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What is the Angle of Polarisation? What is Brewster Law and Derive its Formula?
In 1808 French scientist E L Malus discovered that plane polar-ised light can be produced by reflection. He showed that when ordinary light i.e., unpolarised light is reflected from the surface of a transparent medium such as glass or water, the reflected light becomes partially plane polarised. The degree of polarisation depends upon the angle of incidence.
Definition: For a particular angle of incidence the degree of polarisation by reflection is maximum. This angle of incidence is called angle of polarisation or polarising angle.
The magnitude of this angle depends on the nature of the reflecting surface and the wavelength of the incident light.
For glass, the polarising angle is about 56° and for pure water, it is about 53°.
Experiment of polarisation by Reflection: Let us suppose, a black glass plate is placed perpendicular to a sheet of paper. MM’ is the smooth upper surface of the plane[Fig.]. This is the reflecting surface. As the glass is black, possibility of more than one reflection of the refracted ray is less.
An ordinary ray of light AO is incident on the reflecting surface at an angle 56° and is reflected along OB. The reflected ray OB will be plane polarised. To examine it, a tourmaline crystal (T) is placed perpendicular to the path of the reflected ray OB and looked from point E looked from point E located behind the crystal along BO. Now the crystal is rotated slowly about the reflected ray OB. It will be seen that at a particular position of the crystal, no light is transmitted through the crustal. The crystal is again rotated from this position slowly and when the rotating angle becomes 90°, the intensity of light transmitted by the crystal will be maximum. This proves that the reflected beam OB is polarised.
Explanation: The transverse vibrations of ordinary incident light may be supposed to consist of two mutually perpendicular vibrations-
- one component is in the plane of the paper i.e., lies in the plane of incidence and
- the other perpendicular to the plane of paper i.e., lies parallel to the reflecting surface.
Whatever be the value of the angle of incidence on MM’ plane, the vibration of the second component will always be parallel to the reflecting surface. As a result, if the incident angle changes, the vibration of that component will be parallel with reflecting plane MM’ but the vibrations of the first component will make varying angles with the reflecting plane. If light is incident at polarising angle, the vibrations will be refracted from air to glass and get absorbed inside the glass i.e., these rays will not be reflected. Only the second component will be reflected. Hence reflected ray OB is the plane polarised light. The plane perpendicular to the sheet of paper is the vibration plane of the polarised light. So, it can be said that plane polarised light can be produced by reflection.
Brewster’s Law
When an unpolarised light is incident at polarising angle on the interface of two media of different refractive indices, a portion of that light is reflected and completely polarised and other portion is refracted and partly polarised.
It has been found from experiment that in the event of such reflection and refraction of an unpolarised light, incident at polarising angle, the reflected ray and the refracted ray become mutually perpendicular[Fig.]. It is to be mentioned here that this polarising angle is also called Brewster’s angle.
In Fig., ∠PON = angle of polarisation (polarising angle) = ip and ∠QON’ is the corresponding angle of refraction = r.
Then, ip + r = 90°
or, r = 90° – ip ……. (1)
Then, according to Snell’s law: \(\frac{\sin i_p}{\sin r}\) = \(\frac{\mu_2}{\mu_1}\)
[where µ1 = refractive Index of the medium of incidence;
µ2 = refractive index of the refracting medium]
∴ \(\frac{\sin i_p}{\cos i_p}\) = \(\frac{\mu_2}{\mu_1}\) [∵ r = 90° – ip]
or, tan ip = \(\frac{\mu_2}{\mu_1}\) …… (2)
If both the incident ray and the reflected ray are in air, then µ1 = 1.
In that case, if refractive index of the refracting medium is taken as µ, the equation (2) can be written as follows-
tan ip = µ …… (3)
i.e., the tangent of polarising angle is numerically equal to the refractive index of the reflecting medium. This is Brewster’s law.
Since, the refractive index of a medium depends on the wave length of light it can be said that the polarising angle also depends on wavelength.