How to Determine the Focal Length and Nature of the Lens for Image Projection

What are the steps to find the focal length and nature of the lens needed for a clear image projection?

Given that an illuminated object and a screen are placed 90 cm apart, how can we determine the focal length and nature of the lens required to produce a clear image on the screen that is twice the size of the object?

Answer:

For an object and screen 90 cm apart to create an image twice the size of the object, a converging lens with a focal length of 20 cm is required. Using the lens formula and magnification, we deduce that the image is real, inverted, and produced by a convex lens.

Finding the Focal Length and Nature of Lens for Image Projection

To determine the focal length and nature of a lens required to produce a clear image on a screen that is twice the size of the object with the object and screen 90 cm apart, we can use the lens formula:

1/f = 1/do + 1/di

Where f is the focal length, do is the object distance, and di is the image distance from the lens. Given that the image is twice the size of the object, the magnification m is -2 (negative as the image is inverted when it is real and projected).

Magnification is also given by:

m = -di/do

Rearranging for di, we get:

di = -2do

Since do + di = 90 cm and di = -2do, we find that do = 30 cm and di = -60 cm. Now, we substitute do and di back into the lens formula to find the focal length:

1/f = 1/30 + 1/(-60)

f = 20 cm

The negative value for di indicates that the image is on the same side as the object–which is typical for a real image created by a converging lens (also known as a convex lens).

Therefore, the lens required is a converging lens with a focal length of 20 cm to produce an image that is twice the size of the object when both are placed 90 cm apart.