Question 237: Ray Optics

Question

Two convex lenses of focal length $$20 \mathrm{~cm}$$ each are placed coaxially with a separation of $$60 \mathrm{~cm}$$ between them. The image of the distant object formed by the combination is at _____________ $$\mathrm{cm}$$ from the first lens.

Accepted Answer

1. First refraction in L1 (lens 1):
When considering the first lens (L
1
), the object is at infinity, so the image I
1
formed by this lens is at its focal point. Using the lensmaker's equation:
$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$
Given that the object is at infinity, $$d_o = \infty$$, and the focal length of the first lens is $$f = 20 \mathrm{~cm}$$. Plugging in these values, we get:
$$\frac{1}{20} = \frac{1}{\infty} + \frac{1}{d_i}$$
As $$\frac{1}{\infty}$$ is essentially 0, we have:
$$\frac{1}{20} = \frac{1}{d_i}$$
This implies that $$d_i = 20 \mathrm{~cm}$$, meaning that the image I
1
is formed 20 cm from the first lens L
1
.
2. Second refraction in L2 (lens 2):
Now, the image I
1
formed by L
1
becomes the object for L
2
. The distance between the lenses is 60 cm, so the object distance for L
2
(u) is -40 cm, because the object is to the left of the lens (u is negative). The focal length of L
2
(f) is 20 cm. We can use the lensmaker's equation to find the image distance (v) for L
2
:
$$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$$
Plugging in the values:
$$\frac{1}{v} - \frac{1}{(-40)} = \frac{1}{20}$$
$$\frac{1}{v} = \frac{1}{20} - \frac{1}{40} = \frac{2 - 1}{40}$$
$$\frac{1}{v} = \frac{1}{40}$$
Therefore, $$v = 40 \mathrm{~cm}$$
Since the image distance (v) is positive, the image I
2
is formed on the right side of L
2
at a distance of 40 cm. To find the distance of the final image from L
1
, we add the distance between the lenses (60 cm) and the image distance (v) from L
2
.
Final image distance from L
1
= 60 cm + 40 cm = 100 cm
Thus, the correct answer is 100 cm.

Ray Optics

Two convex lenses of focal length $20 \mathrm{~cm} each are placed coaxially with a separation of 60 \mathrm{~cm} between them. The image of the distant object formed by the combination is at _____________ \mathrm{cm}$ from the first lens.

Integer / Numeric Answer Type