Metamaterials $\rightarrow n = – \sqrt{μ_r ε_r}$ - Nishant Rai

The optical properties of a medium are governed by the relative permitivity $(ε_r)$ and relative permeability ( $μ_r)$ . The refractive index is defined as $\sqrt{μ_rε_r} = n$ . For ordinary material $ε_r > 0$ and $μ_r > 0$ and the positive sign is taken for the square root.

In 1964, a Russian scientist V. Veselago postulated the existence of material with $ε_r < 0$ and $μ_r < 0$ . Since then such ‘metamaterials’ have been produced in the laboratories and their optical properties studied.

For such materials $n = - \sqrt{μ_r ε_r}$ . As light enters a medium of such refractive index, the phases travel away from the direction of propagation.

(i) According to the description above show that if rays of light enter such a medium from air (refractive index =1) at an angle $θ$ in $2nd$ quadrant, then the refracted beam is in the $3rd$ quadrant.

(ii) Prove that Snell’s law holds for such a medium.

$(i)$ Suppose the postulate is true, then two parallel rays would proceed as shown in Figure. Assuming $ED$ shows a wave front then all points on this must have the same phase. All points with the same optical path length must have the same phase.

Thus $- \sqrt{ε_rμ_r} AE = BC -\sqrt{ε_rμ_r}CD$ or $BC = \sqrt{ε_rμ_r}(CD - AE)$

$\rightarrow BC > 0, CD > AE$

As showing that the postulate is reasonable. If however, the light proceeded in the sense it does for ordinary material (viz. in the fourth quadrant, Fig. 2)

Then $- \sqrt{ε_rμ_r} AE = BC -\sqrt{ε_rμ_r}CD$ or $BC = \sqrt{ε_rμ_r}(CD - AE)$

As $AE > CD, BC < O$ , showing that this is not possible. Hence the postulate is correct.

$(ii)$ Prove yourself. $\ddot \smile$

Markdown

Appears as

*italics* or _italics_

italics

**bold** or __bold__

bold


- bulleted
- list

bulleted
list


1. numbered
2. list

numbered
list

Note: you must add a full line of space before and after lists for them to show up correctly

paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)

example link

> This is a quote

This is a quote

    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"

# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"

Math

Appears as

Remember to wrap math in $ ... $ or \[ ... \] to ensure proper formatting.

2 \times 3

2 \times 3

2^{34}

2^{34}

a_{i-1}

a_{i-1}

\frac{2}{3}

\frac{2}{3}

\sqrt{2}

\sqrt{2}

\sum_{i=1}^3

\sum_{i=1}^3

\sin \theta

\sin \theta

\boxed{123}

\boxed{123}

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Try these - Easy if you know about Metamaterials!

Easy if you know about Meta-materials -2!

Nishant Rai - 6 years ago

a comprehension based on this was asked in $JEE$

Tanishq Varshney - 6 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Metamaterials \(\rightarrow n = – \sqrt{μ_r ε_r}\)

Comments