Lenses

Concavity and convexity

There are two types of lenses, based on the nature of the rays once they strike the lens, which can be categorised as converging or diverging.

Converging and diverging lenses

The two sides of a convex (converging) lens curve inwards. It is also known as a converging lens because it converges the light rays impact it onto a focal point. On a screen, for example, a converging lens can provide either a real or a virtual image. Convexity is also a property of the human eye’s lens.

Both sides of a concave (diverging) lens are bent outwards. It is also known as a diverging lens because, when parallel rays are incident on it, the emerging rays spread out or diverge. A diverging lens creates a virtual picture but not a real image.

The principal axis is a horizontal line that passes through the lens’s centre and is perpendicular to the lens. Light rays parallel to the primary axis are concentrated – or seem to diverge from it, in the case of a diverging lens – at the focal point F. The focal length f refers to the distance between the lens’s centre and the focal point. A lens’s focal length is partially controlled by its shape: a lens with a short focal length would be heavily curved (with a small radius of curvature).

The lens of the eye

The lens of the eye converges incident light rays from an object, bringing them to a point of focus on the retina. The focal length of the lens must fluctuate in order to concentrate light from both distant and close objects. The muscles change the curvature of the lens to achieve this.

The lens is connected to the ligaments, which, in turn, are connected to the ciliary muscles, which allow the lens to change its shape and size.

Accommodation is the capacity of the lens of the eye to vary its focal length so that objects at different distances can be brought into focus on the retina. The lens’s focusing power (on closer objects) rises when it has a highly curved surface. A refracting surface’s power is calculated as below.

\[Power = \frac{1}{f}\]

Here, f is the focal length in metres. The dioptre (D) is a unit of power that is equivalent to m-1. Positive power is assigned to a converging lens, whereas negative power is assigned to a diverging lens. The sum of the refracting powers of a group of lenses or surfaces is their total refracting power.

Ray diagrams

A ray diagram can be used to determine where an image will be created by a lens of known power. The lens’s axis is depicted as a straight vertical line, the primary axis as a horizontal line, the focal point F as dots on the principal axis on each side of the lens, and an expanded object as an arrow standing on the principal axis. The illustration is done to scale.

Three distinct rays emerge from the object. According to the criteria listed below each figure, the passage of these through the lens may be anticipated, and the focused picture can be located. The picture will be scaled in terms of distance from the lens and size. In ray diagrams, the rays are depicted to change direction solely at the lens axis.

In the ray diagram, there are three rays that we take into consideration. One ray passes through the optical centre without being refracted. A ray of light travelling parallel to the principal axis refracts through the lens and passes through the focal point on the other side. A ray of light travelling from the focal point to the lens is refracted through the lens and continues its way on a path parallel to the principal axis.

The lens formula

The lens formula may also be used to compute the location of a focused picture. The following formula ties a lens’s focal length, f, to the object distance, u, and image distance, v:

\[\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\]

There is a sign convention when using the lens formula. It is possible to have a negative image distance, v. In this case, the picture is virtual, on the same side of the lens as the object, and upright if the value of v is negative. When the object distance is smaller than the focal length, it will be a virtual image on the same side of the lens as the actual object. The situation for convex lenses is shown in the following diagram.

Magnification

A lens’s image might be smaller or larger than the object it is forming. The ratio of image height to object height determines the magnification of an image generated by a lens. So, in the figure below, we have:

\[Magnification = \frac{A'B'}{AB} = \frac{v}{u}\]

If v is negative, magnification must also be negative, indicating that the picture is virtual and upright rather than actual and inverted.