Gaussian Formula

= Derivation of Gaussian Formula =

Carl Friedrich Gauss established a formula based on the relation between the object distance(u), image distance (v) and the focal length of a spherical mirror. That is,

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

Now this can be derived as follows:Let, an object be placed at the centre of curvature of a spherical mirror.So, the object distance will be equivalent to twice the focal length, that is,

:$u\; equiv\; \{2\}\{f\}$ ------ {1}

Also,:$v\; equiv\; \{2\}\{f\}$

We know that,the sign convention is based on the Cartesian Plane,Therefore, in a spherical mirror for a real image, the following points comes out from the Cartesian Plane:(i) the object distance (u) is always taken as negative.(ii) the image distance (v) is always taken as negative(iii) the focal length (f) and radius of the curvature (R) are always negative.

Now,From (1), using the above points we get,

$-u\; equiv\; \{-2\}\{f\}$

Cancelling the negative signs from both sides, we get

$u\; equiv\; \{2\}\{f\}$

$frac\; \{1\}\{f\}\; =\; frac\; \{2\}\{u\}$

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

now replacing (u) by (v), we get

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

Thus, we get the derivation of this formula.

This formula is valid in any situations for a spherical mirror for any position of an objectitions of an object.With this formula we can get any of the values of u, v and f with respect to a given mirror. But,either of the two should be there to compute the required value of any these quantities.