# Propagation of Error for Focal Length

• Browntown
In summary, the conversation discusses the calculation of focal length for a lens based on measured object and image distances, with associated uncertainties. The speaker is unsure of how to propagate the error from these values into the focal length calculation and is seeking guidance. Two approaches are suggested: a statistical approach and an engineering approach. The speaker is advised to rearrange the equation to account for dependencies between the measurements before using statistical formulas.
Browntown
Homework Statement: Propagation of Error for Focal Length
Homework Equations: f = (s'*s) / (s' + s)

In my lab, we had to calculate the focal length of a lens based on object distances (s) and image distances (s') that we measured. The object distances were measured with an uncertainty of Delta s = +/- 1mm and image distances were measured with an uncertainty of Delta s' = +/- 2.5 mm.

Since we had to add, multiply and divide values, I'm not quite sure what to do to propagate the error from those two values into the one for focal length.

Any help would be much appreciated.

Thank you.

[Moderator's note: Moved here as it is of general interest and not a specific homework exercise.]

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Browntown said:
Homework Statement: Propagation of Error for Focal Length
Homework Equations: f = (s'*s) / (s' + s)

In my lab, we had to calculate the focal length of a lens based on object distances (s) and image distances (s') that we measured. The object distances were measured with an uncertainty of Delta s = +/- 1mm and image distances were measured with an uncertainty of Delta s' = +/- 2.5 mm.

Since we had to add, multiply and divide values, I'm not quite sure what to do to propagate the error from those two values into the one for focal length.

Any help would be much appreciated.

Thank you.
There are two approaches to this.
In engineering, where tolerance limits may be crucial, you simply plug in combinations of extreme values and look at the results that come out.
In science, it is standard to use a statistical approach. See http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm for the formulae that apply to products and sums. However, the formulae there assume the measurements are independent. If you treat the s' in the numerator as independent of the s' in the denominator you will get a greater error than is correct. So to use these formulas you need to put the equation into the form ##f=\frac 1{\frac 1s+\frac 1{s'}}##.

Oh ok, thank you, I'll give that a try

## 1. What is the concept of "Propagation of Error" for focal length?

The "Propagation of Error" for focal length is a mathematical concept that describes how uncertainties in the measurement of different variables can affect the final calculated value for the focal length of a lens. It takes into account the uncertainties in the measurements of the object distance, image distance, and the refractive index of the medium.

## 2. How is the "Propagation of Error" for focal length calculated?

The "Propagation of Error" for focal length is calculated using the formula: Δf = f * √((Δo/o)^2 + (Δi/i)^2 + (Δn/n)^2), where Δf is the uncertainty in the focal length, f is the focal length, Δo is the uncertainty in the object distance, o is the object distance, Δi is the uncertainty in the image distance, i is the image distance, Δn is the uncertainty in the refractive index, and n is the refractive index.

## 3. What factors can contribute to the "Propagation of Error" for focal length?

There are several factors that can contribute to the "Propagation of Error" for focal length, including errors in the measurement of the object and image distances, errors in the refractive index of the medium, and errors in the calculation of the focal length formula.

## 4. How does "Propagation of Error" for focal length affect the accuracy of the final calculated value?

The "Propagation of Error" for focal length can significantly affect the accuracy of the final calculated value. The more uncertainties there are in the measurements, the larger the uncertainty in the final calculated value will be. Therefore, it is essential to minimize errors in the measurements to obtain a more accurate value for the focal length.

## 5. Can the "Propagation of Error" for focal length be reduced?

The "Propagation of Error" for focal length can be reduced by improving the accuracy of the measurements and minimizing errors in the calculation. This can be achieved by using more precise measuring tools, taking multiple measurements and averaging them, and using more accurate formulas for calculating the focal length.

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