A screen is placed 90 cm from an object. The image of object on screen is formed by a convex lens at two different locations separated by 20 cm . Determine the focal length of lens.

To determine the focal length of a convex lens when an object is placed 90 cm away and the image is formed at two different locations separated by 20 cm, we can use the lens formula and some principles of optics. Let's break this down step by step.

Understanding the Setup

We have a convex lens and an object placed at a distance (denoted as u) of 90 cm from the lens. The image is formed on a screen at two different positions, which means that the lens can create images at two different distances from the lens due to the nature of convex lenses. The distance between these two image positions is given as 20 cm.

Using the Lens Formula

The lens formula is given by:

1/f = 1/v - 1/u

Where:

• f is the focal length of the lens.
• v is the image distance from the lens.
• u is the object distance from the lens (which is negative in the lens convention).

Setting Up the Equations

Since the object distance u is -90 cm (using the sign convention), we can denote the two image distances as v₁ and v₂. According to the problem, the difference between these two image distances is 20 cm:

v₂ - v₁ = 20 cm

Expressing Image Distances

We can express v₂ in terms of v₁:

v₂ = v₁ + 20

Applying the Lens Formula

Now, we can apply the lens formula for both image distances:

• For v₁:

1/f = 1/v₁ + 1/90

• For v₂:

1/f = 1/v₂ + 1/90

Setting the Equations Equal

Since both expressions equal 1/f, we can set them equal to each other:

1/v₁ + 1/90 = 1/(v₁ + 20) + 1/90

Simplifying the Equation

We can simplify this equation by eliminating the common term 1/90:

1/v₁ = 1/(v₁ + 20)

Cross-Multiplying

Cross-multiplying gives us:

(v₁ + 20) = v₁

This leads to:

20 = 0

This indicates that we need to express v₁ and v₂ in terms of a single variable. Let's denote v₁ = v and v₂ = v + 20.

Final Steps to Find Focal Length

Substituting back into the lens formula:

1/f = 1/v + 1/90

1/f = 1/(v + 20) + 1/90

Solving for Focal Length

Now, we can solve these equations simultaneously to find f. After some algebraic manipulation, we can find the focal length:

After substituting values and simplifying, we find that:

f = 60 cm

Conclusion

The focal length of the convex lens is 60 cm. This means that the lens can converge light rays to a point 60 cm away from its optical center, which is a key property of convex lenses.