Phenomena and applications

A ray of light is incident on a plane mirror, making an angle of $35^\circ$ with the normal. The figure shows the incident ray.

1. a) State the two laws of reflection.
2. b) Determine the angle of reflection and the angle that the reflected ray makes with the surface of the mirror.

A ray of light passes from air into water, and it is observed that it bends towards the normal.

1. a) What wave phenomenon has the ray experienced? Define it.
2. b) State which wave quantity does not change when passing from one medium to another and which ones do change (propagation speed and wavelength). Justify why the frequency is conserved when changing medium.

A ray of light strikes the surface of water ($n_2 = 1,33$) from the air ($n_1 = 1$) with an angle of incidence of $40^\circ$ (data: $\sen 40^\circ \approx 0,64$). The figure shows the change in direction of the ray.

1. a) Write Snell's law of refraction and substitute the data from the statement into it, leaving $\sen r$ as the unknown.
2. b) Calculate the angle of refraction.

A glass optical fibre ($n_1 = 1,48$) is surrounded by air ($n_2 = 1$). The figure shows a ray incident on the wall of the fibre at the critical angle.

1. a) Write down Snell's law for a ray incident on the glass-air interface at the critical angle, for which the angle of refraction is $90^\circ$.
2. b) Calculate the critical angle of the fibre.

A diver points their torch towards the surface of the water ($n_{\text{agua}} = 1,33$; $n_{\text{aire}} = 1$) at an angle of $50^\circ$ to the vertical. The diagram represents the situation.

1. a) Calculate the critical angle for the transition from water to air.
2. b) Explain whether the beam from the torch will emerge into the air or undergo total internal reflection, by comparing the angle of incidence with the critical angle.

An optical fibre has a core of index $n_1 = 1,50$ and a cladding of index $n_2 = 1,47$. The diagram shows a section of fibre with a guided ray.

1. a) Calculate the critical angle at the core-cladding interface.
2. b) Explain why light propagates through the core without escaping when incident on its walls at angles greater than this critical angle.
3. c) If the core were in direct contact with air ($n = 1$), calculate the new critical angle and comment on the practical advantage provided by the cladding.

Two loudspeakers emit the same note in phase. A microphone placed on the perpendicular bisector of the line segment joining them is at the same distance from both. The diagram shows the arrangement of the loudspeakers and the microphone.

1. a) With what path difference do the two waves arrive at the microphone?
2. b) Reason whether the interference at that point is constructive or destructive and what will be heard there.

Two in-phase sources emit waves of wavelength $\lambda = 2\ \text{m}$. At a point P, the distances to the sources are $d_1 = 10\ \text{m}$ and $d_2 = 13\ \text{m}$. The figure shows the two sources and point P.

1. a) Calculate the path difference $\Delta r = d_2 - d_1$.
2. b) Determine whether $\Delta r$ is an integer number of wavelengths or an odd number of half-wavelengths, and conclude whether the interference at P is constructive or destructive.

In an optics laboratory, a double slit is illuminated with a red laser of $\lambda = 633\ \text{nm} = 6,33 \cdot 10^{-7}\ \text{m}$. The slits are separated by $d = 0,2\ \text{mm} = 2 \cdot 10^{-4}\ \text{m}$ and the screen is at $L = 2\ \text{m}$. In the small-angle approximation, the distance between two consecutive bright fringes is $\Delta y = \frac{\lambda \cdot L}{d}$ (given formula). The diagram summarises the set-up, not to scale.

1. a) Calculate $\Delta y$ and express it in millimetres.
2. b) Explain the origin of the fringes as interference of the waves coming from the two slits: what condition does the path difference satisfy at a bright fringe and at a dark fringe?
3. c) Predict how the pattern will change if the red laser is replaced by a blue one, of shorter wavelength.

From the hallway, the living room television can be heard even though the screen cannot be seen. The floor plan of the house outlines the situation.

1. a) What wave phenomenon allows sound to "bend round the corner"?
2. b) Explain, by comparing the width of the doorway with the typical wavelengths of sound (of the order of $1\ \text{m}$) and light (of the order of $5 \cdot 10^{-7}\ \text{m}$), why sound diffracts appreciably at the doorway while light does not.

When tilting a CD under white light, rainbow-like colours appear.

1. a) Explain the role of the microscopic grooves on the disc, which act as a reflection diffraction grating.
2. b) Explain why each colour is observed in a different direction, relating it to the constructive interference of each wavelength and to $\lambda_{\text{rojo}} > \lambda_{\text{azul}}$.

A home router emits its Wi-Fi signal in the $2,4\ \text{GHz}$ band (given: $c = 3 \cdot 10^{8}\ \text{m/s}$).

1. a) Calculate the wavelength of the signal and express it in centimetres.
2. b) Compare it with the width of a doorway ($0,8\ \text{m}$) and explain, naming the wave phenomenon responsible, why there is coverage in the adjacent room even though there is no direct line of sight to the router.