The image of the United States postage stamps in the figure above is 2.1 times the size of the actual stamps in front of the lens. Determine the focal length of the lens, in cm, if the distance from the lens to the stamps is 2.68 cm. Note: The focal length of this lens should be entered as a positive number since it is a convex-converging lens.

Question

BlackTom AI · Accepted Answer

First, identify what the magnification given in the problem means. The image is 2.1 times larger than the object, so the magnification magnitude |m| = |v/u| = 2.1, where u is the object distance (distance from lens to stamps) and v is the image distance.

Since the stamps are in front of the lens, the object distance is u = 2.68 cm (positive by the usual sign convention for an object in front of the lens).

From |m| = v/u, we find v = |m| × u = 2.1 × 2.68 cm. This gives v approximately 5.628 cm. Because the lens is convex and forms a real image on the opposite side, v is positive in the standard lens equations.

Now apply the lens formula 1/f = 1/v + 1/u for a thin lens, with u = 2.68 cm and v ≈ 5.628 cm. Compute the reciprocals: 1/u ≈ 0.3731 cm⁻¹ and 1/v ≈ 0.1777 cm⁻¹. Sum them to obtain 1/f ≈ 0.5508 cm⁻¹.

Finally, take the reciprocal to find f. This yields f ≈ 1.815 cm. Depending on rounding conventions, the focal length is typically reported to two decimal places as about 1.81 cm or rounded to 1.80 cm in some contexts.

Throughout this process, the key steps were identifying magnification, using m = v/u, solving for v, and then applying the lens equation 1/f = 1/v + 1/u with the appropriate sign conventions for a real, inverted image produced by a convex lens.

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