finding the intensity of light transmitted through a polariser using malus law and by taking initial intensity and angle between polariser and axis as input description malus s law which is named after tiennelouis malus says that when a perfect polarizer is placed in a polarized beam of light the irradiance i of the light that passes through is given by ii cos where i is the initial intensity and is the angle between the light s initial polarization direction and the axis of the polarizer a beam of unpolarized light can be thought of as containing a uniform mixture of linear polarizations at all possible angles since the average value of cos is 12 the transmission coefficient becomes ii 12 in practice some light is lost in the polarizer and the actual transmission will be somewhat lower than this around 38 for polaroidtype polarizers but considerably higher 49 9 for some birefringent prism types if two polarizers are placed one after another the second polarizer is generally called an analyzer the mutual angle between their polarizing axes gives the value of in malus s law if the two axes are orthogonal the polarizers are crossed and in theory no light is transmitted though again practically speaking no polarizer is perfect and the transmission is not exactly zero for example crossed polaroid sheets appear slightly blue in colour because their extinction ratio is better in the red if a transparent object is placed between the crossed polarizers any polarization effects present in the sample such as birefringence will be shown as an increase in transmission this effect is used in polarimetry to measure the optical activity of a sample real polarizers are also not perfect blockers of the polarization orthogonal to their polarization axis the ratio of the transmission of the unwanted component to the wanted component is called the extinction ratio and varies from around 1 500 for polaroid to about 1 106 for glantaylor prism polarizers reference https en wikipedia orgwikipolarizermalus slawandotherproperties roundmaluslaw10 45 2 5 0 roundmaluslaw100 60 2 25 0 roundmaluslaw50 150 2 37 5 roundmaluslaw75 270 2 0 0 roundmaluslaw10 900 2 traceback most recent call last valueerror in malus law the angle is in the range 0360 degrees roundmaluslaw10 900 2 traceback most recent call last valueerror in malus law the angle is in the range 0360 degrees roundmaluslaw100 900 2 traceback most recent call last valueerror the value of intensity cannot be negative roundmaluslaw100 180 2 100 0 roundmaluslaw100 360 2 100 0 handling of negative values of initial intensity handling of values out of allowed range finding the intensity of light transmitted through a polariser using malus law and by taking initial intensity and angle between polariser and axis as input description malus s law which is named after étienne louis malus says that when a perfect polarizer is placed in a polarized beam of light the irradiance i of the light that passes through is given by i i cos²θ where i is the initial intensity and θ is the angle between the light s initial polarization direction and the axis of the polarizer a beam of unpolarized light can be thought of as containing a uniform mixture of linear polarizations at all possible angles since the average value of cos²θ is 1 2 the transmission coefficient becomes i i 1 2 in practice some light is lost in the polarizer and the actual transmission will be somewhat lower than this around 38 for polaroid type polarizers but considerably higher 49 9 for some birefringent prism types if two polarizers are placed one after another the second polarizer is generally called an analyzer the mutual angle between their polarizing axes gives the value of θ in malus s law if the two axes are orthogonal the polarizers are crossed and in theory no light is transmitted though again practically speaking no polarizer is perfect and the transmission is not exactly zero for example crossed polaroid sheets appear slightly blue in colour because their extinction ratio is better in the red if a transparent object is placed between the crossed polarizers any polarization effects present in the sample such as birefringence will be shown as an increase in transmission this effect is used in polarimetry to measure the optical activity of a sample real polarizers are also not perfect blockers of the polarization orthogonal to their polarization axis the ratio of the transmission of the unwanted component to the wanted component is called the extinction ratio and varies from around 1 500 for polaroid to about 1 106 for glan taylor prism polarizers reference https en wikipedia org wiki polarizer malus s_law_and_other_properties round malus_law 10 45 2 5 0 round malus_law 100 60 2 25 0 round malus_law 50 150 2 37 5 round malus_law 75 270 2 0 0 round malus_law 10 900 2 traceback most recent call last valueerror in malus law the angle is in the range 0 360 degrees round malus_law 10 900 2 traceback most recent call last valueerror in malus law the angle is in the range 0 360 degrees round malus_law 100 900 2 traceback most recent call last valueerror the value of intensity cannot be negative round malus_law 100 180 2 100 0 round malus_law 100 360 2 100 0 handling of negative values of initial intensity handling of values out of allowed range
import math




def malus_law(initial_intensity: float, angle: float) -> float:
    

    if initial_intensity < 0:
        raise ValueError("The value of intensity cannot be negative")
        
    if angle < 0 or angle > 360:
        raise ValueError("In Malus Law, the angle is in the range 0-360 degrees")
        
    return initial_intensity * (math.cos(math.radians(angle)) ** 2)


if __name__ == "__main__":
    import doctest

    doctest.testmod(name="malus_law")