Error in the focal length of a lens
f = u v/(u+v). f is the focal length of a convex lens, u the object distance and v the image distance, both taken positive.
Error in a formula
Maximum error in f
0.278
f = 20.00 ± 0.281.4%
- u0.444 × 0.50.222
- v0.111 × 0.50.0556
- Totaladd them0.278
This mixes a sum with a product, so it is worked out with the values: the sum of |slope| times error, the worst case.
f = 20.00 ± 0.28
Δf = |∂f/∂u| Δu + |∂f/∂v| Δv
This mixes a sum with a product, so it is worked out with the values: the sum of |slope| times error, the worst case. The lens formula 1/f = 1/u + 1/v has a sum in it, so the error is worked out with the values.
Other formulas
Measured the same thing several times? Use repeated readings, or find the least count of your instrument.
Error in the focal length of a lens: quick answers
What is the formula for the error in the focal length of a lens?
f is the focal length of a convex lens, u the object distance and v the image distance, both taken positive. With f = u v/(u+v), the maximum error is given by Δf = |∂f/∂u| Δu + |∂f/∂v| Δv. This mixes a sum with a product, so it is worked out with the values: the sum of |slope| times error, the worst case.
How do I work out the error with real numbers?
Take u = 30 ± 0.5 and v = 60 ± 0.5. The maximum error in f is 0.222 + 0.0556 = 0.278, so f = 20.00 ± 0.28.
Why are the errors added and never subtracted?
The error shown is the largest possible one. In the worst case every measurement is off in the direction that pushes the result the same way, so the errors pile up. They are added even when the quantity is divided or subtracted.