A convex lens is needed to produce an image that is 0.55 times the size of the object and located 30 cm from the lens on the other side. What focal length should be specified?
Thin Lens Equation
1/d +1/f = 1/F,
where d is the object distance,
f is the image distance
F is the focal length
A is the size of the object
B is the size of the image.
B = 0.55•A,
From the similarity of triangles
d/A = f/B
d = (A/B) •f = f/0.55
F = d•f/(d+f) =f/1.55=0.3/1.55 = 0.194 m
To determine the required focal length of the convex lens, we can utilize the thin lens formula, which relates the focal length (f) of a lens to the object distance (do) and the image distance (di) as follows:
1/f = 1/do + 1/di
Given:
- The image is 0.55 times the size of the object, which implies that the image distance (di) is negative (-30 cm).
- The magnification (m) is defined as the ratio of the image height (hi) to the object height (ho), and in this case, m = -0.55.
Since m = -di/do, we can solve for do and substitute it back into the lens formula to find the focal length.
Step 1: Find the object distance (do)
Using the magnification formula: m = -di/do
Substituting the given values: -0.55 = -30/do
Simplifying the equation: 0.55 * do = 30
Dividing both sides by 0.55: do = 54.54 cm
Step 2: Substitute the object distance (do) into the lens formula
1/f = 1/do + 1/di
Substituting the values: 1/f = 1/54.54 + 1/-30
Simplifying the equation: 1/f = (1 - 1.818) / 54.54
Calculating: 1/f = -0.818 / 54.54
Dividing both sides: f = -54.54 / 0.818
Calculating: f ≈ -66.69 cm
Therefore, the focal length that should be specified for the convex lens is approximately -66.69 cm. Note that the negative sign indicates that the lens is a converging lens (convex).