Question

A convex lens, of focal length \[30cm\], a concave lens of focal length \[120cm\], and a plane mirror are arranged as shown. For an object kept at a distance of \[60cm\] from the convex lens, the final image, formed by the combination, is a real image, at a distance of:

A. 60 cm from the concave lens
B. 70 cm from the convex lens
C. 60 cm from the convex lens
D. 70 cm from the concave lens

Answer 1

Hint: Even if it is a combination of optical systems, we can separate each for the easy calculation. Use lens formula to find the distance of image formation for the concave and convex lens. In this question, we have to find out the real image position, not the virtual image position.

Formula used: \[\dfrac{1}{f}=\dfrac{1}{v}-\dfrac{1}{u}\], where f is the focal length, u is the distance of the object from the lens and v is the image distance from the lens.

Complete Step-by-Step solution:
We are considering an image formed by the combination of concave, convex and a plane mirror. We can write the given information from the question to find out the distance of image formation.
Focal length of the convex lens is 30 cm.
Focal length of the concave lens is 120 cm.
The distance between the object and convex lens is 60 cm.
The distance between the convex lens and concave lens is 20 cm.
The distance between the convex lens and plane mirror is 70 cm.
The focal length of a plane mirror is always infinity.
The following diagram shows the image formation in this combination system.
We can split this problem into several parts. First, we can find the image from the convex lens only.
According to the lens formula,
\[\dfrac{1}{f}=\dfrac{1}{v}-\dfrac{1}{u}\], where f is the focal length, u is the distance of the object from the lens and v is the image distance from the lens.
We can assign the given values into the equation.
\[\dfrac{1}{30}=\dfrac{1}{v}-\dfrac{1}{-60}\], Here we have given negative sign to the distance of the object from the lens. Since we are measuring these distances from the lens.
\[\dfrac{1}{30}=\dfrac{1}{v}+\dfrac{1}{60}\]
\[\dfrac{1}{v}=\dfrac{1}{30}-\dfrac{1}{60}\]
\[\dfrac{1}{v}=\dfrac{1}{60}\]
Thereby v will be 60 cm. So, the image will form at a distance 60 cm from the convex lens.
Next, we can consider the case of a concave lens. Here the u will be 40 cm to the right of the concave lens. Since, the image formed by the convex lens is at 60 cm to the right. 20 cm is the distance between the convex and concave lenses. So, the remaining distance is 40 cm to the right. Hence u will be a positive quantity.
We can again use the lens formula to find the image distance from the concave lens.
\[\dfrac{1}{f}=\dfrac{1}{v}-\dfrac{1}{u}\], where f is the focal length, u is the distance of the object from the lens and v is the image distance from the lens.
We can assign the given values into the equation.
\[\dfrac{1}{-120}=\dfrac{1}{v}-\dfrac{1}{40}\], Here we have used negative value for focal length of concave lens. For concave lenses focal length is always negative and for convex lens, it is always positive.
\[\dfrac{1}{v}=\dfrac{1}{40}-\dfrac{1}{120}\]
\[\dfrac{1}{v}=\dfrac{1}{60}\]
So, the image will form at 60 cm from the concave lens.
Now we have to do the calculation with some logic. Here the image will form at 60 cm from the concave lens to the right direction. But the distance between the concave lens and plane mirror is,
\[70cm-20cm=50cm\].
The image formed by a concave lens will always be virtual, erect and diminished. So, the virtual image is formed at 10 cm to the right of the plane mirror. This will behave like an object for a plane mirror. Thus, the plane mirror will form a real image at the same distance opposite to the direction. Then the image will form at 10 cm left to the plane mirror.
We have already given the distance between the convex lens and plane mirror. It is actually 70 cm.
From this, we can find out the distance between the image formed by the combination of lenses and plane mirror and the convex lens.
\[70cm-10cm=60cm\]
So, the image will form at 60 cm from the convex lens. Hence the option C is correct. In the diagram you can find this image at point v.

Note: In optics, the signs have very importance. We have to take v and u as positive or negative according to the system that we are considering. Try to remember that, the focal length of a convex lens is always positive and for a concave lens it is always negative. In this question it is not mentioned, so you have to consider it yourself. If we are considering the virtual image position, then the answer will be wrong. It will lead to other options.