Why is Bessel's method more accurate than thin lens equation when determining focal length?

Question

I have just completed a physics practical where we had to measure the focal lengths of various lenses using the thin lens equation and then Bessel's method. The task then asked for an explanation of which method is more accurate. From googling around I see that Bessel's method is usually stated as more accurate, but I am unsure as to why, as it requires more measurements (that are not averaged), which should lead to a higher uncertainty on the determined focal length. Could anybody enlighten me?

For reference, here is the focal length $f$ from the thin lens equation:

$f=\frac{bg}{b+g}$

where $b$ and $g$ are the distances of the image and object from the lens, respectively.

Here is Bessel's method:

$f = \frac{d^2-e^2}{4d}$

where $d$ is the distance between the image and the object, and $e$ is the distance between the two lens positions that yield a focussed image.

As a final note, I am struggling to understand the point of the second measurement for Bessel's method at all, as the derivation we were given states that the second position of the lens can simply be determined from the first. The derivation (matching the image above) is as follows:

$\frac{1}{f} = \frac{1}{g_1} + \frac{1}{b_1} = \frac{1}{g_2} + \frac{1}{b_2}$

and:

$g_1 + b_1 = g_2 + b_2 = d$

$b_1 = g_2; \hspace{10pt} g_1 = b_2$

therefore:

$b_1 - g_1 = e$

giving:

$b_1 = \frac{d+e}{2}; \hspace{10pt} g_1 = \frac{d-e}{2}$

$f = \frac{d^2-e^2}{4d}$

So if we first measure $b_1$ and $g_1$, and we already know that $b_1 = g_2$ and $g_1 = b_2$, whats the point in measuring $b_2$ and $g_2$?

Any help would be greatly appreciated!

Answer 1

If you have a thin lens and perfect measurements, both methods give the same result, and as you note, you don't need the second measurement. However, imagine you can't measure the position of the center of the lens accurately, for example if it is asymmetric, so one side has a different curvature than the other, or any circumstance where it's absolute position is not easy to determine. Bessel's method, by using two measurements, just needs to measure the displacement of the lens, so you can measure from any point on the lens to the corresponding point of the displaced lens.