Calculating Uncertainty for Focal Length: 50mm + 3mm & 200mm + 5mm

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SUMMARY

The discussion focuses on calculating the uncertainty in focal length using the formula 1/u + 1/v = 1/f, where u is 50mm with a 3mm uncertainty and v is 200mm with a 5mm uncertainty. The initial calculation yielded an uncertainty of Δf = 4.68mm, but the correct uncertainty is 2.1mm. The discrepancy arises from the method used; the correct approach involves calculating the maximum and minimum focal lengths (fmax and fmin) and averaging them. This highlights the importance of understanding error propagation in optical measurements.

PREREQUISITES
  • Understanding of optical formulas, specifically 1/u + 1/v = 1/f
  • Knowledge of uncertainty propagation techniques
  • Familiarity with independent errors and their quadrature addition
  • Basic skills in algebra for rearranging equations
NEXT STEPS
  • Study the method for calculating maximum and minimum values in uncertainty analysis
  • Learn about error propagation in physics, focusing on independent and correlated errors
  • Explore practical applications of uncertainty calculations in optical systems
  • Review the concept of quadrature addition of uncertainties in measurements
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Physics students, optical engineers, and anyone involved in precision measurement and uncertainty analysis in experimental setups.

hexathane
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Hi!

So I've stumbled upon this question asking us to calculate the uncertainty associated with the focal length.
1/u + 1/v = 1/f
u = 50mm with uncertainty 3mm
v= 200mm with uncertainty 5mm

F is calculated as 40mm, what's the uncertainty in this value?

My first approach was to rearrange the formula to make f the subject, i.e. f= uv/u+v, then i applied the formula
Δf/f = Δu/u + Δv/v +Δ(u+v)/(u+v)

Substituting values,

Δf/40 = 3/50 + 5/200 + 8/250
Δf = 4.68

However, the correct answer is 2.1. From the solutions, i understand that another approach to this question would be to calculate fmax and fmin and then add them up and divide them by two.

Puzzled as to why my earlier steps doesn't yield the correct answer though, any help would be greatly appreciated!

Thank you :-)
 
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Independent errors are added in quadrature: http://www.physics.rutgers.edu/ugrad/389/errors.pdf
 

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