A convex lens of focal length 30 cm and a concave lens of focal length 10 cm are kept at a distance d apart in the air. If light rays parallel to the principal axis are falling on the convex lens, the value of d for which light rays going out from the concave lens are also parallel to the principal axis is ______ (give your answer in cms) [the answer is an integer]Correct answer is '20'. Can you explain this answer?

Question

Answer

Given data:
- Focal length of convex lens, f1 = 30 cm
- Focal length of concave lens, f2 = -10 cm (negative sign indicates concave lens)
- Distance between the lenses, d = ?

Concept:
When a parallel beam of light passes through a convex lens, it converges at a point called the focal point. Similarly, when a parallel beam of light passes through a concave lens, the light rays diverge as if they are coming from a point called the focal point.

When the convex lens and concave lens are placed at a certain distance from each other, the converging effect of the convex lens can be balanced by the diverging effect of the concave lens. This results in the light rays leaving the concave lens parallel to the principal axis.

Calculating the distance between the lenses (d):
To find the value of d, we need to consider the lens formula and the lens-maker's formula.

Lens formula:
The lens formula relates the object distance (u), image distance (v), and the focal length (f) of a lens. It is given by:

1/v - 1/u = 1/f

For the convex lens, the object distance (u1) is infinity because the light rays are parallel. Therefore, the equation becomes:

1/v1 - 1/infinity = 1/f1
1/v1 = 1/f1
v1 = f1

Lens-maker's formula:
The lens-maker's formula relates the focal length (f), refractive index of the lens material (μ), and the radii of curvature of the lens surfaces (R1 and R2). It is given by:

1/f = (μ - 1) * (1/R1 - 1/R2)

For the concave lens, the focal length (f2) is given as -10 cm. Since the concave lens is a diverging lens, the focal length is negative. Therefore, the equation becomes:

1/-10 = (μ - 1) * (1/R1 - 1/R2)
1/-10 = (μ - 1) * (1/R1 + 1/R2)

Assuming the radii of curvature are equal for simplicity (R1 = R2 = R), we can rewrite the equation as:

1/-10 = (μ - 1) * (2/R)

Simplifying further:

-1/10 = (μ - 1) * (2/R)
-1/10 = (μ - 1) * (2/2R)
-1/10 = (μ - 1) * (1/R)

Balancing the converging and diverging effects:
To balance the converging effect of the convex lens with the diverging effect of the concave lens, the image formed by the convex lens should act as the object for the concave lens. Therefore, the image distance (v1) calculated earlier becomes the object distance (u2) for the concave lens.

Applying the lens formula for the concave lens:

1/v2 - 1/u2 = 1/f2
1/v2 - 1/v1 = 1/f2
1/v2 - 1/f1 = 1/f2

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