User:Inconspicuum/Physics (A Level)/Lenses

Curvature of Wavefronts

Light can be viewed as beams travelling between points. However, from most light sources, the light radiates outwards as a series of wavefronts. Light from a light source is bent – wavefronts of light have a property known as curvature.

As light travels further away from its source, its curvature decreases. Consider a sphere expanding gradually from a point, which represents a given wavefront of light. As the sphere expands, the curvature of its surface decreases when we look at any part of the surface with a constant area. It should be noted at this point that light from a source infinitely far away has 0 curvature - it is straight. This is useful, as ambient light (light from a source that is far away) can be assumed to have a curvature of 0, as the difference between this and its actual curvature is negligible.

The curvature of a wavefront is given as:

$C={\frac {1}{v}}$ ,

where v is the distance from the wavefront to the in-focus image depicted by the light. Curvature is measured in dioptres (D).

Power of lenses

The function of a lens is to increase or decrease the curvature of a wavefront. Lenses have a 'power'. This is the curvature which the lens adds to the wavefront. Power is measured in dioptres, D, and is given by the formula:

$P={\frac {1}{f}}$ ,

where f equals the focal length of the lens. This is the distance between the lens and the point where an image will be in focus, if the wavefronts entering the other side of the lens are parallel.

The Lens Equation

Overall, then, the formula relating the curvature of the wavefronts leaving a lens to the curvature of the wavefronts entering it is:

${\frac {1}{v}}={\frac {1}{u}}+{\frac {1}{f}}$

where v is the distance between the lens (its centre) and the in-focus image formed, u is the distance between the lens (its centre) and the object which the in-focus image is of, and f is the focal length of the lens. The power of the lens can be substituted in for the reciprocal of f, as they are the same thing.

The Cartesian Convention

If we were to place a diagram of the lens on a grid, labelled with cartesian co-ordinates, we would discover that measuring the distance of the object distance is negative, in comparison to the image distance. As a result, the value for u must always be negative. This is known as the Cartesian convention.

This means that, if light enters the lens with a positive curvature, it will leave with a negative curvature unless the lens is powerful enough to make the light leave with a positive curvature.

Types of Lens

There are two types of lens:

Converging lenses add curvature to the wavefronts, causing them to converge more. These have a positive power, and have a curved surface which is wider in the middle than at the rim.

Diverging lenses remove curvature from the wavefronts, causing them to diverge more. These have a negative power, and have a curved surface with a dip in the middle.

Magnification

Magnification is a measure of how much an image has been enlarged by a lens. It is given by the formula:

$M={\frac {h_{2}}{h_{1}}}$

where h₁ and h₂ are the heights of the image (or object) before and after being magnified, respectively. If an image is shrunk by a lens, the magnification is between 0 and 1.

Magnification can also be given as:

$M={\frac {v}{u}}$

where v and u are the image and object distances. Therefore:

$M={\frac {h_{2}}{h_{1}}}={\frac {v}{u}}$

An easy way to remember this in the middle of an exam is the formula:

$I=AM$

where I is image size, A is actual size of the object M is the magnification factor.