In the article “Temperature-Dependent Optical Constants of Water Ice in the Near Infrared: New Results and Critical Review of the Available Measurements” (B. Rajaram, D. Glandorf, et al., Applied Optics, 2001:4449-4462), the imaginary index of refraction of water ice is presented for various frequencies and temperatures. At a frequency of 372.1 cm-1 and a temperature of 166 K. the index is estimated to be 0.00116. At the same frequency and at a temperature of 196 K, the index is estimated to be 0.00129. The uncertainty is reported to be 10-4 for each of these two estimated indices. The ratio of the indices is estimated to be 0.00116/0.00129 = 0.899. Find the uncertainty in this ratio.

Solution 10E

Step1 of 3:

We have a frequency of 372.1 cm-1 and a temperature of 166 K. the index is estimated to be 0.00116. and the same frequency and at a temperature of 196 K, the index is estimated to be 0.00129.

That is

And also we have uncertainties .

Let r =

=

= 0.8992

We need to find the uncertainty of r.

Step2 of 3:

Consider the ratio

r =

Differentiate above equation with respect to “” then

=

=

=

=

= 775.1937

Hence, = 775.1937.

Again consider the ratio

r =

Differentiate above equation with respect to “” then

=

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