Question

To a fish in an aquarium, the $4.00-\mathrm{mm}$-thick walls appear to be only $3.50\mathrm{mm}$thick. What is the index of refraction of the walls?

Short answer

The index of refraction of the walls is$1.52$

Step-by-step solution

By given data.

To find the refractive index of the aquarium walls, use the equation that shows the relationship between the refractive indices of the two media, the real thickness, and the apparent thickness.

Deriving the equation for the refractive index of the aquarium

The relationship between the wall's true thickness $S$and its apparent thickness $s^{\mathrm{\prime }}$is given by,

$s^{\mathrm{\prime }}=\left(\frac{n_{\mathrm{w}}}{n_{\mathrm{aq}}}\right)s$

The refractive index of water is $n_{\mathrm{w}}$, and the refractive index of aquarium walls is $n_{\mathrm{aq}}$.

Rearrange the equation for the aquarium's refractive index.

$n_{\mathrm{aq}}=\left(\frac{s}{s^{\mathrm{\prime }}}\right)n_{\mathrm{w}}$

To find the refraction of the walls

The refractive index of the aquarium's walls is,

$n_{\mathrm{aq}}=\left(\frac{s}{s^{\mathrm{\prime }}}\right)n_{\mathrm{w}}$

Substituting $4.00\mathrm{mm}$for $S$, $3.50\mathrm{mm}$for $s^{\mathrm{\prime }}$, and $1.33$for $n_{\mathrm{w}}$.

role="math" localid="1651391770543" $n_{aq}=\left(\frac{4.00\mathrm{mm}}{3.50\mathrm{mm}}\right)1.33$

$=1.52$

As a result, the aquarium's walls have a refraction index of$1.52$