Question

Imagine you have two identical perfect linear polarizers and a source of natural light. Place them one behind the other and position their transmission axes at 0 and 50 degrees respectively. Now insert between them a third linear polarizer with its transmission axis at 25 degrees. If 1000 W/cm2 of natural light is incident, how much will emerge with and without the middle polarizer in place?

Answer 1

Without the third polarizer inserted in the middle of the first and second polarizer the irradiance of light that would emerge from the second polarizer is

`I_2=321.39 W/cm^2`

With the insertion of the third polarizer at the middle of the first and second polarizer the irradiance of light that would emerge from the second polarizer is

`I_4 = I(\theta_d_1) = 337.3 W/cm^2`

Step-by-step explanation:

From the question we are told that the

The angle between the transmission axis of first linear polarizer and the vertical axis is
`\theta_1 = 0^o`

The angle between the transmission axis of second linear polarizer and the vertical axis is
`\theta_2 = 50^o`

The angle between the transmission axis of third linear polarizer and the vertical axis is
`\theta_2 = 25^o`

The irradiance of the incident natural light is
`I = 1000 W/cm^2`

Generally a linear polarizer divides the irradiance of a natural light by 2

So For the first polarizer the irradiance of the natural light would become

`I_1 = (I)/(2)`

Substituting values

`I_1 = (1000)/(2)`

`=500 \ W/cm^2`

Now looking at the question we can deduce that the angle between the transmission axis of the and second polarizer is

`\theta_d = \theta _2 - \theta_1`

Substituting values

`\theta_d = 50^o - 0^o`

`=50^o`

According to Malus Law the irradiance of light that would come out from the second polarizer is obtained by this mathematical expression

`I(\theta_d) = I_1 cos^2(\theta_d)`

Substituting values

`I_2 = I(\theta_d) = 500 *cos (50)`

`=321.39 W/cm^2`

When the third polarizer is inserted between the first and second polarizer, we have that

The angle between the first polarizer and the third polarizer is mathematically evaluated as

`\theta_d_1 = \theta_3 - \theta_1`

Substituting the values

`\theta_d_1 = 25^o -0^o`

`= 25^o`

According to Malus Law the irradiance of light that would come out from the third polarizer is obtained by this mathematical expression

`I(\theta_d_1) = I_1 cos^2(\theta_d_1)`

`I(\theta_d_1) = I_1 cos^2 (25)`

Substituting values

`= 500 cos^2(25)`

`I_3 = I(\theta_d_1) = 410.69 W/cm^2`

The angle between the third polarizer and the second polarizer is mathematically evaluated as

`\theta_d_2 = \theta_2 - \theta_3`

Substituting the values

`\theta_d_2 = 50^o -25^o` [Note the third polarizer is placed at the

`= 25^o` The middle of the first and second

polarizer

According to Malus Law the irradiance of light that would come out from the second polarizer is obtained by this mathematical expression

`I(\theta_d_2) = I_3 cos^2(\theta_d_2)`

`I(\theta_d_1) = I_3 cos^2 (25)`

Substituting values

`= 410.69 cos^2(25)`

`I_4 = I(\theta_d_1) = 337.3 W/cm^2`