Question

A convex lens of power +12 D is used as a magnifier to examine a wildflower. What is the angular magnification if the final image is at (a) infinity or (b) the near point of \(25 \mathrm{cm} ?\)

Short answer

The angular magnification when the final image is at infinity is 3, and when the final image is at the near point (25 cm), it is 2.

Step-by-step solution

Calculate the focal length of the lens

We are given the power of the lens (P) and we can find the focal length (f) using the formula: \( f = \frac{1}{P} \) In our case, P = +12 D. So the focal length of the lens is: \( f = \frac{1}{12} = 0.0833 \mathrm{m} \)

Find the object distance (u) when the image is at infinity

When the final image is at infinity, the object is placed at the focal point of the lens (f). In this case, the object distance u is equal to the focal length (0.0833 m): \( u = f = 0.0833 \mathrm{m} \)

Calculate the angular magnification (a)

We can now find the angular magnification using the formula: \( M = \frac{d}{u} \) Here, d is the least distance of distinct vision, which is 25 cm or 0.25 m. So the angular magnification for case (a) is: \( M = \frac{0.25}{0.0833} = 3 \) Thus, the angular magnification when the final image is at infinity is 3.

Find the object distance (u) when the image is at the near point

When the image is at the near point (25 cm), we need to rearrange the lens formula and solve for u: \( \frac{1}{u} = \frac{1}{f} - \frac{1}{v} \) We know the focal length (f) and the image distance (v) is equal to the near point distance, which is 25 cm or 0.25 m. Substituting these values, we get: \( \frac{1}{u} = \frac{1}{0.0833} - \frac{1}{0.25} = 8 \) Thus, the object distance (u) when the image is at the near point is: \( u = \frac{1}{8} = 0.125 \mathrm{m} \)

Calculate the angular magnification (b)

Now, we can find the angular magnification for case (b) using the formula: \( M = \frac{d}{u} \) Using the object distance (u) calculated in Step 4, we have: \( M = \frac{0.25}{0.125} = 2 \) Thus, the angular magnification when the final image is at the near point is 2. In conclusion, the angular magnification of the convex lens is 3 when the final image is at infinity and 2 when the final image is at the near point (25 cm).