A converging lens (f = 12.2 cm) is located 33.4 cm to the left of a diverging lens (f = -4.64 cm). A postage stamp is placed 31.4 cm to the left of the converging lens. (a) Locate the final image of the stamp relative to the diverging lens. (b) Find the overall magnification.
To find the final image of the stamp relative to the diverging lens and the overall magnification, we can use the lens formula and the magnification formula.
(a) To locate the final image of the stamp relative to the diverging lens, we will calculate the position and nature (real or virtual) of the image formed by each lens and then combine them.
1. For the converging lens, we can use the lens formula:
1/f = 1/v - 1/u,
where f is the focal length of the lens, v is the image distance, and u is the object distance.
Given: f1 = 12.2 cm and u1 = -31.4 cm
By substituting these values into the lens formula and solving for v1:
1/12.2 = 1/v1 - 1/-31.4
31.4 - 12.2 = 12.2v1 - 1(-31.4)
19.2 = 12.2v1 + 31.4
12.2v1 = -12.2
v1 = -1 cm
The image formed by the converging lens is real and located 1 cm to the left of the lens.
2. For the diverging lens, we can use the lens formula again:
1/f = 1/v - 1/u,
where f is the focal length of the lens, v is the image distance, and u is the object distance.
Given: f2 = -4.64 cm and u2 = -33.4 cm
By substituting these values into the lens formula and solving for v2:
1/-4.64 = 1/v2 - 1/-33.4
-33.4 + 4.64 = -4.64v2 - 1(-33.4)
28.76 = -4.64v2 + 33.4
-4.64v2 = 4.64
v2 = -1 cm
The image formed by the diverging lens is virtual and located 1 cm to the left of the lens.
Since both images are located at the same position and have the same orientation (real and upright), they can be considered as a single image. Therefore, the final image of the stamp relative to the diverging lens is located 1 cm to the left of the diverging lens and is real.
(b) To find the overall magnification, we can calculate the product of the magnifications of each lens:
Magnification (overall) = Magnification (converging lens) × Magnification (diverging lens)
1. For the converging lens:
Magnification (converging lens) = -v1/u1 = -(-1)/(-31.4) = -1/31.4
2. For the diverging lens:
Magnification (diverging lens) = -v2/u2 = -(-1)/(-33.4) = -1/33.4
Therefore, the overall magnification is:
Magnification (overall) = (-1/31.4) × (-1/33.4) = 1/(31.4 × 33.4)
Now you have located the final image of the stamp relative to the diverging lens and calculated the overall magnification.