Brief overview on Errors and Error Propagation
Geometrical Optics
Brief overview on Errors and Error Propagation
Ideal Case: Suppose we have a set of measures of an observable n
n
1.3
1.35
1.49
2.37
1.29
1.38
1.42
1.32
Mean value, standard deviation
and standard error:
Notes
hni =
v
u
u
=t
Too large
Result:
n = hni ±
n
N
1 X
ni
N i=1
1
N
1
N
X
(hni
i=1
n= p
N
ni )
2
Brief overview on Errors and Error Propagation
Geometrical Optics
Propagation of Errors
Many times we got to measure things indirectly
To measure A, we measure x1 , x2 , . . .
A is a known function of x1 , x2 , . . . .
We measure directly x1 , x2 , . . . . So we have: hxi i and their
uncertainties xi
The uncertainty for our measurement of A is:
s✓
◆2
✓
◆2
@A
@A
2
A=
( x1 ) +
( x2 ) 2 + . . .
@x1
@x2
Our result to report is:
A = A (hx1 i, hx2 i, . . . ) ±
A
Brief overview on Errors and Error Propagation
Geometrical Optics
Example of propagation of errors
Consider the index of refraction n of a glass.
We are going to measure angles ✓i and ✓t (the incident and
transmitted angles) n = sin ✓i / sin ✓t .
Suppose that experimentally we found:
✓i = 32.0 ± 1.0
✓t = 23.3 ± 0.8
from where we get the mean value of n to be:
n = 1.34
How much is the uncertainty in the index of refraction?
Brief overview on Errors and Error Propagation
Geometrical Optics
Applying the formula for error propagation:
s✓
◆2 ✓
@n
@n
n=
✓i +
@✓i
@✓t
Using:
✓t
◆2
@n
cos ✓i
=
@✓i
sin ✓t
and something similar for the derivative with respect to ✓t we find:
s✓
◆2
✓
◆2
cos ✓i
sin ✓i cos ✓t
n=
✓i2 +
✓t2
sin ✓t
sin2 ✓t
r
⇣ ⇡ ⌘2
⇣ ⇡
⌘2
= 4.60 ⇥
1 + 9.68 ⇥
0.8
180
180
= 0.06
Then the result to report is:
n = 1.34 ± 0.06
Brief overview on Errors and Error Propagation
Geometrical Optics
Other ways to present data
Discrepancy
Sometimes we have an expected value or a given value (say Ag ) of an
observable A.
We report how different this value is compared to the value hAi we
find experimentally:
Absolute Discrepancy:
A.D. = |Ag
hAi|
|Ag
hAi|
Ag
Relative Discrepancy:
R.D. =
Relative Discrepancy, %:
R.D. =
|Ag
hAi|
⇥ 100%
Ag
Brief overview on Errors and Error Propagation
Geometrical Optics
Geometrical Optics
Ligth:
Electromagnetic wave:
associated a wavelength
electricmagnetic field.
and amplitudes E0 and B0 of the
frequency ⌘ f =
c
Low energy
In Optics we concentrate in the visible spectrum:
Low energy (keV or so).
Not too low wavelengths.
Brief overview on Errors and Error Propagation
Geometrical Optics
Geometrical Optics: rays approximation
If d ⌘ av. size of objects in our experiment
If
⇠d
If ⌧ d
rays.
) Wave effects are important.
) Wave effects are negligible, light can be treated as
With the equipment and laser beam we have for project # 1, give a
number of how good is the geometrical approximation
Brief overview on Errors and Error Propagation
Geometrical Optics
Reflection and Refraction
When Light hits a surface there is a certain amount that is reflected
from the surface and certain amount that is transmitted through the
surface.
For the transmitted light:
For the reflected light:
Law of Reflection
✓i = ✓r
Snell’s Law
ni sin(✓i ) = nt sin(✓t )