The optical properties of a medium are governed by the relative permittivity  (εr) and relative permeability (μ_r). The refractive index is defined as $\sqrt[]{{\mathit{\mu }}_{\mathit{r}}{\mathit{\epsilon }}_{\mathit{r}}}$ = 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 εr < 0 and μr < 0. Since then such ‘metamaterials’ have been produced in the laboratories and their optical properties studied. For such materials n = – $\sqrt[]{{\mathit{\mu }}_{\mathit{r}}{\mathit{\epsilon }}_{\mathit{r}}}$   . 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, them the refracted beam is in the 3rd quadrant.  (ii) Prove that Snell’s law holds for such a medium.

This is a Long Answer Type Questions as classified in NCERT Exemplar

Explanation-All points with the same optical path length must have the same phase.

So – $\sqrt[]{{\mu }_r{\epsilon }_r}AE$   =BC-–  $\sqrt[]{{\mu }_r{\epsilon }_r}CD$

BC= $\sqrt[]{{\mu }_r{\epsilon }_r}$ (CD-AE)

BC>0, si must be greater than AD

But in other figure

–  $\sqrt[]{{\mu }_r{\epsilon }_r}AE=BC–\mathrm{ }\mathrm{ }\sqrt[]{{\mu }_r{\epsilon }_r}CD$

So BC= –  $\sqrt[]{{\mu }_r{\epsilon }_r}\left(CD-AE\right)$

But clearly here BE is less than zero

To proving snells law we know that

BC=ACsin $\theta$ and CD-AE=ACsin $\theta$

So n= sini/sinr

<p>This is a Long Answer Type Questions as classified in NCERT Exemplar</p><p><strong>Explanation-</strong>All points with the same optical path length must have the same phase.</p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1743139525phpeC0Xlm.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1743139525phpeC0Xlm.jpeg" alt="" width="493" height="178"></picture></div></div><p>So –<span contenteditable="false"> <math> <mroot> <mrow> <mrow> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> </msub> <msub> <mrow> <mrow> <mi>ε</mi> </mrow> </mrow> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow></mrow> </mroot> <mi>A</mi> <mi>E</mi> </math> </span>&nbsp;&nbsp;=BC-–&nbsp; <span contenteditable="false"> <math> <mroot> <mrow> <mrow> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> </msub> <msub> <mrow> <mrow> <mi>ε</mi> </mrow> </mrow> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow></mrow> </mroot> <mi>C</mi> <mi>D</mi> </math> </span></p><p>BC=<span contenteditable="false"> <math> <mroot> <mrow> <mrow> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> </msub> <msub> <mrow> <mrow> <mi>ε</mi> </mrow> </mrow> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow></mrow> </mroot> </math> </span> (CD-AE)</p><p>BC&gt;0, si must be greater than AD</p><p>But in other figure</p><p>–&nbsp; <span contenteditable="false"> <math> <mroot> <mrow> <mrow> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> </msub> <msub> <mrow> <mrow> <mi>ε</mi> </mrow> </mrow> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow></mrow> </mroot> <mi>A</mi> <mi>E</mi> <mo>=</mo> <mi>B</mi> <mi>C</mi> <mo>–</mo> <mi mathvariant="normal">&nbsp;</mi> <mi mathvariant="normal">&nbsp;</mi> <mroot> <mrow> <mrow> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> </msub> <msub> <mrow> <mrow> <mi>ε</mi> </mrow> </mrow> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow></mrow> </mroot> <mi>C</mi> <mi>D</mi> </math> </span></p><p>So BC= –&nbsp; <span contenteditable="false"> <math> <mroot> <mrow> <mrow> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> </msub> <msub> <mrow> <mrow> <mi>ε</mi> </mrow> </mrow> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow></mrow> </mroot> <mfenced separators="|"> <mrow> <mrow> <mi>C</mi> <mi>D</mi> <mo>-</mo> <mi>A</mi> <mi>E</mi> </mrow> </mrow> </mfenced> </math> </span></p><p>But clearly here BE is less than zero</p><p>To proving snells law we know that</p><p>BC=ACsin <span contenteditable="false"> <math> <mi>θ</mi> </math> </span>and CD-AE=ACsin<span contenteditable="false"> <math> <mi>θ</mi> </math> </span></p><p>So n= sini/sinr</p>

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