Question

A beam of parallel light, \(1.00 \mathrm{~mm}\) in diameter passes through a lens with a focal length of \(10.0 \mathrm{~cm}\). Another lens, this one of focal length \(20.0 \mathrm{~cm},\) is located behind the first lens so that the light traveling out from it is again parallel. a) What is the distance between the two lenses? b) How wide is the outgoing beam? 33.41 How large does a \(5.0-\mathrm{mm}\) insect appear when viewed with a system of two identical lenses of focal length \(5.0 \mathrm{~cm}\) separated by a distance \(12 \mathrm{~cm}\) if the insect is \(10.0 \mathrm{~cm}\) from the first lens? Is the image real or virtual? Inverted or upright?

Short answer

Answer: The distance between the two lenses is 20 cm.

Step-by-step solution

Apply the lens formula for the first lens

The thin lens formula is given by \(\frac{1}{f} = \frac{1}{d_o}+\frac{1}{d_i}\), where \(f\) is the focal length, \(d_o\) is the object distance from the lens, and \(d_i\) is the image distance from the lens. For the first lens, it has a focal length of \(f_1 = 10.0 \mathrm{~cm}\). In this problem, we want to find the distance between the lenses (\(d_{12}\)), so we can rewrite the lens formula as: \(d_{i1} = \frac{f_1d_o}{(d_o - f_1)}\) Since we know that light coming out of the first lens is parallel, the image formed by the first lens is at its focal point \(f_1\).

Image formed by the first lens is at its focal point

As we know the image formed by the first lens is at its focal point, we can write the image distance as \(d_{i1} = f_1\). Then, by substituting the values, we can find the value of \(d_{12}\): \(d_{12} = d_{i2} - d_{i1}\)

Apply the lens formula for the second lens

For the second lens, with a focal length of \(f_2 = 20.0 \mathrm{~cm}\), we can apply the lens formula, and rewrite it as: \(d_{i2} = \frac{f_2d_o}{(d_o - f_2)}\) The object distance for the second lens, \(d_{o2}\), is equal to the sum of the image distance for the first lens, \(d_{i1}\), and the distance between the two lenses, \(d_{12}\).

Calculate the distance between the lenses

Substitute the known values into the formula to solve for \(d_{12}\): \(d_{12} = d_{i2} - d_{i1} = \frac{f_2(f_1 + d_{12})}{(f_1 + d_{12} - f_2)} - f_1\) After some algebraic operations, we get: \(d_{12} = \frac{f_1f_2}{(f_2 - f_1)}\) Plug in values: \(d_{12} = \frac{10 \times 20}{(20 - 10)} = 20 \ \text{cm}\)

Key concepts

Lens Formula

The lens formula is a fundamental concept in optical physics that relates the focal length of a lens to the distances of the object and the image from the lens. This formula is central to determining how lenses form images, and it is represented as \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), where \(f\) is the focal length, \(d_o\) is the object distance, and \(d_i\) is the image distance, all measured from the lens.

When using the lens formula, it's important to follow sign conventions for distances. Conventionally, distances are measured positive when they're in the direction of the incoming light and negative when they're opposite. The lens formula provides a clear method to solve problems involving lenses, like finding the position of the image formed by a lens when the object position and focal length are known.

Focal Length

Focal length is a crucial concept in optical physics, indicating how strongly a lens converges or diverges light. It is the distance from the lens to the point where parallel rays of light converge to a single point, known as the focal point. Lenses with a short focal length bend light rays more than lenses with a long focal length, leading to a greater degree of magnification.

Focal length is not just a physical property of a lens but it also determines the lens's optical power. The shorter the focal length, the higher the optical power and the more sharply it can bend light rays, allowing images to be formed over a shorter distance.

Image Distance

Image distance, denoted by \(d_i\), is the distance from the lens to the point where an image is formed. It can be positive or negative depending on whether the image is formed on the same side as the incoming light (real image) or opposite side (virtual image).

In the context of combined lenses, such as the textbook exercise, the image formed by the first lens can become the object for the second lens, making the calculation of the image distance an interesting step in solving multi-lens systems. Real images have positive image distances, while virtual images, which cannot be projected on a screen, have negative image distances.

Object Distance

Object distance, represented by \(d_o\), is the distance between the object and the lens. It is an essential parameter in applying the lens formula to determine the nature and position of the resulting image.

In practical problems involving multiple-lens systems, the object distance for one lens might become the image distance from the previous lens, and careful accounting for these distances is key to a correct solution. For instance, when calculating sequential lens setups, the object distance for the second lens is dependent on the image distance of the first, as illustrated in the exercise with the two-lens system.