# What is the angle between the electric field and the axis of the filter?

This question aims to find the angle between the electric field and the axis of a filter when a polarized light wave passes through the polarizing filter.

The property of the electromagnetic radiation passing through a polarizing filter in a certain direction having a magnitude of vibrating electric field is called polarization. The light waves are originally unpolarized and move in more than one direction along both electric and magnetic planes.

The unpolarized light waves always move perpendicular in each plane. When this unpolarized light passes through the polarizer, it takes the direction of the electric field and starts to move in a single plane. This phenomenon is called polarization.

Unpolarized light can be polarized by many phenomena when these light waves hit a certain material. These phenomena are reflection, refraction, diffraction, etc.

Only 25% of the polarized light wave passes through the polarizing filter and the other light waves do not cross the filter.

The intensity of light passing through the polarizer is given by the following equation:

$I _ {transmitted} = I _ {incident} ( cos \theta ) ^ 2$

By re-arranging the equation:

$\frac { I _ {transmitted} } { I _ {incident} } = \frac { 25 } { 100 } = cos ^ 2 \theta$

$\frac { I _ {transmitted} } { I _ {incident} } = \frac { 1 } { 4 } = cos ^ 2 \theta$

$cos ^ 2 \theta = \frac { 1 } { 4 }$

$cos \theta = \frac { 1 } { 2 }$

$\theta = cos ^ -1 ( \frac { 1 } { 2 } )$

$\theta = 60 °$

## Numerical Solution

The angle between the electric field and the axis of a filter when polarized light passes through the polarizing filter is 60 °.

## Example

Find the angle between the polarizing filter and the electric field when only 30% light wave passes through the polarizing filter.

The intensity of light passing through the polarizer is given by the following equation:

$I _ {transmitted} = I _ {incident} ( cos \theta ) ^ 2$

By re-arranging the equation:

$\frac { I _ {transmitted} } { I _ {incident} } = \frac { 30 } { 100 } = cos ^ 2 \theta$

$\frac { I _ {transmitted} } { I _ {incident} } = \frac { 3 } { 10 } = cos ^ 2 \theta$

$cos ^ 2 \theta = \frac { 3 } { 10 }$

$cos \theta = \frac { 3 } { 10 }$

$\theta = cos ^ -1 ( \frac { 3 } { 10 } )$

$\theta = 72.5 °$

The angle between the electric field and the axis of a filter when polarized light passes through the polarizing filter is 72.5 °.

Image/Mathematical drawings are created in Geogebra.

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