Topic
16. Refraction and Snell’s Law
When light strikes the boundary
between two different media, a portion is reflected according to the laws of
reflection, and another portion enters the medium. The ray which crosses the
boundary and enters the second medium has been refracted, and the
process at the boundary is called refraction.
The velocity of light depends on the properties
of the medium it is traveling in, and especially on the density of the
material. The velocity of light decreases when light enters a denser medium. The
ratio of the speed of light in vacuum to the speed of light in any material is
called the index of refraction of that material. If c is the speed of light in
vacuum and v is the speed in some material, the index of refraction is
Since
v is always less than c, the index of refraction is always greater than 1. Note
that the index of refraction is a ratio of two speeds and therefore has no
units. Another way of expressing the same relationship is:
That
is, the speed of light in any medium is the speed in vacuum divided by the
index of refraction.
The
index of refraction depends on the detailed properties of the material and may
also vary with external parameters such as ambient temperature, etc. However,
we will generally ignore these variations and assume that the index of refraction
is a constant that is a characteristic of the material. In addition, although
the index of refraction of the atmosphere is about 1.0003, we will often take
this index to be exactly 1, so that the speed of light in air is equal to its
vacuum value. (As we will see in the next section, the index also varies with
the frequency or wavelength of the incident light.)
Using
Huygens’ principle, it is easy to show that when the velocity of light
decreases at the boundary then the refracted light is bent towards the normal.
That is, the angle between the refracted ray and the normal is smaller than the
angle between the incident ray and the normal. Conversely, when light crosses a
boundary into a medium where its velocity is greater, the angle of the
refracted ray with respect to the normal is larger than the angle of incidence.
This relationship is called Snell’s Law; the mathematical form of the
law (which we will not discuss) relates the incident angle, the refracted angle
and the indices of refraction of the two media at the boundary. (The
relationship actually involves the sines of the angles rather than the angles
themselves. See Appendix B, page 416 in the textbook if you are interested in
this.)
Since
light that leaves a denser medium and enters a less dense one (glass to air,
for example) is bent away from the normal, the angle of refraction will
always be greater than the angle of incidence in this case. Therefore, the
angle of refraction can exceed 90 degrees if the angle of incidence is large
enough. There is no refracted wave at this angle, which is called the critical
angle – the light is only reflected. This is called total internal reflection. There is also no
refracted wave for all angles of incidence larger than the critical angle. This
effect is the basis for optical fibers, which effectively “trap” light inside
the fiber because of total internal reflection.
There are a number of optical effects that are caused by the variation of the index of refraction. A common effect is a mirage, which is caused by the variation of the index of refraction of air with temperature. The air near the surface of a roadway is often much hotter than the air at higher elevations, and its density and therefore its index of refraction are smaller. Light incident on this air from above is therefore bent away from the normal. The light that finally strikes the road is almost perfectly reflected, since the angle of incidence is close to 90 degrees, and these reflected rays will appear to come from a virtual object that is below the surface of the roadway. Since this effect is common over water, our eyes often assume that the hot roadway is actually a body of water.
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