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.

## Expert Answer

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**.*