9.3 Refraction

When a beam of light encounters another transparent medium, a part of light gets reflected back into the first medium while the rest enters the other. A ray of light represents a beam. The direction of propagation of an obliquely incident \(\left(0^{\circ}<i<90^{\circ}\right)\) ray of light that enters the other medium, changes at the interface of the two media. This phenomenon is called refraction of light. Snell experimentally obtained the following laws of refraction:

(i) The incident ray, the refracted ray and the normal to the interface at the point of incidence, all lie in the same plane.
(ii) The ratio of the sine of the angle of incidence to the sine of angle of refraction is constant. Remember that the angles of incidence ( \(i\) ) and refraction ( \(r\) ) are the angles that the incident and its refracted ray make with the normal, respectively. We have
\(
\frac{\sin i}{\sin r}=n_{21} \dots(9.10)
\)
where \(n_{21}\) is a constant, called the refractive index of the second medium with respect to the first medium. Equation (9.10) is the well-known Snell’s law of refraction. We note that \(n_{21}\) is a characteristic of the pair of media (and also depends on the wavelength of light), but is independent of the angle of incidence.

From Eq. (9.10), if \(n_{21}>1, r<i\), i.e., the refracted ray bends towards the normal. In such a case medium 2 is said to be optically denser (or denser, in short) than medium 1. On the other hand, if \(n_{21}<1, r>i\), the refracted ray bends away from the normal. This is the case when incident ray in a denser medium refracts into a rarer medium.

Note: Optical density should not be confused with mass density, which is mass per unit volume. It is possible that mass density of an optically denser medium may be less than that of an optically rarer medium (optical density is the ratio of the speed of light in two media). For example, turpentine and water. Mass density of turpentine is less than that of water but its optical density is higher.

If \(n_{21}\) is the refractive index of medium 2 with respect to medium 1 and \(n_{12}\) the refractive index of medium 1 with respect to medium 2 , then it should be clear that
\(
n_{12}=\frac{1}{n_{21}} \dots(9.11)
\)
It also follows that if \(n_{32}\) is the refractive index of medium 3 with respect to medium 2 then \(n_{32}=\) \(n_{31} \times n_{12}\), where \(n_{31}\) is the refractive index of medium 3 with respect to medium 1.

Some elementary results based on the laws of refraction follow immediately. For a rectangular slab, refraction takes place at two interfaces (airglass and glass-air). It is easily seen from Fig. 9.9 that \(r_2=i_1\), i.e., the emergent ray is parallel to the incident ray-there is no deviation, but it does suffer lateral displacement/shift with respect to the incident ray. Another familiar observation is that the bottom of a tank filled with water appears to be raised (Fig. 9.10). For viewing near the normal direction, it can be shown that the apparent depth \(\left(h_1\right)\) is real depth \(\left(h_2\right)\) divided by the refractive index of the medium (water).