FBISE, NCP & NBF-aligned resources for Classes 9–12
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Physics Chapter 9 : Topic 6
Malus’s Law
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Malus’s law describes the relationship between:
Reasoning: Malus’s law gives the intensity of plane-polarized light after passing through a polarizing filter.
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The mathematical expression of Malus’s law is:
Reasoning: Malus’s law states that transmitted intensity is proportional to the square of the cosine of the angle between polarizer axes.
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In Malus’s law, $\mathbf{I}_{\mathbf{o}}$ represents:
Reasoning: $\mathbf{I}_{\mathbf{o}}$ is the intensity of the polarized light incident on the second polarizer.
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In Malus’s law, $\mathbf{(}\mathbf{I}\mathbf{)}$ represents:
Reasoning: $\mathbf{(I}\mathbf{)}$ is the intensity of light emerging from the analyzer.
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The angle $\mathbf{\theta}$ in Malus’s law is the angle between:
Reasoning: The angle between the two polarizing filters determines the transmitted intensity.
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When the transmission axes of two polarizers are parallel, the transmitted intensity is:
Reasoning: For $\mathbf{\theta =}\mathbf{0}^{\mathbf{o}}$ $\mathbf{\Longrightarrow}$ $\mathbf{I}\mathbf{=}\mathbf{I}_{\mathbf{o}}\mathbf{\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ \theta}$ $\mathbf{\Longrightarrow}$ $\mathbf{I =}\mathbf{I}_{\mathbf{o}}\mathbf{\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{0}^{\mathbf{o}}\mathbf{=}\mathbf{I}_{\mathbf{o}}$
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When two polarizers are crossed at $\mathbf{90}^{\mathbf{o}}$, the transmitted intensity is:
Reasoning: $\mathbf{I =}\mathbf{I}_{\mathbf{o}}\mathbf{\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{\theta}$ $\mathbf{\Longrightarrow}$ $\mathbf{I =}\mathbf{I}_{\mathbf{o}}\mathbf{\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{90}^{\mathbf{o}}\mathbf{= 0}$
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If the angle between two polarizers increases from $\mathbf{0}^{\mathbf{o}}$ to $\mathbf{90}^{\mathbf{o}}$, the transmitted intensity:
Reasoning: The value of $\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{\theta}$ decreases as $\mathbf{\theta}$ increases from $\mathbf{0}^{\mathbf{o}}$ to $\mathbf{90}^{\mathbf{o}}$.
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Maximum transmission through two polarizers occurs when their axes are:
Reasoning: Parallel axes allow all vibrations of the polarized wave to pass.
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Minimum transmission occurs when the axes are:
Reasoning: At $\mathbf{90}^{\mathbf{o}}$, no component of vibration passes through the analyzer.
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Plane-polarized light of intensity $\mathbf{100\ }\mathbf{W/m}^{\mathbf{2}}$ passes through an analyzer at $\mathbf{0}^{\mathbf{o}}$. The transmitted intensity is:
Reasoning: $\mathbf{I =}\mathbf{I}_{\mathbf{o}}\mathbf{\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ \theta}$ $\mathbf{\Longrightarrow}$ $\mathbf{I}\mathbf{= 100\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{0}^{\mathbf{o}}$ $\mathbf{\Longrightarrow}$ $\mathbf{I}\mathbf{= 100}$ $\left( \frac{\mathbf{W}}{\mathbf{m}^{\mathbf{2}}} \right)$
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A polarized light beam has intensity $\mathbf{200\ }\mathbf{W/m}^{\mathbf{2}}$. After passing through a filter at $\mathbf{60}^{\mathbf{o}}$, the intensity becomes:
Reasoning: $\mathbf{I =}\mathbf{I}_{\mathbf{o}}\mathbf{\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ \theta}$ $\mathbf{\Longrightarrow}$ $\mathbf{I}\mathbf{= 200\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{60}^{\mathbf{o}}\mathbf{= 200\ }\mathbf{(0.5)}^{\mathbf{2}}\mathbf{= 50\ W/}\mathbf{m}^{\mathbf{2}}$
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A polarized beam of intensity $\mathbf{400\ }\mathbf{W/m}^{\mathbf{2}}$ passes through an analyzer at $\mathbf{45}^{\mathbf{o}}$. The transmitted intensity is:
Reasoning: $\mathbf{I =}\mathbf{I}_{\mathbf{o}}\mathbf{\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{\theta}$$\mathbf{I}\mathbf{= 400\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{45}^{\mathbf{o}}\mathbf{= 400\ }\mathbf{(0.707)}^{\mathbf{2}}\mathbf{= 200\ W/}\mathbf{m}^{\mathbf{2}}$
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If $\mathbf{I}_{\mathbf{o}}\mathbf{= 800\ }\mathbf{W/m}^{\mathbf{2}}$ and $\mathbf{\theta}\mathbf{=}\mathbf{90}^{\mathbf{o}}$, transmitted intensity is:
Reasoning: $\mathbf{cos\ }\mathbf{90}^{\mathbf{o}}\mathbf{= 0}$ $\mathbf{\Longrightarrow}$ Therefore, $\mathbf{I = 0}$
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A polarizer reduces the intensity of polarized light to half. The angle between axes is:
Reasoning: $\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{\theta}\mathbf{=}$ $\frac{\mathbf{1}}{\mathbf{2}}$ $\mathbf{\Longrightarrow}$ $\mathbf{\theta}\mathbf{=}\mathbf{45}^{\mathbf{o}}$
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When a third polarizer is placed between two crossed polarizers at $\mathbf{45}^{\mathbf{o}}$, the transmitted light:
Reasoning: The middle polarizer allows a component of vibration to pass and changes the polarization direction gradually.
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A series of polarizing filters affects light intensity by:
Reasoning: Each analyzer transmits only the component along its transmission axis.
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Two polarizers have axes at $\mathbf{30}^{\mathbf{o}}$. The transmitted intensity is:
Reasoning: $\mathbf{I =}\mathbf{I}_{\mathbf{o}}\mathbf{\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{\theta}$ $\mathbf{\Longrightarrow}$ $\mathbf{I}\mathbf{=}\mathbf{I}_{\mathbf{o}}\mathbf{\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{30}^{\mathbf{o}}\mathbf{=}\mathbf{I}_{\mathbf{o}}\mathbf{\ }\left( \mathbf{0.866} \right)^{\mathbf{2}}\mathbf{=}\mathbf{0.75\ }\mathbf{I}_{\mathbf{o}}$
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Malus’s law applies to:
Reasoning: The law describes transmission of already polarized electromagnetic waves.
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Two polarizers are arranged at $\mathbf{(}\mathbf{60}^{\mathbf{o}}\mathbf{)}$. The transmitted intensity is:
Reasoning: $\mathbf{I =}\mathbf{I}_{\mathbf{o}}\mathbf{\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{\theta}$ $\mathbf{\Longrightarrow}$ $\mathbf{I}\mathbf{=}\mathbf{I}_{\mathbf{o}}\mathbf{\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{60}^{\mathbf{o}}\mathbf{=}\mathbf{0.25\ }\mathbf{I}_{\mathbf{o}}$
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If the angle between polarizers is doubled, the transmitted intensity:
Reasoning: Intensity depends on the square of cosine of the angle.
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Increasing the angle between polarizers reduces intensity because:
Reasoning: Only the component of vibration along the analyzer axis is transmitted.
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At $\mathbf{45}^{\mathbf{o}}$, the transmitted intensity is:
Reasoning: $\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{45}^{\mathbf{o}}\mathbf{=}$ $\frac{\mathbf{1}}{\mathbf{2}}$
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If transmitted intensity is equal to incident polarized intensity, the angle must be:
Reasoning: Maximum transmission occurs when axes are parallel.
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A polarizer and analyzer are used together because:
Reasoning: Their relative angle controls the amount of polarized light transmitted.
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The intensity after a polarizer depends on:
Reasoning: Malus’s law relates intensity to the angle between transmission axes.
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A polarized light beam passes through an analyzer. If the angle changes from $\mathbf{0}^{\mathbf{o}}$ to $\mathbf{60}^{\mathbf{o}}$, the intensity changes from:
Reasoning: At $\mathbf{60}^{\mathbf{o}}$ $\mathbf{\Longrightarrow}$ $\mathbf{I}\mathbf{=}\mathbf{I}_{\mathbf{o}}\mathbf{\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{\theta\ \ \ \ \Longrightarrow \ \ }\mathbf{\ \ }\mathbf{I}\mathbf{=}\mathbf{I}_{\mathbf{o}}\mathbf{\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{60}^{\mathbf{o}}\mathbf{= 0.25\ }\mathbf{I}_{\mathbf{o}}$
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A polarizer transmits maximum light when the electric field vibrations are:
Reasoning: The component parallel to the transmission axis passes through.
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A polarized wave with intensity $\mathbf{I}_{\mathbf{o}}$ passes through an analyzer at $\mathbf{90}^{\mathbf{o}}$. This demonstrates:
Reasoning: Crossed polarizers block all transmitted vibrations.
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Malus’s law proves that the intensity of transmitted polarized light depends on:
Reasoning: The law is: $\mathbf{I}\mathbf{=}\mathbf{I}_{\mathbf{o}}\mathbf{\ }\mathbf{\cos}^{\mathbf{2}}\mathbf{\ }\mathbf{\theta}$
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