Question

A telescope consisting of a +3.0-cm objective lens and a +0.80-cm eyepiece is used to view an object that is 20 m from the objective lens. Part A: What must be the distance between the objective lens and eyepiece to produce a final virtual image 100 cm to the left of the eyepiece? Part B: What is the total angular magnification?

Answer 1

To find the required distance between the lenses, one must use the lens formula considering the distance where the virtual image is formed. Part B's total angular magnification is the product of the magnifications of each lens.

Step-by-step explanation:

For Part A, we need to calculate the distance between the objective lens and the eyepiece. The lens formula is 1/f = 1/di + 1/do, where f is the focal length, di is the image distance, and do is the object distance. Since the final image is formed at 100 cm to the left of the eyepiece, and this position is virtual, we take di as -100 cm (negative for virtual image). The focal length of the eyepiece is +0.80 cm. Using the lens formula, 1/f = 1/-100 - 1/do gives us the do of the eyepiece (which is the image distance for the objective lens). After finding do for the objective lens, we can use the lens formula again with the focal length of the objective lens (+3.0 cm) to find di for the objective lens. The distance between the eyepiece and the objective lens will be the absolute value of the difference between these two image distances.

For Part B, the total angular magnification of the telescope is the product of the magnifications of the objective and the eyepiece. Magnification of the objective (Mo) is given by -di/do (negative sign indicates inverted image), and magnification of the eyepiece (Me) is given by -25/fe, where 25 cm is the near point of relaxed vision for a standard human eye and fe is the focal length of the eyepiece. The total angular magnification (M) is then M = Mo * Me.

Answer 2

To determine the separation between lenses for a telescope, we apply the lens formula separately for the objective and the eyepiece, and add their respective distances to get the total length. The total angular magnification is the product of the individual magnifications of the objective lens and the eyepiece.

Step-by-step explanation:

To find the distance between the objective lens and the eyepiece for the telescope to produce a final virtual image 100 cm to the left of the eyepiece, we apply the lens formula 1/f = 1/do + 1/di, where f is the focal length, do is the object distance for the lens, and di is the image distance from the lens. For the objective lens (f = 3.0 cm, do = 20 m), we find the image distance di_o. Since the final image is virtual and located 100 cm to the left of the eyepiece, its image distance is di_e = -100 cm. The total distance between the lenses is the sum of the absolute value of the image distance formed by the objective lens and the object distance for the eyepiece, so L = |di_o| + do_e.

For Part B, the total angular magnification of the telescope is given by the magnification of the objective lens (Mo) multiplied by the magnification of the eyepiece (Me), with Mo being the ratio of the focal length of the objective to the image distance (f_o/di_o) and Me being the negative ratio of the image distance to the focal length of the eyepiece (di_e/f_e). The negative sign indicates that the final image is virtual and upright relative to the original object.