I need to find the focal length of a lens by using equation 1/u + 1/v=1/f
I have: u= 50+-3 mm
v= 200+-5 mm
I calculate the value of f as 40mm. Now i need to find the uncertainty in this value. I have two approaches, but only the second one is correct. I do not know what is wrong with the first one.
FIRST APPROACH : since f=(uv)/(u+v)
Delta f/f= Fractional error of f= fractional error of u+ fractional error of v + fractional error of (u+v)
From this the uncertainty is 4.7 mm
SECOND APPROACH:we have
Fractional error of 1/f = fractional error of f
So delta( 1/f) = delta(f)/f^2 (*)
Similarly (*) is true for u and v in place of f
We have : delta(1/f) = delta(1/u) + delta(1/v)
So delta(f)/f^2= delta(u)/u^2 + delta(v)/v^2
From this delta(f) is 2.1mm which is correct
What is wrong with my first attempt?
Best Answer
The problem with your first approach is that you are assuming that the uncertainties in $u$, $v$ and $u+v$ are independent, when clearly they are not, they are highly positively correlated (when they are all positive). Hence you overestimate the uncertainty.
I should just add that I think both of your approaches are incorrect if you understand the error bar to mean the standard deviation of your estimate. Independent uncertainties should be combined in quadrature. I get $\delta F= 1.9$ mm.