^Malus law

Let I₀ be the intensity and ‘A’ the amplitude of the unpolarized light falling on the polarizer & θ be the angle between the pass axis of the polaroid & amplitude vector, then the component Acosθ, parallel to the pass axis (y) of polarizer passes through it & the component Asinθ, perpendicular to pass axis is blocked by the polarizer, thus the intensity of light transmitted by the polarizer is

I = k (A cosθ)² = k A² cos² θ

or         I = I₀ cos² θ        [called Malus law

Actually the phase angle for a beam of unpolarised light can have any value from 0⁰ to 360⁰. Now as the average value of cos² θ for the range 0⁰ to 360⁰ is 0.5, thus the average intensity of light transmitted by a polarizer is only 0.5 I₀.  Following observations can be explained on the basis of Malus law.

1. Two polarizes with parallel pass axis (y & y)

2. Two polarizes with crossed pass axis (y & z)

^How do polarisers work 

A Polaroid is a thin commercial sheet containing long chain of hydrocarbons which become conducting at optical frequencies & absorb the electric field parallel to chains & transmit the electric field perpendicular to the chain length. This is why a direction perpendicular to the chain length is called the transmission axis or pass axis of the polarizer. For shown drawing y axis is pass axis.

^Plane polarized light

If the electric field vector of a light wave vibrates just in one direction perpendicular to the direction of wave propagation, then it is said to be linearly polarized along that direction. Human eyes can’t distinguish between an unpolarised light and a polarized light, but the eyes of a bee can.

^Polarisation

A wave having vibrations is only in one direction & that is normal to its direction of propagation is called a polarized wave & the phenomenon of restricting the oscillation of a wave to just one direction in a direction transverse plane or normal to its direction of propagation is called polarisation of waves. Only transverse waves can be polarized or longitudinal waves can’t be polarized. Any transverse wave can be linearly polarized in any direction perpendicular to the direction of propagation.

^Resolving power of a compound microscope

Here, λ = wavelength of light used

μ = refractive index of the medium between the object & objective lens

θ = Half angle of the cone of light from each point object.

Also μ sinθ is called numerical aperture.

From above relation it is clear that the resolving power of a compound microscope can be increased by increasing refractive index of the medium. It is due to this reason that oil immersion objectives are used to increase the resolving power. Also, higher the value of numerical aperture, larger will be the resolving power of the microscope.