angular magnification, the height of the image, and the focal lengths of the telescope's components. However, without directly calculating using the focal lengths, the key insight is that the angular size of the image is determined by the magnification and the size of the object image formed by the objective lens.Since the specific calculation steps to arrive at the exact answer (0.17 rad) are not detailed in the explanation due to the initial setup, the correct choice is identified based on understanding the principles of optical magnification and the geometry of the situation. The actual calculation would involve using the magnification formula and the geometry of the image formation to find the subtended angle, which would require additional steps not fully outlined here.

The Angular Magnification of a Refracting Telescope Is 40 X