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Ankur Verma

Refraction

Refraction means the bending of a wave resulting from a change in its velocity as its moves from one medium to another. Since the frequency of a wave cannot change, independent of the source changing its frequency when it originally emits a wave, this change in wave velocity must result from a change in its wavelength in the second medium.

As shown in the above diagram, when the waves encounter an oblique interface, both their direction and wavelength change. In the instance illustrated, the wavelengths shorten and the refracted rays "bend towards the normal" as the waves enter the shallow, or slower, medium: ?_r < ?_i.

This diagram illustrates that when an incident wave crosses an interface and its wavefronts are parallel to the interface, the wave will still exhibit a change in wavelength but there will be no change in direction since its rays are parallel to the normals.

Now we will extend our discussion of refraction to light waves. To quantify the degree of refraction, we will introduce a dimensionless quantity called the index of refraction, n.

n = c/v

In this formula,

c is the speed of alight in a vacuum, 3 x 10⁸ m/sec
v represents the average speed of light in the optically dense medium
n is the medium's index of refraction

This defining formula can be easily modified to describe the changes that occur in wavelength during refraction.

n = c/v
n = (f ?)/(f ?_n)

n = ?/?_n

Notice that the frequency, f, cancels since it is an invariant and does not depend on the medium through which the wave is traveling.

Some common indices of refraction for a midrange wavelength of light (589 nm, a prominent line in the emission spectrum of sodium) are:

vacuum	1.00000	fused quartz	1.46
air (STP)	1.00029	crown glass	1.52
water (20ºC)	1.33	polystyrene	1.55
acetone	1.36	carbon disulfide	1.63
ethyl alcohol	1.36	flint glass (heavy)	1.65
sugar solution (30%)	1.38	sapphire	1.77
sugar solution (80%)	1.49	diamond	2.42

Snell's law

To determine the degree to which a light ray bends as it obliquely transitions from one medium to another, we will use our knowledge of refraction and Snell's Law..

n₁ sin ?₁ = n₂ sin ?₂

where

n₁ is the index of refraction for the first medium
n₂ is the index of refraction for the second medium
that angles ?₁ and ?₂ are always measured from the normal, NEVER from the interface.

Here is a list of common indices for the 589 nm wavelength in sodium's spectrum.

medium	index	medium	index
vacuum	1.00000	fused quartz	1.46
air (STP)	1.00029	crown glass	1.52
water (20ºC)	1.33	polystyrene	1.55
acetone	1.36	carbon disulfide	1.63
ethyl alcohol	1.36	flint glass (heavy)	1.65
sugar solution (30%)	1.38	sapphire	1.77
sugar solution (80%)	1.49	diamond	2.42

If n₂ > n₁, then the light is entering an optically more dense medium and the ray will bend "towards the normal" as it enters n₂.

This phenomena occurs because the wavelength shortens in the second medium resulting in the light having a slower average velocity.

Note that the ray bends towards the normal as the light enters the glass and that ?_glass is smaller than ?_air.

If n₂ < n₁, then the light is entering an optically less dense medium and the ray bends "away from the normal" when it enters n₂.

This phenomena occurs because the wavelength lengthens in the second medium resulting in the light having a faster average velocity.

Note that the ray bends away from the normal as the light exits the glass as it returns into the air and that ?_air is greater than ?_glass.

Proof of Snell's Law :-

Consider three incident rays of light encountering an interface between two media. In this example, the second medium is the slower medium and the rays are refracted towards the normal - note that angle A is greater than angle B in the diagram.

Since all rays are perpendicular to their respective wavefronts,

m?A + m?1 = 90º
m?B + m?2 = 90º

Since all normals are perpendicular to their respective interfaces,

m?C + m?1 = 90º
m?D + m?2 = 90º

Therefore, m?C = m?A and m?D = m?B and we can now examine the following new relationships:

sin(A) = d₁/L
sin(B) = d₂/L

where L is the distance along the interface between points P₁ and P₂ as shown in the diagram below.

Solving each equation for L yields:

L = d₁/sin(A)
L = d₂/sin(B)

Therefore:

d₁/sin(A) = d₂/sin(B)

If d₁ and d₂ represent the distances traveled in the respective mediums during the same amount of time, then we can replace them with the expressions

d₁ = v₁t
d₂ = v₂t

But v₁ and v₂ represent the speed of the waves in each medium and can be replaced with the expressions

n₁ = c/v₁ v₁ = c/n₁
n₂ = c/v₂ v₂ = c/n₂

where n₁ and n₂ are the respective indices of refraction and c is the speed of light.

At this junction, we can now write

d₁/sin(A) = d₂/sin(B)
v₁t/sin(A) = v₂t/sin(B)
(c/n₁)[t/sin(A)] = (c/n₂)[t/sin(B)]

Canceling the common terms (c and t) yields

sin(A)/n₁ = sin(B)/n₂
n₁ sin(A) = n₂ sin(B)

Or, as Snell's Law is more commonly expressed:

n₁ sin(?₁) = n₂ sin(?₂)

Notice that Snell's Law shows that the index of refraction and the sine of the angle of refraction are inversely proportional - that is, as the refractive index gets larger [n₂ > n₁] the sine of the refracted angle gets smaller [sin(?₂) < sin(?₁)]. Since sin(?) is an increasing function in the first quadrant, as sin(?) decreases, so does ?.

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