Proof of the Thin Lens Equation

The thin lens equationallows us to find the distanceof an image from a lens given an object distanceand the focal length of the lensThe diagram below shows an example of a lens forming an image.

The image is formed by three diagnostic rays of light.

One ray travels from the object O straight through the centre of the lens and Is unrefracted.

One ray travels from the object O, parallel to the axis, and one reaching the lens, is refracted to pass through the focal point F.

One ray travels from the object O, passes through the focal point F to the lens, and is refracted by the lens to pass parallel to the axis.

The the rays of light meet at a single point where the three rays of light meet.

From the diagram aboveand

From the first equationand from the second,Equating these two givesorDividing throughout bygives usAddingto both sides then gives

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