Experiment No.

Malus law

AIM :

To determine the relation ship between the intensity of the transmitted light through two polarizers and
the angle 𝜃, of the axes of the two polarizers and to verify Malus Law.

Apparatus Used :

A diode laser, a polarizer-analyzer pair, photo detector, detector output measuring unit (micro ammeter),
dial fitted to the polarizer and an optical bench .

Formula Used : Intensity of the transmitted light is given by

It = At2 = Ao2 Cos2θ = Io Cos2θ
Where It is the intensity of the light transmitted through the analyzer;
Io is the intensity of the incident plane polarized light and
θ is the angle between the axis of polarizer and analyser

Theory:

The light coming from the Sun, candle light, and light emitted by a bulb is an ordinary light and is
known to be un-polarized. In an un-polarized light electric and magnetic field vectors vibrate in all
possible directions perpendicular to each other and also perpendicular to the direction of propagation of
light. Unpolarised light can be represented as shown in fig. 1(a). The unpolarised light can be considered
to be composed of two linear orthogonal polarization states with complete incoherence.
When unpolarized light is incident on an ideal polarizer, the intensity of the transmitted light is one-half
of the incident light. Also if the polarizer is rotated w.r.t. incident light there is no change in the
irradiance of the transmitted light i.e. its intensity remains half of the incident light.
Polarization
Certain transparent materials such as Nicol, Tourmaline are capable of filtering and allowing light waves
having vibrations in only one plane. Such materials are called Polaroids. This filtering is possible due
the structure of the material that is having its cells arranged in a straight line manner only in one direction
(parallel to the pass axis of polarizer) which is represented in fig. 1(b) & fig. 1(c).
This phenomenon of filtering and producing light waves having vibrations confined to one particular
direction is called polarization. Polarization is a property of a material by which light waves are filtered
and made directional.

Figure 1(a) Figure 1(b) Figure 1(c)

Malus’s Law
When light falls on a polarizer, the transmitted light gets polarized. The polarized light falling on another
Polaroid, called analyzer, transmits light depending on the orientation of its axis with the polarizer. The
intensity of light transmitted through the analyzer is given by Malus’ law. The law describes how the
intensity of light transmitted by the analyzer varies with the angle that its plane of transmission makes
with that of the polarizer. The law can be stated in words as follows:

The intensity of the transmitted light varies as the square of the cosine of the angle between
the two planes of transmission.

Figure 2

If Ao is the amplitude of the incident light and At is amplitude of the light transmitted through the
analyzer; which in inclined at an angle θ with the polarizer then (fig 2 & 3),
At= Ao Cosθ ……………………………………………………1
As Intensity ∝ (amplitude)2
It = At2 = Ao2 Cos2θ = Io Cos2θ ………………………………..2
Where It is the intensity of the light transmitted through the analyzer; and Io is the intensity of the
incident plane polarized light.
Consider the two extreme cases illustrated by this equation:
• If θ is zero, the second polarizer (analyser) is aligned with the first polarizer, and the value of
cos2θ is one. Thus the intensity transmitted by the second filter is equal to the light intensity that
passes through the first filter. This case will allow maximum intensity to pass through.
• If θ is 90˚, the second polarizer (analyser ) is oriented perpendicular to the plane of polarization
2
of the first filter, and the cos (90˚) gives zero. Thus no light is transmitted through the second filter.
This case will allow minimum (zero) intensity to pass through.
The light intensity cannot be measured directly. The light energy is converted into electrical energy using
photo detectors such as a photo cell or light dependent resistor (LDR). In such photo detectors the current
produced is directly proportional to the light intensity.
It ∝ current
It = K *current
The constant K appearing here is nothing but the conversion efficiency of photo detector. Using this
concept Malus’s law (equations 1 and 2) is verified in this experiment. The angles are noted
experimentally from the dial fitted to the Polaroids.
 Figure 3
Experimental Procedure

 Set up the laser, photodiode, the polarizer and analyser as shown in Figure 4 to test Malus’s Law.
 Make sure the polarizer and analyser are normal to the laser beam and that the beam passes through
a “good” portion of the polarizers – look for minimal scattering, etc.
 When the laser is going through polarizer, analyser and then into a detector make sure polarizer and
analyser transmission axes are parallel. That way you can work with an offset from 0 o. To do this,
keep the polarizer fixed and rotate the analyser until you observe a maximum in transmission. Note
down maximum current Imax . At this point the pass axes of polarizer and analyser are parallel.
 Rotate the analyzer in 10o increments from 0o to 360o . Take readings of the intensity at each angle.
The intensity of light beam that passes through polarizer and analyser was measured by the light
sensor. The rotary motion sensor measures the angle that was obtained from rotating the second
analyzer relative to the first polarizer.
 In each case the current is noted and tabulated in Table-1.
 Plot a graph taking the current ‘Iexpt’ along Y-axis and angle of rotation of analyzer on the X-axis.
From the graph the cosine nature of the curve is clearly evident, validating the Malus’ law.
 Cos θ, Cos2θ, Itheo are calculated and presented in Table-1. Plot two more graphs showing the
variation of Iexpt vs Cos2θ and Iexpt vs Itheo
 The slope of straight line in graph Iexpt vs Itheo is calculated. Slope ≈1
 Figure :4

OBSERVATIONS

Maximum Current when both the polarizers have their axes parallel Imax=…….. 𝜇𝑎𝑚𝑝

S.No. Angle between Intensity (in terms of Current ‘Itheo’

axes of polarizers Cosθ Cos θ
2
current ‘Iexpt’ read Itheo= Imax * Cos2θ
‘θ’ (degrees) in 𝝁𝒂𝒎𝒎𝒆𝒕𝒆𝒓) (𝝁𝒂𝒎𝒑)
(𝝁𝒂𝒎𝒑)
1.
2.
3.
4.
..
..
..
..
..

RESULTS

 Experimental results representing the light intensity versus the angle (θ) and the cos2 θ are shown in
figure 5 and 6, respectively. The shape of the graph of the intensity versus the angle is sinusoidal.
This was obtained by positioning both polarizers at the same angle, (zero degree), and then rotating
analyzer from 0° to 360°. The maximum and minimum light intensity corresponds to 0° and 180°,
90° and 270°, respectively.
 We have demonstrated the Malus’ law of polarization of light. The experimental results agreed well
with the predicted relationship between the polarization direction of light and the intensity of that
light transmitted through a second polarizer.
 Figure 5 Figure 6

DISCUSSION

 The current (proportional to light intensity), noted for different angles of rotation of the analyzer,
follows a cosine curve for 360o of rotation, indicating the validity of equation-2. The experimentally
measured current (Iexpt) and (Itheo) that calculated using equation Itheo = Imax Cos2θ agree within the
limits of the experimental error.
 The relative intensity of the light emerging from analyzer is maximum at 0° and 180° and it attains
minimum value at 90° and 270°. In between it varies as a Cosine function as indicated by the graph.
 The light intensity Iexpt versus Cos2θ curve is a straight line, as expected, with unit slope indicating the
correctness of the Malus’s law.

Precautions:

1. Analyzer and Polarizer should be at same horizontal level.

2. Analyzer must be rotated by small angles (5o). Changing values abruptly may cause errors.
3. Experiment should be performed in dark room.
4. Photo detector is a very sensitive device. It should be adjusted well (at appropriate height) to
receive maximum current.

Figure 7