The focal length of a convex lens is 'f' What should be the minimum distance between the object and the screen so that a real image is formed.

CONCEPT:

• Lens: The transparent curved surface which is used to refract the light and make an image of any object placed in front of it is called a lens.
  • Convex lens: ​A lens having two spherical surfaces, bulging outwards is called a double convex lens (or simply convex lens).
  • It is thicker in the middle as compared to the edges.
  • Convex lenses converge light rays and hence, convex lenses are also called converging lenses.

• The lens formula is given by:

\(\frac{1}{v} - \frac{1}{u} = \frac{1}{f} \)

Where V = Distance of image from the optical center, U = Distance of object from the optical center

• Depending upon the situation convex lens produces real and virtual images.

SOLUTION:

• Assume the distance between the object and the real image formed by the convex lens be' X'
• Let the distance of the object from the lens be, U = -Y
• So the image distance from the lens will be, V = X - Y
• The thin lens equation is given by

\(⇒ \frac{1}{f} =\frac{1}{V} -\frac{1}{U}\)

\(⇒\frac{1}{f} =\frac{1}{X-Y} -\frac{1}{-Y}=\frac{1}{X-Y} +\frac{1}{Y}\)

⇒ Y² - XY + fX = 0

Then the value of Y according to the quadratic equation is given by, 

\(\Rightarrow Y = {-X \pm \sqrt{X^2-4fX} \over 2}\)

For the real value of Y, the value \(-X + \sqrt{X^2-4fX}≥ 0\)

⇒ X² - 4fX ≥ 0

⇒ X² ≥  4fx

⇒ X ≥ 4f 

• The minimum distance between the and the real image in a convex lens is 4f. Hence option 4 is the answer.

Additional Information

• Nature, position, and relative size of the image formed by a convex lens for various positions of the object:-