Chapter 23, Problem 85QAP

To determine

The speed of light and refractive index of different optical materials

Answer to Problem 85QAP

The completed table is shown below.

Optical material with percentage of time required for light to pass through compared to an equal length of vacuum	Speed of light(meter/sec)	Index of refraction
100%	$3\times {10}^8$	1
125%	$2.4\times {10}^8$	1.25
150%	$2\times {10}^8$	1.5
200%	$1.5\times {10}^8$	2
500%	$6\times {10}^7$	5
1000%	$3\times {10}^7$	10

Explanation of Solution

Given info:

The given table is shown below,

Optical material with percentage of time required for light to pass through compared to an equal length of vacuum	Speed of light(meter/sec)	Index of refraction
100%
125%
150%
200%
500%
1000%

Formula used:

The speed of light in a given optical material can be expressed as the ratio of the time that is required for light to travel through a length of vacuum to the time required for light to travel through the same length of the optical material.

Time required for light to travel through a length of vacuum = ${\text{t}}_{\text{vacuum}}$

Time required for light to travel through the same length of optical material = ${\text{t}}_{\text{medium}}$

The ratio of time is, $\text{r =}\frac{{\text{t}}_{\text{vacuum}}}{{\text{t}}_{\text{medium}}}$

The velocity of light in the optical medium is,

  ${\text{v =v}}_{\text{vac}}\times \left(\frac{{\text{t}}_{\text{vacuum}}}{{\text{t}}_{\text{medium}}}\right)\text{ = (3}\times {\text{10}}^{\text{8}}\text{)}\times \left(\frac{{\text{t}}_{\text{vacuum}}}{{\text{t}}_{\text{medium}}}\right)\text{m/s}$

Refractive index of an optical material is calculated by using the following relation

  $\text{n= }\frac{{\text{v}}_{\text{vac}}}{\text{v}}=\frac{\text{3}\times {\text{10}}^{\text{8}}\text{ m/s}}{\text{v}}$

Calculation:

For 100%,

  $\text{r =}\frac{{\text{t}}_{\text{vacuum}}}{{\text{t}}_{\text{medium}}}=\frac{{\text{t}}_{\text{vacuum}}}{\frac{100}{100}\times {\text{t}}_{\text{vacuum}}}=\frac{1}{1}=1$

Similarly,

The ratio is calculated for different percentage as shown below.

Optical material with percentage of time required for light to pass through compared to an equal length of vacuum	${\text{t}}_{\text{medium}}=\%\text{of}{\text{t}}_{\text{vacuum}}$	$\text{r =}\frac{{\text{t}}_{\text{vacuum}}}{{\text{t}}_{\text{medium}}}$
100%	1	1
125%	1.25	0.8
150%	1.5	0.67
200%	2	0.5
500%	5	0.2
1000%	10	0.1

Now, the velocity of light in a material is,

  ${\text{v =v}}_{\text{vac}}\times \left(\frac{{\text{t}}_{\text{vacuum}}}{{\text{t}}_{\text{medium}}}\right)\text{ = (3}\times {\text{10}}^{\text{8}}\text{)}\times \left(\frac{{\text{t}}_{\text{vacuum}}}{{\text{t}}_{\text{medium}}}\right)\text{m/s}$

The refractive index is,

  $\text{n= }\frac{{\text{v}}_{\text{vac}}}{\text{v}}=\frac{\text{3}\times {\text{10}}^{\text{8}}\text{ m/s}}{\text{v}}$

So, for 100%,

  $\begin{array}{c}\text{v = r}\times \text{(3}\times {\text{10}}^{\text{8}}\text{)}\text{m/s}\\ \text{=1}\times \text{3}\times {\text{10}}^8\text{m/s}\\ =3\times {10}^8\text{m/s}\end{array}$

  $\text{n=}\frac{\text{3}\times {\text{10}}^{\text{8}}\text{ m/s}}{\text{3}\times {\text{10}}^8\text{m/s}}=1$

Similarly, the speed as well as refractive index for different percentage is shown below.

Optical material with percentage of time required for light to pass through compared to an equal length of vacuum	Speed of light(meter/sec)	Index of refraction
100%	$3\times {10}^8$	1
125%	$2.4\times {10}^8$	1.25
150%	$2\times {10}^8$	1.5
200%	$1.5\times {10}^8$	2
500%	$6\times {10}^7$	5
1000%	$3\times {10}^7$	10