A convex lens with a focal length of 20 cm is placed in front of a convex mirror with a focal length of 7.5 cm. A luminous object is placed in front of the convex lens at a distance of 40 cm from it. What is the distance between the mirror and the lens to get the object's erect image coincident with the object?
To find the distance between the mirror and the lens to get the object's erect image coincident with the object, we can use the concept of refraction and the relationship between the object distance (u), image distance (v), and focal length (f) of lenses and mirrors.
Given Information:
Focal length of convex lens (f1) = 20 cm (positive value indicates convex lens)
Focal length of convex mirror (f2) = -7.5 cm (negative value indicates convex mirror)
Object distance from the convex lens (u) = 40 cm
To solve this problem, we can follow these steps:
Step 1: Find the image formed by the convex lens.
The lens formula relates the object distance (u), image distance (v), and focal length (f) of the lens.
For a convex lens, the lens formula is: 1/f = 1/v - 1/u
Substituting the given values:
1/20 = 1/v - 1/40
Solving this equation will give us the image distance (v) formed by the convex lens.
Step 2: Find the position of the image formed by the convex lens relative to the mirror.
Since the object distance for the mirror is on the same side as the object distance for the lens, the image distance for the lens is also on the same side as the image distance for the mirror. Therefore, the image distance for the mirror (v2) will be the same as the image distance for the lens (v1).
Step 3: Find the object distance for the mirror.
The mirror formula relates the object distance (u), image distance (v), and focal length (f) of the mirror.
For a convex mirror, the mirror formula is: 1/f = 1/v - 1/u
Substituting the given values:
1/-7.5 = 1/v1 - 1/u
Solving this equation will give us the object distance (u2) for the mirror.
Step 4: Find the distance between the mirror and the lens.
The distance between the mirror and the lens can be found by subtracting the object distance for the mirror (u2) from the image distance for the lens (v1): Distance = v1 - u2.
Now, let's calculate the values.
Step 1:
1/20 = 1/v1 - 1/40
1/v1 = 1/20 + 1/40
1/v1 = (2 + 1)/40
1/v1 = 3/40
v1 = 40/3 cm
Step 2:
v2 = v1 (The image distance formed by the lens is the same as the image distance formed by the mirror.)
v2 = 40/3 cm
Step 3:
1/-7.5 = 1/(40/3) - 1/u2
1/u2 = 1/-7.5 + 3/40
1/u2 = (-8 + 3)/120
1/u2 = -5/120
u2 = -120/5 cm
Step 4:
Distance = v1 - u2
Distance = 40/3 - (-120/5)
Distance = 40/3 + 120/5
Distance = (200 + 360)/15
Distance = 560/15 cm
Therefore, the distance between the mirror and the lens to get the object's erect image coincident with the object is 37.33 cm (rounded to two decimal places).