Question

Two sheets of polarizing material are placed with their transmission axes at right angles to one another. A third polarizing sheet is placed between them with its transmission axis at \(45^{\circ}\) to the axes of the other two. (a) If unpolarized light of intensity \(I_{0}\) is incident on the system, what is the intensity of the transmitted light?(b) What is the intensity of the transmitted light when

Short answer

Answer: The intensity of the transmitted light with three sheets is \(\frac{I_{0}}{4}\), and the intensity of the transmitted light with two sheets is \(0\).

Step-by-step solution

Identify the angles between the sheets

As the third sheet is placed at \(45^{\circ}\) to both the first and second sheets, we know that: - Angle between Sheet 1 and Sheet 3: \(\theta_{1} = 45^{\circ}\) - Angle between Sheet 3 and Sheet 2: \(\theta_{2} = 45^{\circ}\)

Apply Malus Law for Sheet 1 and Sheet 3

We'll first find the intensity after Sheet 3 using the input light intensity \(I_{0}\): $$I_3 = I_0 \cos^2(\theta_1) = I_0 (\cos 45^{\circ})^2 = I_0 (\frac{\sqrt{2}}{2})^2 = \frac{I_0}{2}$$

Apply Malus Law for Sheet 3 and Sheet 2

Now, we'll find the transmitted intensity after Sheet 2 using the intensity after Sheet 3: $$I_t = I_3 \cos^2(\theta_2) = \frac{I_0}{2} (\cos 45^{\circ})^2 = \frac{I_0}{2} (\frac{\sqrt{2}}{2})^2 = \frac{I_0}{4}$$ So the intensity of the transmitted light when all three sheets are present is \(\frac{I_{0}}{4}\). ##Step 2: Calculating the intensity for Scenario B with only two polarizing sheets ##

Identify the angle between the remaining sheets

Since the sheets are at right angles to one another, the angle between Sheet 1 and Sheet 2 is: $$\theta_{3} = 90^{\circ}$$

Apply Malus Law for Sheet 1 and Sheet 2

Now, we'll find the transmitted intensity when the third sheet is removed: $$I_t' = I_0 \cos^2(\theta_3) = I_0 (\cos{90^{\circ}})^2 = I_0 (0)^2 = 0$$ So the intensity of the transmitted light when the third sheet is removed is \(0\). In conclusion, the intensity of the transmitted light with three sheets is \(\frac{I_{0}}{4}\), and the intensity of the transmitted light with two sheets is \(0\).