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Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
---|---|---|---|---|---|---|
12.9 | Lenses. Equation of Lenses. Image Formation of Lenses |
In this Physics tutorial, you will learn:
Why do people use optical glasses? What is the shape of these glasses?
Why normal glasses such as window glasses cannot be used to see better?
What happens when you direct a magnifying glass towards sunlight?
What happens to sunlight when it passes through a magnifying glass?
This tutorial is entirely focuses on lenses as the most important tools in optical systems. Since their mathematical apparatus is not very different from curved mirrors, we will deal more with some technical details that were not discussed in the previous tutorial Mirrors, Equation of Curved Mirrors and Image Formation in Plane and Curved Mirrors. Therefore, the tutorial we will explain now is an extension of the previous tutorial mentioned above.
Lenses are optical tools used to enlarge or reduce the size of images. This is made possible through refraction occurring when light passes through them.
In a certain sense, lens is an extension of the concept of curved mirrors because unlike the latter, a lens can be used in both sides, as it is a piece of transparent material, usually circular in shape, with two polished surfaces, either or both of which is/are curved.
There are two main categories of lenses: converging (concave) and diverging (convex). They are shown in the figure below.
From the figure, you can see that a converging lens is thicker at middle and thinner in extremities, while diverging lenses are thinner at middle and thicker in extremities.
The lenses shown in the figure are both regular, i.e. they behave equally in both directions. However, there is a variety of lenses with non-identical sides. Some of them are shown in the figure below.
All lenses except plane-concave and plane-convex ones have two focîbecause they are formed by joining two spherical parts. As a result, they have two centres as well (one in each side). They can be at different distances from the lens depending on their curvature but for simplicity, we will only consider lenses that have identical sides, i.e. which have the same value for focîand centres in both directions, as shown in the figure below.
Like in spherical mirrors, we have to use the special rays to build up the image in lenses. However, in this case, two special rays are enough to build the image. They are:
The two special rays used in lenses are shown in the figure below.
Converging lenses are very similar in concept to concave mirrors. Therefore, we have again six possible cases of image formation in converging lenses depending to the position of object in respect to the lens.
Like in convex mirrors, the image formation in diverging lenses has only one case. The image is formed closer to the mirror than focus. It is erect and diminished. Since the image is obtained from rays extensions, it is virtual.
The image produced by a lens is shown in the figure below.
If the incident rays coming from the object refract to the lens from left to right, determine:
The equation of lenses is identical to that of curved mirrors. Thus, if we denote by do the position of object in respect to the lens, by dîthe position of image and by F the focus (focal length), we obtain
The sign rules are identical to those used in spherical mirrors, i.e.
An object is placed 12 cm on the left of a converging lens of focal length equal to 8 cm.
to find the position of image in respect to the lens. Thus, substituting the values (given that do = + 12 cm and F = + 8 cm) we obtain
Therefore dî= 24 cm. This result means the image is produced at 24 cm on the right of the lens as shown in the figure below.
The most important feature of lenses (for which they are produced) is the magnification they provide. The approach is the same as magnification in curved mirrors. This means we can use two formulae for the calculation of magnification:
where h stands for height, and
For example, if we take the height of the object in the previous example equal to 3 cm, we can work out the image's height by combining the two formulae of magnification:
Substituting the known values, we obtain for the image's height:
Thus,
When we use lenses, light passes from air to another medium (usually glass) and refracts through it. It is a known fact that all lenses have their own thickness, which is different in various part of them. We have stated earlier that converging lenses are thicker at middle and diverging lenses are thicker at edges.
Aberration is the non-regular deviation of light rays through lenses due to non-uniform thickness, causing images of objects to be blurred.
In an ideal system, every point on the object will focus to a point of zero size on the image. However, in reality this does not occur, because lenses are not ideal optical tools. As a result, parallel rays do not converge at a single dimensionless point as assumed earlier, but in a zone around focus, as shown in the figure.
In curved mirrors, aberrations are less visible as light does not enter inside the glass but it is reflected by the mirroring surface. Therefore, curved mirrors are more preferable than lenses to be used in powerful optical systems such as telescopes and microscopes.
We can combine optical tools such as plane and curved mirror with lenses to produce new optical systems, like we did in the previous tutorial where two curved mirrors were combined to produce new images. Let's see an example in this regard.
An optical magnifying device is composed by a converging lens and a concave mirror as shown in the figure.
What is the total magnification produced by this optical system if the object is placed at the distance shown?
First, let's calculate the position of the image produced by the converging lens. From the figure, we can extract the following clues:
Applying the equation
we obtain
This means the image produced by the converging lens is 24 cm on its left. Thus, the magnification produced by the converging lens is
This means the image produced by the converging lens is 4 times larger than the object.
This image acts as an object for the concave mirror. Its distance from the mirror is
We have:
Thus, the second image is 40 cm on the left of the converging lens. In this case, there is no magnification as
Thus, the mirror is used only to turn the image upright.
Therefore, the total magnification produced by this system is
This means the final image is 4 times larger than the object.
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