Three Unlabelled Mirrors
An optician's new assistant is asked to sort three unlabelled mirrors before the shop opens. Mirror P shows a customer's face erect and smaller however far away he stands, and the assistant can see a wide sweep of the street behind her in it. Mirror Q, mounted on a stand, gives an erect, magnified image when a pin is held 8 cm in front of it, and that image appears 24 cm behind the mirror when measured with a scale. The same mirror throws an inverted image on a card when the pin is moved to 20 cm. Mirror R gives an image of exactly the same size as the object, and the image is as far behind the mirror as the object is in front. The shopkeeper wants one mirror for a wall-mounted security view and one for close-up work, and asks for the focal length and magnification of Q at the 8 cm position.
1. Identify the type of each mirror P, Q and R, and state which one the shopkeeper should use for the security view and which for close-up work, with reasons.
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P is a convex mirror. Its reflecting surface bulges towards the object, so it always forms a virtual, erect, diminished image and its field of view is wide. That is why a whole street can be seen in it, and it is the correct choice for the security view.
Q is a concave mirror. It gives an erect, magnified, virtual image when the object is within its focal length, and an inverted real image on a card when the object is beyond the focus. That behaviour fits the two observations. It is the correct choice for close-up work such as a dentist's mirror or a shaving mirror, because the image is enlarged when the face is close.
R is a plane mirror: its image is virtual, erect, of the same size, and as far behind the mirror as the object is in front. It is the ordinary looking-glass and is not suitable for either special job.
2. Using the sign convention, calculate the focal length of mirror Q from the 8 cm observation, and calculate the magnification. Show each step.
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Sign convention: distances are measured from the pole, and the direction of the incident light is taken as positive. The object is in front of the mirror, so u = −8 cm. The image is erect and behind the mirror, so it is virtual and v = +24 cm.
Mirror formula: 1/f = 1/v + 1/u 1/f = 1/24 + (−1/8) = 1/24 − 3/24 = −2/24 f = −12 cm.
The negative sign means the focal length is measured in front of the mirror, as for a concave mirror. So Q has a focal length of 12 cm and a radius of curvature of 24 cm.
Magnification: m = −v/u = −(+24)/(−8) = +3.
The image is erect, virtual and three times the size of the pin. The positive sign agrees with the observation that the image is erect; a real image would have given a negative magnification, which is a useful check on the arithmetic.
3. For mirror Q, predict the position, nature and magnification of the image when the pin is 6 cm in front of it.
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The object is still within the focal length (6 cm < 12 cm), so the image should again be virtual, erect and magnified, but larger than before and farther from the mirror.
u = −6 cm, f = −12 cm. 1/v = 1/f − 1/u = −1/12 − (−1/6) = −1/12 + 1/6 = +1/12 v = +12 cm.
So the image is 12 cm behind the mirror, virtual and erect. m = −v/u = −(+12)/(−6) = +2.
The image is twice the size of the pin. As the pin is brought closer to the pole, the image also moves closer to the pole and the magnification falls towards 1; moving the pin towards the focus would push the image towards infinity, which is why an object exactly at the focus gives no image on a screen.