Question

Read the following question and choose.
A. If both the statements are true and statement-2 is the correct explanation of the statement-1
B. If both the statements are true and statement-2 is not the correct explanation of the statement-1
C. If statement-1 is true and statement-2 is false
D. If statement-1 is false and statement-2 is true
Statement-1: For determining the focal length of a convex lens by displacement method, the minimum separation between the fixed object and the screen should be \[4f\] .
Statement-2: Minimum separation between an object and its real image is \[4f\] .

Answer 1

Hint: First of all, we will draw the diagram which completely illustrates the method. After that we will find relations between the object distance, image distance, distance between the object and the screen and difference between the two positions of the lens to obtain a clear image on the screen. We will use the lens formula and manipulate accordingly to obtain the answer.

Complete step by step answer:
In the given question, we are supplied the following data:
It says about the displacement method which is employed to determining the focal length of a convex lens.
There are two statements given which give various situations, regarding the displacement method.
We are required to choose one of them which suits the best.

In the displacement method, with the object at one end and a screen to obtain the image at the other end, the convex lens is placed on a calibrated scale. In order to get a clear image on the screen, the lens is first placed closer to the object and its position is adjusted. It notes the position of the lens and the distance between the object and the picture. The lens is now moved towards the screen at the same distance between the object and the image and its position is adjusted until a clear image is obtained on the screen. This lens position is also noted.Let the distance between the object and the screen be \[D\] and \[d\] for the difference between the two positions of the lens to obtain a clear image on the screen. To have a better understanding, let us draw a diagram which is given below:

We will repeat this for the various values of distances between the object and the screen.
Let the distance between the object and the lens be \[u\] and \[v\] be the distance between the image and the lens.
Now, we can write:
\[D = u + v\] …… (1)
\[d = v - u\] …… (2)
Solving the two equations we get:
\[v = \dfrac{{D + d}}{2}\] and
\[u = \dfrac{{D - d}}{2}\]
Now, we use the lens formula,
$\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f} \\
\Rightarrow \dfrac{1}{{\dfrac{{D + d}}{2}}} - \dfrac{1}{{\dfrac{{D - d}}{2}}} = \dfrac{1}{f} \\
\Rightarrow \dfrac{{4D}}{{{D^2} - {d^2}}} = \dfrac{1}{f} \\
\Rightarrow f = \dfrac{{{D^2} - {d^2}}}{{4D}} \\$
\[ \therefore 4f = \dfrac{{{D^2} - {d^2}}}{D}\]
Hence, the following result suggests that the minimum separation between the fixed object and the screen should be \[4f\] . Again, in the case of a convex lens, the image obtained is always real if the object is placed beyond the focus.

Therefore, both the statements are true and statement-2 is the correct explanation of the statement-1.The correct option is A.

Note:While solving this problem, remember that the image formed by a convex lens is always real if the object is placed beyond the focus. If the object is placed inside the focus, we will get a virtual image. A concave lens gives a virtual image.