Question

The focal length of a mirror is given by `1/f=1/u+1/v` where u and v represent object and image distances respectively,The maximum relative error in f is ?

Answer 1

Given expression is

`1/f=1/u+1/v`
`=>1/f=(v+u)/(uv)`
`=>f=(uv)(v+u)^-1`

Taking `log` of both sides we get

`logf=logu+logv+log(v+u)^-1`
`=>logf=logu+logv-log(v+u)`

Differentiating with respect to each variable.

`(Deltaf)/f=(Deltau)/u+-(Deltav)/v+-(Delta(u+v))/(v+u)`
Here `Deltaf` represents error in `f` and so on. It is assumed that errors in each are independent, random, and sufficiently small

`=>(Deltaf)/f=(Deltau)/u+-(Deltav)/v+-(Deltau+Deltav)/(v+u)`
`=>(Deltaf)/f=(Deltau)/u+-(Deltav)/v+-(Deltau)/(v+u)+-(Deltav)/(v+u)`

We calculate maximum average error as below

`Deltaf=barf+-sqrt(((Deltau)/u)^2+((Deltav)/v)^2+((Deltau)/(v+u))^2+((Deltav)/(v+u))^2)`