Question

(a) What is the focal length of a magnifying glass that gives an angular magnification of 8.0 when the image is at infinity? (b) How far must the object be from the lens? Assume the lens is held close to the eye.

Short answer

Answer: The focal length of the magnifying glass is approximately 3.57 cm, and the object should be placed at a distance of approximately 1.39 cm from the lens.

Step-by-step solution

Write the formula for angular magnification and lens formula

The formula for angular magnification (M) of a magnifying glass when the image is at infinity is given by: M = 1 + (D / f) where, M = Angular magnification D = The distance of the nearest clear vision (25 cm) f = Focal length of the lens The lens formula is given by: 1 / f = 1 / u + 1 / v where, f = Focal length of the lens u = Object distance from the lens v = Image distance from the lens (a) We need to find the focal length (f) when angular magnification (M) is 8.0. (b) After finding the focal length, we will find the object distance (u) by using the lens formula.

Solve the angular magnification formula for the focal length (f)

Rearrange the formula for angular magnification to find the value of focal length (f): f = D / (M - 1) Now substitute the given values: M = 8.0 D = 25 cm f = 25 / (8 - 1) f = 25 / 7 f = 3.57 cm So, the focal length of the magnifying glass is 3.57 cm.

Use the lens formula to find the object distance (u)

When the image is at infinity, the image distance (v) is considered to be infinite. The lens formula will become: 1 / f = 1 / u Now, rearrange the formula to find the value of the object distance (u): u = f / (1 - f) Substitute the value of the focal length (f = 3.57 cm): u = 3.57 / (1 - 3.57) u = 3.57 / -2.57 u = -1.39 cm Since the distance cannot be negative, we'll take the absolute value of the result: u = 1.39 cm So, the object must be 1.39 cm away from the lens.