A convex spherical mirror has a radius of curvature of 10.0 cm. Calculate the location and size of the image formed of an 8.00-mm-tall object whose distance from the mirror is
(a) 15.0 cm, (b) 10.0 cm, (c) 2.50 cm, and (d) 10.0 m.
To determine the location and size of the image formed by a convex spherical mirror, you can use the mirror formula and magnification formula.
The mirror formula is given by:
1/f = 1/dᵢ + 1/dₒ
where f is the focal length of the mirror, dᵢ is the image distance, and dₒ is the object distance.
The magnification formula is given by:
m = -dᵢ/dₒ
where m is the magnification of the image.
In this case, the radius of curvature (R) of the mirror is given as 10.0 cm. For a convex mirror, the focal length (f) is half the radius of curvature: f = R/2 = 10.0 cm / 2 = 5.0 cm.
Now, let's calculate the location and size of the image for each case:
(a) When the object distance (dₒ) is 15.0 cm:
Using the mirror formula: 1/5.0 = 1/dᵢ + 1/15.0
Simplifying: 3/dᵢ = 1/5.0
Rearranging: dᵢ = 5.0/3 ≈ 1.67 cm
Using the magnification formula: m = -dᵢ/dₒ = -1.67/15.0 ≈ -0.11
Therefore, the image is formed at approximately 1.67 cm from the mirror, and it is about 11% of the size of the object, but inverted.
(b) When the object distance (dₒ) is 10.0 cm:
Using the mirror formula: 1/5.0 = 1/dᵢ + 1/10.0
Simplifying: 2/dᵢ = 1/5.0
Rearranging: dᵢ = 5.0/2 = 2.50 cm
Using the magnification formula: m = -dᵢ/dₒ = -2.50/10.0 = -0.25
Therefore, the image is formed at 2.50 cm from the mirror, and it is 25% of the size of the object, but inverted.
(c) When the object distance (dₒ) is 2.50 cm:
Using the mirror formula: 1/5.0 = 1/dᵢ + 1/2.50
Simplifying: 7.50/dᵢ = 1/5.0
Rearranging: dᵢ = 7.50/5.0 = 1.50 cm
Using the magnification formula: m = -dᵢ/dₒ = -1.50/2.50 = -0.60
Therefore, the image is formed at 1.50 cm from the mirror, and it is 60% of the size of the object, but inverted.
(d) When the object distance (dₒ) is 10.0 m:
Using the mirror formula: 1/5.0 = 1/dᵢ + 1/1000.0
Simplifying: 2000.0/dᵢ = 1/5.0
Rearranging: dᵢ = 2000.0/5.0 = 400.0 cm
Using the magnification formula: m = -dᵢ/dₒ = -400.0/1000.0 = -0.40
Therefore, the image is formed at 400.0 cm (or 4.00 m) from the mirror, and it is 40% of the size of the object, but inverted.