Question

Callum is examining a square stamp of side \(3.00 \mathrm{cm}\) with a magnifying glass of refractive power \(+40.0 \mathrm{D}\). The magnifier forms an image of the stamp at a distance of \(25.0 \mathrm{cm} .\) Assume that Callum's eye is close to the magnifying glass. (a) What is the distance between the stamp and the magnifier? (b) What is the angular magnification? (c) How large is the image formed by the magnifier?

Short answer

$$ f = \frac{1}{40.0\,\text{D}} = 0.025\,\text{m} = 2.5\,\text{cm} $$ So, the distance between the stamp and the magnifier (focal length) is 2.5 cm. #tag_title#Step 2: Find the angular magnification#tag_content#The angular magnification M of the magnifying glass can be found using the formula: $$ M = 1 + \frac{D}{f} $$ where \(D = 25\,\text{cm}\) is the near point distance (the closest distance without strain that a person can focus on an object). Plug in the values: $$ M = 1 + \frac{25\,\text{cm}}{2.5\,\text{cm}} $$ #tag_title#Step 3: Compute the angular magnification#tag_content#Now, compute the angular magnification: $$ M = 1 + \frac{25\,\text{cm}}{2.5\,\text{cm}} = 1 + 10 = 11 $$ So, the angular magnification is 11 times. #tag_title#Step 4: Find the size of the image formed#tag_content#Using the angular magnification, we can find the size of the image formed by the magnifier using the formula: $$ H_{\text{image}} = M \times H_{\text{object}} $$ where \(H_{\text{image}}\) is the size of the image, \(H_{\text{object}} = 2.5\,\text{mm}\) is the size of the stamp and \(M = 11 \) is the angular magnification. Plug in the values: $$ H_{\text{image}} = 11 \times 2.5\,\text{mm} $$ #tag_title#Step 5: Calculate the size of the image formed#tag_content#Compute the size of the image formed: $$ H_{\text{image}} = 11 \times 2.5\,\text{mm} = 27.5\,\text{mm} $$ The size of the image formed by the magnifier is 27.5 mm.

Step-by-step solution

Find the distance between the stamp and the magnifier (focal length)

We are given the refractive power of the magnifying glass: \(P = +40.0\,\text{D}\). We can find the focal length by taking the inverse of the refractive power: $$ f = \frac{1}{P} $$