91影视

The incident angle, ${胃}_1=50掳$

The length of the medium 1, $L_1=20渭\text{m}$

The length of the medium 3, role="math" localid="1663069065664" $L_3=25渭\text{m}$

The refractive index of medium 1, $n_1=1.6$

The refractive index of medium 3,$n_3=1.45$

Snell's law of refraction states that: The incident ray, the refracted ray, and the normal at the point of incidence all lie in the same plane. The ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for a pair of given environments.

The expression for Snell鈥檚 law is given as follows.

$n\sin {胃}_1=n_1\sin {胃}_2=n_2\sin {胃}_3.........$

The expression to calculate the time is given as follows.

$t=\frac{n_{3}L}{c}$ .......(1)

Let assume angle ${胃}_2,{胃}_3,{胃}_4,{胃}_5$ and ${胃}_6$ are the refracted angle in first, second, third, fourth, fifth, and air respectively.

Use Snell鈥檚 law for the air and first medium interface.

$n_{air}\sin {胃}_1=n_1\sin {胃}_2$

Use the Snell鈥檚 law for the first and second medium.

$n_1\sin {胃}_2=n_2\sin {胃}_3$

Substitute $n_{air}\sin {胃}_1$ for $n_2\sin {胃}_3$into above equation.

$n_{air}\sin {胃}_1=n_3\sin {胃}_4$

Use the Snell鈥檚 law for the second and third medium.

$n_2\sin {胃}_3=n_3\sin {胃}_4$

Use the Snell鈥檚 law for the third and fourth medium

$n_3\sin {胃}_4=n_4\sin {胃}_5$

Substitute $n_{air}\sin {胃}_1$for $n_3\sin {胃}_4$into above equation.

$n_{air}\sin {胃}_1=n_4\sin {胃}_5$

Use the Snell鈥檚 law for the fourth and fifth medium.

$n_4\sin {胃}_5=n_5\sin {胃}_6$

Substitute $n_{air}\sin {胃}_1$for $n_4\sin {胃}_5$into above equation.

$n_{air}\sin {胃}_1=n_5\sin {胃}_6$

Use the Snell鈥檚 law for the fifth and air medium.

$n_5\sin {胃}_6=n_{air}\sin {胃}_7$

Substitute $n_{air}\sin {胃}_1$for $n_5\sin {胃}_6$into above equation

$\begin{array}{rcl}{n}_{air}\sin {胃}_1& =& n_{air}\sin {胃}_7\\ \sin {胃}_1& =& \sin {胃}_7\\ {胃}_1& =& {胃}_7\end{array}$

Substitute $50掳$ for ${胃}_1$ into above equation.

${胃}_7=50掳$

Therefore, the angle of immerge of the light into air is 50 degree.

Consider the distance travelled by the light in third medium.

Recall the equation (2),

$\begin{array}{rcl}{n}_{air}\sin {胃}_1& =& n_3\sin {胃}_4\\ \sin {胃}_4& =& \frac{n_{air}}{n_3}\sin {胃}_1\end{array}$

Substitute 1 for $n_{air}$ , $1.45$for ${胃}_3$and $50掳$for ${胃}_1$ into above equation.

$\begin{array}{rcl}{胃}_4& =& \frac{1}{1.45}\sin 50掳\\ \sin {胃}_4& =& 0.528\\ {胃}_4& =& {\sin }^{-1}\left(0.528\right)\\ {胃}_4& =& 31.89掳\end{array}$

From the figure,

$\begin{array}{rcl}{胃}_4& =& \frac{L_3}{L}\\ L& =& \frac{L_3}{\cos {胃}_4}\end{array}$

Substitute$25渭\text{m}$ for $L_3$ and $31.89掳$ for ${胃}_4$ into above equation.

$$\begin{gathered} L=\frac{25脳{10}^{-6}}{\cos 31.89掳} \\ L=2.944脳{10}^{-5}\text{m} \end{gathered}$$

Calculate the time taken by the light to travel through medium 3.

Substitute $2.944脳{10}^{-5}\text{m}$ for$L$, $1.45$ for $n_3$ and $3脳{10}^8\raisebox{1ex}{$\text{m}$}\!\left/ \!\raisebox{-1ex}{$\text{s}$}\right.$ for c into equation (1).

$\begin{array}{rcl}t& =& \frac{1.45脳2.944脳{10}^{-5}}{3脳{10}^8}\\ & =& \frac{4.2688脳{10}^{-13}}{3}\\ & =& 1.42脳{10}^{-13}\\ & =& 0.142\text{ps}\end{array}$

Hence the time taken by the light to travel through medium 3 is $0.142\text{ps}$.