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Problem 23 Hard Difficulty

A person looking into an empty container is able to see the far edge of the container’s bottom, as shown in Figure P22.23a. The height of the container is $h,$ and its width is $d$ . When the container is completely filled with a fluid of index of refraction $n$ and viewed from the same angle, the person can see the center of a coin at the middle of the container's bottom, as shown in Figure P 22.23b . (a) Show that the ratio $h / d$ is given by
$$\frac{h}{d}=\sqrt{\frac{n^{2}-1}{4-n^{2}}}$$
(b) Assuming the container has a width of 8.00 cm and is filled with water, use the expression above to find the height of the container.

Answer

a. $\begin{array}{l}{\text { Using Snell's law and simple trigonometric functions it can be proved that }} \\ {\qquad \frac{h}{d}=\sqrt{\frac{n^{2}-1}{4-n^{2}}}}\end{array}$
b. $\text { The height of the container is } h=4.73 \mathrm{cm} .$

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Video Transcript

we can show the given formula from starting from the Snell's law. That is, in one time, science data. One is 1/4 and two down. Sign that, too. Let's always accordion from the figure. Then we can write. Scientific data will be called D, divided by square root off each square, plus the square. Ah, then sign that dish will be called to D 52 Um, it's weird off red square, plus de divide by two whole square, then in one week in sub student wanted to be in and that I want to be that dish and to anyone and better to be better. Then the bow expression will become end. Is it cool to D, who to divided by swirled off X square, plus de divided by two full square that isn't cool to D, divided by throwed off its square plus the square. So writing both equations together and squared, divided by four times each square, plus the square divide before is equal to one over. It's Queer Plus de Square. Um, then Father isn't defined. We have n Square right by four D square into each or D hold square, plus one over four this is equal to one or D square edge over D Square plus one, then, ah, following the steps, we can right, um, and square edge or de sole square. Plus, uh, and square is equal to fall into each over de whole square plus one. We can further simplify this and write four minus and square into itch or D swear. Physical toe end square minus one. From here, we can just ah, Saul forage or D. This is Jordi will become end square minus and divided by four minus and square square root fence. We showed this formula, but be when, um, the musical to eight and and is equal to 1.333 Ah, then our actual become. It will be that you were. It is physical, too, simply putting those values in their former mother. Krishna showed 1.3 to the three square minus one, divided by four minus 1.333 square. This simplify someone applying eight year this simplifies into, um four foreign 73 centimeter. So the height of container is a 4.73 centimeters

Wesleyan University
Top Physics 103 Educators
Elyse G.

Cornell University

Marshall S.

University of Washington

Farnaz M.

Other Schools

Jared E.

University of Winnipeg