A thin, double-convex lens has a refractive index of 1.50. The radius of curvature of the front surface is 15 cm and that of the rear surface is 10 cm. See sketch. (a) How far from the lens would an image of the sun be formed? (b) How far from the lens would an image of a toy figure 24 cm from the lens be formed? (c) How do the answers to (a) and (b) change if you flip the lens over?
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To answer these questions, we can use the lens formula and the magnification formula. The lens formula gives us the relationship between the object distance (u), the image distance (v), and the focal length (f) of the lens. The formula is:
1/f = 1/v - 1/u
where f is the focal length, v is the image distance, and u is the object distance.
The magnification formula gives us the relationship between the height of the object (h1) and the height of the image (h2). The formula is:
magnification (m) = h2/h1 = -v/u
where h1 is the height of the object and h2 is the height of the image. The negative sign in front of the magnification (m) indicates that the image is inverted.
(a) To find the distance from the lens where an image of the sun is formed, we need to determine the object distance (u) and the image distance (v). Since the object is the sun and is located at infinity, we can assume that the object distance (u) is very large (~infinity), which means that 1/u can be considered negligible compared to 1/v in the lens formula:
1/f = 1/v
The focal length (f) can be calculated using the lens-maker's formula:
1/f = (n - 1)((1/R1) - (1/R2))
where n is the refractive index of the lens, R1 is the radius of curvature of the front surface, and R2 is the radius of curvature of the rear surface.
Substituting the values given:
1/f = (1.50 - 1)((1/15) - (1/10))
Simplifying:
1/f = 0.50((1/15) - (1/10))
= 0.50((2/30) - (3/30))
= 0.50(-1/30)
= -1/60
Taking the reciprocal of both sides:
f = -60 cm
Now we can substitute the value of f into the lens formula to find the image distance (v):
1/(-60) = 1/v
Simplifying:
1/v = -1/(-60)
= 1/60
Taking the reciprocal of both sides:
v = 60 cm
The image of the sun would be formed 60 cm from the lens.
(b) To find the distance from the lens where an image of a toy figure 24 cm from the lens is formed, we need to determine the object distance (u) and the image distance (v). In this case, the object distance (u) is given as 24 cm.
Substituting the values into the lens formula:
1/f = 1/v - 1/u
We already know the value of f (-60 cm) from the previous calculation. Substituting:
1/(-60) = 1/v - 1/24
Simplifying:
1/v = 1/24 - 1/(-60)
= 1/24 + 1/60
= (5/120) + (2/120)
= 7/120
Taking the reciprocal of both sides:
v = 120/7 cm
The image of the toy figure would be formed approximately 17.14 cm from the lens.
(c) If we flip the lens over, the sign of the focal length changes (since the curvature of the lens surfaces is reversed), but the object distance and image distance remain the same. Therefore, the answers to parts (a) and (b) would remain the same even if we flip the lens over.