# Lenses. Equation of Lenses. Image Formation of Lenses Revision Notes

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12.9Lenses. Equation of Lenses. Image Formation of Lenses

In these revision notes for Lenses. Equation of Lenses. Image Formation of Lenses, we cover the following key points:

• What are lenses?
• How many types of lenses are there?
• What are the main features of lenses?
• How the image is formed in converging and diverging lenses?
• What is the equation of lenses?
• How to find the magnification produced by lenses?
• What are aberrations in lenses?
• How to combine various optical tools in an optical system?

## Lenses. Equation of Lenses. Image Formation of Lenses Revision Notes

Lenses are optical tools used to enlarge or reduce the size of images by means of refraction of light. A lens 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). Converging lenses are thicker at middle and thinner in extremities, while diverging lenses are thinner at middle and thicker in extremities.

All lenses except plane-concave and plane-convex ones have two foci, because they are formed by joining two spherical parts. As a result, they have two centres (one in each side) as well.

Like in spherical mirrors, we have to use the special rays to build up the image in lenses. However, unlike in curved mirrors, two special rays are enough to build the image in lenses. They are:

1. The ray that originates from the top of object, is incident to the lens in parallel to the principal axis and after refracting through lens, it passes through focus in converging lenses while in diverging lenses the extension passes through focus.
2. The ray that originates from the top of object, passes through the middle of lens (point O) then moves away without changing direction, as is normally incident to the lens surface.

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.

1. The object is beyond the centre of curvature. In this case, the image is formed at the other side of lens, between focus and centre of curvature; it is diminished and inverted. The image is also real because it is produced from the two reflected rays and not from their extensions.
2. The object is at centre of curvature. In this case, the image will form at the other centre of curvature; it has the same size as the object and is inverted. The image is real because it is produced from the two reflected rays and not from their extensions.
3. The object is between centre of curvature and focus. The image is formed at the other side of lens, beyond the other centre of curvature. The image is magnified and inverted; it is real because is produced from the two reflected rays and not from their extensions.
4. The object is at focus. In this case, there is no image as the refracted rays are parallel.
5. The object is close to the lens than focus. In this case, the reflected rays diverge and therefore, we take their extension to build up the image. As a result, the image is at the same side to the object; it is enlarged, erect and virtual, as it is obtained by considering the rays' extensions.
6. The object is at infinity. In this case, the image has no dimensions. It is a bright point at focus as all parallel rays coming from the object converge at focus.

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 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

1/do + 1/dî = 1/F

The sign rules are identical to those used in spherical mirrors, i.e.

1. The object's distance do is always taken as positive.
2. The image's distance dîis taken as positive when the image is real, otherwise it is negative.
3. Focal length F is positive for converging lenses and negative for diverging ones.

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:

M = himage/hobject

where h stands for height, and

M = dimage/dobject

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.

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.

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