What Percentage Of The Incident (Unpolarized) Light Will Pass Through The First Filter?

Polarization

Polarization is a phenomenon peculiar to transverse waves. Longitudinal waves such every bit sound cannot exist polarized. Lite and other electromagnetic waves are transverse waves made up of mutually perpendicular, fluctuating electric and magnetic fields. In the diagram below an EM wave is propagating in the x-management, the electrical field oscillates in the xy-aeroplane, and the magnetic field oscillates in the xz-plane. A line traces out the electric field vector as the wave propagates.

EM wave

An unpolarized electromagnetic moving ridge traveling in the x-direction is a superposition of many waves. For each of these waves the electric field vector is perpendicular to the x-centrality, but the bending information technology makes with the y-centrality is unlike for different waves. For unpolarized low-cal traveling in the x-direction Eastward_y and Due east_z are randomly varying on a timescale that is much shorter than that needed for observation.

unpolarized light
Unpolarized lite: Natural lite is, in full general, unpolarized.

For a linearly polarized electromagnetic wave traveling in the 10-direction, the bending the electric field makes with the y-centrality is unique.

polarizer transmission axis An ideal polarizer is a material that passes only EM waves for which the electrical field vector is parallel to its transmission axis. The electrical field is a vector and can be written in terms of the components parallel and perpendicular to the polarizer's manual centrality.
E = E _parallel + Due east _{perpendicular}.
An ideal polarizer passes E _parallel and absorbs Eastward _{perpendicular}.

If E ₀ is the incident field vector and the bending between Due east ₀ and the transmission centrality is θ, so the magnitude of transmitted field vector is E₀cosθ and its direction is the direction of the transmission axis. The intensity I of an electromagnetic moving ridge is proportional to the square of the magnitude of the electrical field vector. We therefore have

I_transmitted= I₀cosⁱⁱθ.

This is chosen the law of Malus. If θ = xc^o the transmitted intensity is zero.

crossed polarizers
The lines indicate the direction of the transmission axis.

When unpolarized light passes through a polarizer, the intensity is reduced by a factor of ½. The average of cos^twoθ, averages over all angles θ is ½.
I_transmitted= I₀<cos^twoθ>_{all angles} = ½I₀.

Problem:

three polarizers A beam of unpolarized light of intensity I₀ passes through a serial of ideal polarizing filters with their transmission centrality turned to various angles, as shown in the effigy.
(a) What is the calorie-free intensity (in terms of I₀) in regions A, B, and C?
(b) If nosotros remove the centre filter, what will be the light intensity at point C?

Solution:

Reasoning:
When unpolarized lite passes through a polarizer, the intensity is reduced past a factor of ½. The transmitted low-cal is polarized forth the axis of the polarizer.
When polarized light of intensity I₀ is incident on a polarizer, the transmitted intensity is given by I = I₀cos^twoθ, where θ is the angle between the polarization direction of the incident calorie-free and the axis of the filter. The transmitted light is polarized along the axis of the polarizer.
Details of the calculation:
In this problem nosotros accept 3 polarizing filters. For the second polarizer θ = 30^o between the polarization management of the calorie-free incident of the filter and the axis of the filter. For the tertiary polarizer θ = 90^o - 30^o = lx^o between the polarization direction of the light incident of the filter and the axis of the filter.
We then accept that:
(a) In region A the intensity is I₀/ii and the low-cal is polarized along the vertical direction.
In region B the intensity is (I₀/2)cosⁱⁱ30^o, = 0.375 I₀, and the light is polarized forth the centrality of the second polarizer.
in region C the intensity is (0.375 I₀)cos²60^o = 0.0938 I₀ and the light is horizontally polarized.
(b) If nosotros remove the middle filter, for the terminal filter we now have that θ = 90^o. Thus I = 0.
Information technology is important to visualize the fact that adding the middle filter increases the transmitted intensity!
This "paradoxical" effect is a signature of moving ridge phenomena in general.

There are different polarization mechanisms. The nearly common method of producing polarized calorie-free is to use polaroid material, made from chains of organic molecules, which are anisotropic in shape. Light transmitted is linearly polarized perpendicular to the direction of the chains. The transmission axis is perpendicular to the bondage.

A polarizer produces linearly polarized light. Information technology is often user-friendly to orient the transmission centrality of a polarizer vertically or horizontally to produce low-cal with vertical or horizontal linear polarization.

vertically and horizontally polarized light
Vertical and horizontal polarization

Polarization past reflection:

polarization by reflection When unpolarized calorie-free is incident on a boundary between two dielectric surfaces, for case on an air-water purlieus, so the reflected and transmitted components are partially polarized. The reflected wave is 100% linearly polarized when the incident bending is equal to an angle called the Brewster angle.
For h2o this bending is is ~53^o with respect to the normal or 37^o with respect to the water surface.
For are considerable angular range around the Brewster angle the reflected light is highly polarized in the horizontal direction.

polarized sunglasses When the sunday is at a low angle in the sky, the sunlight reflecting off the surface of h2o is well-nigh 100% horizontally polarized because the angle of incidence is close to the Brewster angle.
Glare-reducing sunglasses are coated with a polarizer with a vertical manual centrality and therefore block the reflected light.

Link: Polarization of light (Youtube)
Link: Polarization and reflection (Youtube)

Demonstrations

The way low-cal is transmitted through some materials depends on its polarization.

Certain birefringent crystalline substances curve light trough an bending that depends upon the state of incident polarization. The have an optic axis. Unpolarized light entering a birefringent crystal not along the optic axis of the crystal is split into beams which are bend by different amounts.

birefringent crystal
Double refraction in a calcite crystal. The vertical lines are imaged twice.
The polarization of the lite producing one paradigm is perpendicular to the
polarization of the calorie-free producing the other image.

Some materials plow birefringent when stressed. Past placing transparent materials between two polarizers, we can perform stress analysis tests.

stressed glass

Optically active or circular birefringent materials rotate the direction of polarization of linearly polarized light. The amount of rotation depends on the wavelength of the calorie-free. Sugar molecules have a handedness (chirality) and in solution are optically agile. If we polarize white light and pass it through sugar syrup, the direction of polarization of the light emerging from the syrup will exist different for the different colour components. If the light then passes through a second polarizer, its colour changes with the orientation of the manual centrality of this polarizer.

Sugar syrup between crossed polarizers