# Refraction at Plane Surface Formulas

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## Refraction at Plane Surface Formulae Sheet

If c → Velocity of light in air
V → Velocity of light in a medium
µ → Absolute refractive index of medium.
µm → Permeability of medium.
εm → Permittivity of a medium.
µ0 → Permeability of free space.
0 → Permittivity of free space.
c = $$\frac{1}{\sqrt{\varepsilon_{0} \mu_{0}}}$$, v = $$\frac{1}{\sqrt{\varepsilon_{m} \mu_{m}}}$$
µ = $$\frac{c}{v}=\sqrt{\frac{\varepsilon_{m} \mu_{m}}{\varepsilon_{0} \mu_{0}}}=\sqrt{\varepsilon_{r} \mu_{r}}$$
refractive index of 2 w.r.t. 1
1µ2 = $$\frac{\mu_{2}}{\mu_{1}}=\frac{\mathrm{c} / \mathrm{v}_{2}}{\mathrm{c} / \mathrm{v}_{1}}=\frac{\mathrm{v}_{1}}{\mathrm{v}_{2}}$$

1. Laws of refraction • Frequency (Colour) and phase not change (while wavelength and velocity changes) when light ray go from one medium to another medium.
• I, R & N are in a same plane.
• Snell’s law $$\frac{\sin i}{\sin r}={ }_{1} \mu_{2}=\frac{\mu_{2}}{\mu_{1}}=\frac{v_{1}}{v_{2}}=\frac{\lambda_{1}}{\lambda_{2}}$$

2. Total internal reflection

It occur when i > θc (Critical angle) At θc, the refracted ray just grazes the boundary between two media.
sin µ1θc = µ2 sin 90°
θc = sin-1$$\frac{\mu_{2}}{\mu_{1}}$$
If µ2 = 1 (air), θc = sin-1$$\left(\frac{1}{\mu}\right)$$

3. Brewester law

When partially reflected & refracted rays makes 90° angle then both get polarised.
and ip = tan-1µ

4. Refraction by slab Lateral displacement d
d = t $$\frac{\sin (i-r)}{\cos r}$$
In fig. I & II lines are parallel.

5. Apparent depth (Dapp) dapp = $$\frac{\mathrm{d}_{1}}{\mu_{1}}+\frac{\mathrm{d}_{2}}{\mu_{2}}+\frac{\mathrm{d}_{3}}{\mu_{3}} \ldots \ldots \frac{\mathrm{d}_{\mathrm{n}}}{\mu_{\mathrm{n}}}$$
Apparent shift (s)
s = d – dapp

6. Equivalent µ
µ = $$\frac{d_{\text {actual }}}{d_{\text {app. }}}=\frac{d_{1}+d_{2} \ldots \ldots . d_{n}}{\frac{d_{1}}{\mu_{1}}+\frac{d_{2}}{\mu_{2}} \ldots \ldots . \frac{d_{n}}{\mu_{n}}}=\frac{d}{\sum_{i=1}^{i=n} \frac{d_{i}}{\mu_{i}}}$$

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