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Optics - Diffraction and Polarization of Light

Grade 12ICSEPhysics

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Diffraction is the phenomenon of bending of light waves around the edges of an obstacle or aperture. It is most prominent when the size of the obstacle aa is comparable to the wavelength of light λ\lambda.

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In Fraunhofer Single Slit Diffraction, the central maximum is formed at the center of the screen where path difference is zero. Secondary maxima and minima are formed on either side.

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Polarization is the process of restricting the vibrations of a light wave to a single plane. This phenomenon demonstrates that light is a transverse wave.

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Unpolarized light contains vibrations in all possible planes perpendicular to the direction of propagation. When passed through a Polarizer, it becomes plane-polarized with intensity I=I0/2I = I_0/2.

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Brewster's Law states that when unpolarized light is incident at a specific angle ipi_p (Brewster's angle), the reflected light is completely plane-polarized and the reflected ray is perpendicular to the refracted ray.

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Malus' Law describes the intensity of light transmitted through an analyzer when the transmission axis of the analyzer is at an angle θ\theta to the polarizer.

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Resolving Power is the ability of an optical instrument to produce distinct images of two objects placed very close to each other.

📐Formulae

asin⁡θ=nλ(Condition for nth secondary minima)a \sin \theta = n\lambda \quad \text{(Condition for } n^{th} \text{ secondary minima)}

asin⁡θ=(2n+1)λ2(Condition for nth secondary maxima)a \sin \theta = (2n + 1)\frac{\lambda}{2} \quad \text{(Condition for } n^{th} \text{ secondary maxima)}

W=2λDa(Linear width of central maximum)W = \frac{2\lambda D}{a} \quad \text{(Linear width of central maximum)}

2θ=2λa(Angular width of central maximum)2\theta = \frac{2\lambda}{a} \quad \text{(Angular width of central maximum)}

μ=tan⁡ip(Brewster’s Law)\mu = \tan i_p \quad \text{(Brewster's Law)}

I=I0cos⁡2θ(Malus’ Law)I = I_0 \cos^2 \theta \quad \text{(Malus' Law)}

Limit of resolution (Telescope)=Δθ=1.22λD\text{Limit of resolution (Telescope)} = \Delta \theta = \frac{1.22 \lambda}{D}

💡Examples

Problem 1:

A slit of width 0.1 mm0.1\,mm is illuminated by monochromatic light of wavelength 600 nm600\,nm. Calculate the angular width of the central maximum.

Solution:

Given: Slit width a=0.1 mm=10−4 ma = 0.1\,mm = 10^{-4}\,m, Wavelength λ=600 nm=6×10−7 m\lambda = 600\,nm = 6 \times 10^{-7}\,m. The angular width of the central maximum is given by 2θ=2λa2\theta = \frac{2\lambda}{a}. Substituting the values: 2θ=2×6×10−710−4=12×10−3 rad2\theta = \frac{2 \times 6 \times 10^{-7}}{10^{-4}} = 12 \times 10^{-3}\,rad.

Explanation:

The angular width is the angle subtended by the first order minima on either side of the central maximum at the slit.

Problem 2:

The refractive index of a certain glass is 1.51.5. Calculate the Brewster's angle for this glass and the corresponding angle of refraction.

Solution:

Using Brewster's Law: μ=tan⁡ip\mu = \tan i_p. Given μ=1.5\mu = 1.5, so 1.5=tan⁡ip⇒ip=tan⁡−1(1.5)≈56.31∘1.5 = \tan i_p \Rightarrow i_p = \tan^{-1}(1.5) \approx 56.31^\circ. At the Brewster's angle, ip+r=90∘i_p + r = 90^\circ. Therefore, r=90∘−56.31∘=33.69∘r = 90^\circ - 56.31^\circ = 33.69^\circ.

Explanation:

At the polarizing angle, the reflected and refracted rays are perpendicular to each other, which leads to the relation ip+r=90∘i_p + r = 90^\circ.

Problem 3:

Two polaroids are oriented such that their pass axes are at an angle of 60∘60^\circ. If unpolarized light of intensity I0I_0 is incident on the first polaroid, what is the intensity of light emerging from the second polaroid?

Solution:

After the first polaroid, the intensity becomes I1=I02I_1 = \frac{I_0}{2}. Applying Malus' Law for the second polaroid (analyzer): I2=I1cos⁡2θI_2 = I_1 \cos^2 \theta. I2=I02cos⁡2(60∘)=I02×(12)2=I08I_2 = \frac{I_0}{2} \cos^2(60^\circ) = \frac{I_0}{2} \times (\frac{1}{2})^2 = \frac{I_0}{8}.

Explanation:

The first polaroid reduces the intensity of unpolarized light by half, and the second reduces it further based on the square of the cosine of the angle between axes.

Diffraction and Polarization of Light Revision - Class 12 Physics ICSE