Questions: Linear and Angular Magnification in Optical Systems

Question 1

A lens produces an image with linear magnification m = −3. A student says: 'The image is 3 times smaller than the object and right-side-up.' Is she correct?

• A. Yes — m = −3 means the image is 3 times smaller in linear size
• B. No — m = −3 means the image is 3 times larger and inverted
• C. No — m = −3 means the image is 3 times smaller and inverted
• D. Yes — the negative sign indicates a real image, which is always smaller than the object

Answer: (B)

Linear magnification m = −i/o, so |m| = 3 means the image is 3 times the physical size of the object — larger, not smaller. The negative sign encodes orientation: m < 0 means the image is inverted (upside down), while m > 0 means upright. A positive |m| > 1 means the image is enlarged; |m| < 1 means reduced. The student confused the sign (which encodes orientation) with the magnitude (which encodes size ratio).

Question 2

A simple magnifying glass has a focal length of 5 cm and is used with the eye relaxed (image formed at infinity). What is the angular magnification?

• A. 5×, since M = 25 cm / f = 25/5
• B. 0.2×, since M = f / 25 cm = 5/25
• C. −5×, since the magnifying glass inverts the image
• D. Infinite, because the object is placed at the focal point and the image forms at infinity

Answer: (A)

For a relaxed eye (image at infinity), the object is placed at the focal point f. The angular magnification is M = 25 cm / f = 25/5 = 5×, where 25 cm is the conventional near-point distance. The image forming at infinity is not a problem — it means the eye views the image with no accommodation (relaxed). The magnification is finite because it compares the angle subtended with the instrument to the angle subtended at the near point without the instrument. The negative sign for inverted images applies to linear magnification, not angular magnification.

Question 3

A shorter focal length produces greater angular magnification for a simple magnifying glass.

• T. True
• F. False

Answer: True

Angular magnification M = 25 cm / f, so M increases as f decreases. A 2 cm focal length lens gives M = 12.5×, while a 10 cm lens gives M = 2.5×. This makes sense geometrically: a shorter focal length allows you to place the object very close to the lens and still form a usable image, making the object subtend a much larger angle than it would at the 25 cm near point.

Question 4

For a telescope observing a distant star, linear magnification is the relevant measure of how much the telescope improves visibility.

• T. True
• F. False

Answer: False

For objects effectively at infinity (like stars), linear magnification is not meaningful — the image is also effectively at infinity, and 'how many times bigger physically' becomes undefined. What matters for a telescope is angular magnification: how many times larger does the star field appear, and how well can the telescope resolve two nearby stars. Angular magnification M = f_objective / f_eyepiece is the relevant quantity. A telescope that produces a large physical intermediate image but no angular magnification increase would be useless.

Question 5

Why is angular magnification, rather than linear magnification, the relevant quantity for evaluating optical instruments like microscopes and telescopes?

Model answer: The human visual system judges the apparent size of objects by the angle they subtend at the eye, not their physical size at some plane in space. A nearby coin appears larger than a distant building because it subtends a larger angle, even though the coin is physically smaller. Optical instruments produce images that the eye observes — what matters is how large that image appears to the observer, which is its angular size. Linear magnification tells you the physical size of an intermediate image at a particular plane, which is irrelevant if the eye views it from varying distances. Angular magnification directly captures the perceptual gain: how many times larger does the viewed object appear compared to the unaided eye?

This distinction becomes critical in multi-element systems. A microscope's objective provides high linear magnification (the intermediate image is much larger than the object), but the eyepiece then provides angular magnification of that image to the eye. Total angular magnification = M_objective × M_eyepiece. For a telescope, where the object is at infinity and linear magnification is undefined, angular magnification M = f_obj/f_eye is the only meaningful measure.