Question

The thin glass shell shown in (Figure 1) has a spherical shape with a radius of curvature of 14.0 cm , and both of its surfaces can act as mirrors. A seed 3.30 mm high is placed 15.0 cm from the center of the mirror along the optic axis, as shown in the figure.Part A- Calculate the location of the image of this seed. Express your answer in centimeters.

Part B- Calculate the height of the image of this seed. Express your answer in millimeters.
Part C- Suppose now that the shell is reversed. Find the location of the seed's image. Express your answer in centimeters.
Part D- Find the height of the seed's image. Express your answer in millimeters.

Answer 1

Part A: The location of the image of the seed is 5.45 cm away from the mirror.

Part B: The height of the image of the seed is 1.37 mm.

Step-by-step explanation:

Part A: To find the location of the image of the seed, we can use the mirror formula for spherical mirrors:
`\( (1)/(f) = (1)/(d_o) + (1)/(d_i) \),` where f is the focal length,
`\(d_o\)` is the object distance, and
`\(d_i\)` is the image distance. Given that the radius of curvature
`\(R = 14.0 \, \text{cm}\),` the focal length
`\(f = (R)/(2) = 7.0 \, \text{cm}\)`, and the object distance
`\(d_o = 15.0 \, \text{cm}\)`, we can rearrange the formula to solve for
`\(d_i\)`. Substituting the values gives
`\( (1)/(7.0) = (1)/(15.0) + (1)/(d_i) \),` which gives
`\(d_i = 5.45 \, \text{cm}\).`

Part B: To find the height of the image of the seed, we can use the magnification formula:
`\( (h_i)/(h_o) = -(d_i)/(d_o) \), where \(h_i\)` is the image height,
`\(h_o\)` is the object height,
`\(d_i\)` is the image distance, and
`\(d_o\)` is the object distance. Given that the object height
`\(h_o = 3.30 \, \text{mm}\)` and we've already calculated
`\(d_i = 5.45 \, \text{cm}\), and \(d_o = 15.0 \, \text{cm}\),` substituting the values gives
`\( (h_i)/(3.30) = -(5.45)/(15.0) \), which gives \(h_i = 1.37 \, \text{mm}\).`

In summary, the image of the seed is formed 5.45 cm away from the mirror, and its height is 1.37 mm. These calculations consider the mirror formula and the magnification formula for spherical mirrors.