Measuring Refractive Index

Collecting angle pairs to measure the refractive index of a material

To measure a refractive index you simply send a ray into the material at known angles, record where it bends, and apply Snell's law to the angle pairs.

A standard procedure to determine the refractive index of a glass block:

• Place the block on paper and trace its outline.
• Direct a single narrow ray at the entry face at a measured angle of incidence and mark the incident and emerging rays.
• Remove the block, draw the normal, and measure the angle of refraction inside the glass.
• Repeat for a range of incidence angles (e.g. $20°, 30°, 40°, 50°, 60°$).
• For each pair, $n = \dfrac{\sin\theta_1}{\sin\theta_2}$ (with air as medium 1, $n_1 = 1.00$), or plot $\sin\theta_1$ against $\sin\theta_2$ and take the gradient.

To measure refractive index: trace the block, send a narrow ray in at several measured incidence angles, record the matching refraction angles, then use $n = \dfrac{\sin\theta_1}{\sin\theta_2}$ for each pair or take the gradient of a $\sin\theta_1$ vs $\sin\theta_2$ graph.

Pause, copy the highlighted method into your book before moving on.

Turning a measured angle pair into a refractive index

We just saw how to collect angle pairs. That raises a question: given one measured pair, what value of $n$ does it give, and how do we check it against the accepted value? This card answers it → apply Snell's law, then calculate the percentage error.

For each angle pair you divide the sine of the incidence angle by the sine of the refraction angle, that ratio is the refractive index of the glass.

With air as medium 1 ($n_1 = 1.00$), Snell's law gives $1.00 \times \sin\theta_1 = n \times \sin\theta_2$, so $n = \dfrac{\sin\theta_1}{\sin\theta_2}$.

For air→glass, $n = \dfrac{\sin\theta_1}{\sin\theta_2}$. Compare with the accepted value using percentage error $= \dfrac{|n_{\text{measured}} - n_{\text{accepted}}|}{n_{\text{accepted}}} \times 100\%$, a small percentage error means a more accurate result.

Add the highlighted calculation and error formula to your notes before the check below.

Judging how trustworthy the measured refractive index is

We just learned to turn data into a value of $n$. That raises a question: how good is that value, and what could have pushed it away from the true number? This card answers it → distinguishing accuracy from reliability and identifying error sources.

Assessing accuracy means asking two separate things: how consistent were the repeats (reliability) and how close was the result to the accepted value (accuracy), and the two can disagree.

Common sources of error in this experiment, and how to reduce them:

• Thick ray / fuzzy marks (random): use the thinnest ray possible and mark ray centres; repeat and average.
• Protractor reading and parallax (random): read at eye level and measure each angle twice.
• Mis-drawn normal or block shifting (systematic): draw the normal carefully at $90°$ and keep the block fixed while marking.
• Plotting a graph of $\sin\theta_1$ vs $\sin\theta_2$ and taking the line-of-best-fit gradient averages out random scatter and improves reliability.

A small percentage error (a few per cent) indicates good accuracy. Consistent repeats indicate good reliability, but if a systematic error is present, the result can be reliable yet inaccurate, so both must be assessed.

Assess the measurement two ways: accuracy (small percentage error from the accepted value) and reliability (consistent repeats). Random errors (thick rays, parallax) are reduced by repeating, averaging and plotting a line of best fit; systematic errors (mis-drawn normal) shift every reading and can make a result reliable but inaccurate.

Pause, copy the highlighted accuracy-vs-reliability summary into your book before the check below.