Q: A beam of X-rays with wavelength $$0.1\ \mathrm{nm} is $$incident on a crystal with interplanar spacing $$d = 0.2\ \mathrm{nm}. At $$what angle $$\theta ($$relative to the crystal planes) will the first-order ($$m=1) $$diffraction maximum occur?

$$\theta = \arcsin\left(\frac{0.1}{2 \times 0.2}\right) = 14.5^\circ$$

Q: According to Brewster’s Law, at what angle $$\theta_p$$ will reflected light be completely polarized for an air-water interface ($$n_1 = 1.00$, n_2$ = 1.33$$)?

$$\theta_p = \arctan\left(\frac{n_2$}{n_1$}\right) = \arctan\left(\frac{1.33}{1.00}\right) = 53.1^\circ$$

Question 1

Which equation describes the condition for constructive interference (maxima) in X-ray diffraction from a crystal lattice?

Accepted Answer

$$m\lambda = 2d\sin	heta$$

Question 2

A beam of X-rays with wavelength $$0.1\ \mathrm{nm} is $$incident on a crystal with interplanar spacing $$d = 0.2\ \mathrm{nm}. At $$what angle $$	heta ($$relative to the crystal planes) will the first-order ($$m=1) $$diffraction maximum occur?

Accepted Answer

$$	heta = \arcsin\left(\frac{0.1}{2 	imes 0.2}ight) = 14.5^\circ$$

Question 3

Which of the following best explains why X-rays are used for crystal structure analysis rather than visible light?

Accepted Answer

X-rays have wavelengths comparable to interatomic spacings in crystals.

Question 4

In the operation of an X-ray tube, what is the primary mechanism for X-ray production?

Accepted Answer

High-speed electrons striking a metal target emit X-rays.

Question 5

A linearly polarized wave passes through a polarizing filter whose transmission axis is perpendicular to the wave's polarization direction. What is the intensity of the transmitted wave?

Accepted Answer

$$0$$

Question 6

According to Malus’ Law, what is the transmitted intensity $$I$$ when polarized light of intensity $$I_0$$ passes through an analyzer at an angle $$	heta to $$the polarization direction?

Accepted Answer

$$I = I_0$ \cos^2$ 	heta$$

Question 7

Unpolarized light of intensity $$I_0$$ passes through a polarizer. What is the intensity of the transmitted light?

Accepted Answer

$$\frac{1}{2}I_0$

Question 8

A pilot flying at a height $$H = 1000\ \mathrm{m}$$ observes two objects on the ground using green light ($$\lambda = 555\ \mathrm{nm}) $$entering her eye through a pupil of diameter $$D = 2.5\ \mathrm{mm}. $$What is the minimum distance $$s_{min}$$ between the objects that can be resolved?

Accepted Answer

$$s_{min} = 1.22 \frac{\lambda}{D} H = 0.27\ \mathrm{m}$$

Question 9

What is the physical meaning of the Rayleigh criterion in optical instruments?

Accepted Answer

It defines the minimum angular separation at which two point sources can be distinguished.

Question 10

If unpolarized sunlight is scattered by atmospheric molecules, what is the polarization state of the scattered light observed at 90° to the incident direction?

Accepted Answer

The scattered light is completely polarized perpendicular to the incident direction.

Question 11

A beam of unpolarized light with intensity $$I_0$$ passes through two ideal polarizers. The transmission axis of the second polarizer is at $$60^\circ to $$the first. What is the intensity of the transmitted light?

Accepted Answer

$$I = \frac{1}{2}I_0$ \cos^2$ 60^\circ = \frac{1}{8}I_0$

Question 12

According to Brewster’s Law, at what angle $$	heta_p$$ will reflected light be completely polarized for an air-water interface ($$n_1 = 1.00$, n_2$ = 1.33$$)?

Accepted Answer

$$	heta_p = \arctan\left(\frac{n_2$}{n_1$}ight) = \arctan\left(\frac{1.33}{1.00}ight) = 53.1^\circ$$

Question 13

A light beam is partially polarized and partially unpolarized. If the minimum transmitted intensity through a polarizer is $$2.0\ \mathrm{W/m^2$}$$ and the maximum is $$8.0\ \mathrm{W/m^2$}$$, what is the intensity of the unpolarized component in the incident beam?

Accepted Answer

$$I_{0,unpolar} = 4.0\ \mathrm{W/m^2$}$$

Question 14

Which of the following best describes the effect of inserting a third polarizer at an angle between two crossed polarizers?

Accepted Answer

Some light will be transmitted through all three polarizers.

Question 15

Which of the following is a practical application of polarization by reflection?

Accepted Answer

Polaroid sunglasses reducing glare from horizontal surfaces.

Question 16

Which of the following statements about the electromagnetic spectrum is correct?

Accepted Answer

X-rays have shorter wavelengths than visible light but longer than gamma rays.

Question 17

If the diameter of the aperture in an optical instrument is doubled, what happens to the minimum resolvable angle $$	heta_{min}$$ according to the Rayleigh criterion?

Accepted Answer

$$	heta_{min} is $$halved.

Question 18

Which of the following best explains why 3-D movies use polarized glasses?

Accepted Answer

Each eye receives light polarized in a different direction, allowing the brain to perceive depth.

Lecture 20

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