3.3 Interference

Two coherent sources, A and B, which are in phase with each other, emit microwaves of wavelength 40.0 mm. The amplitude of waves from source B is twice that of source A. 

A detector is placed at the point P where it is 0.93 m from A and 1.19 m from B, as shown in Figure 1. The centre axis is normal and a bisector to the straight line joining A and B.

Figure 1

With reference to the phase of the microwaves, deduce the magnitude of the detected signal at P and explain your reasoning.

Wave intensity I is proportional to the square of the amplitude A. This can be written as: 

                     I ∝ A²

Use this result to determine the ratio of the intensity at P to the intensity at O.

Discuss, with suitable calculations, what happens to the detected signal as the detector is moved from P to O.

The source B is altered such that it emits waves that are 180^º out of phase with source A. 

Deduce the type of interference that now occurs at point P and explain your reasoning.

Figure 1 shows two loudspeakers  and  which are placed in an open field on a still day.

Figure 1

M is a microphone placed in the same horizontal plane as the loudspeakers.  and  are separated by 3.22 m and  is 10 m away from M. 

The lines   and  are perpendicular to each other. 

Discuss the conditions for wave coherence and suggest how it can be achieved by the setup shown in Figure 1.

When the loudspeakers are switched on, sound of frequency f = 1650 Hz is emitted in phase. 

Given that the speed of sound in air is 330 m s^–1, determine whether the sound detected by the microphone M is at minimum or maximum amplitude. 

You may treat the loudspeakers and the microphone as point objects.

The frequency of the signal is slowly varied to a value . 

During this change, the microphone M detected two whole cycles of change in amplitude. 

Calculate the two possible values of .

A soap bubble is known as a ‘thin film’. There is a thin layer of water trapped between soap molecules on either side, as shown in Figure 1a. 

Light that hits the bubble behaves in very predictable ways, resulting in visually interesting and colourful effects.

Figure 1a

Blue light of wavelength 400 nm is incident at an angle θ on a bubble where it splits into a ray that is reflected (ray α) and a ray which refracts into the bubble (ray β). Ray β reflects from the other side of the film, and then leaves the bubble again. 

A simplified diagram of this situation is shown in Figure 1b, with labelled points A, B, C and D.

Figure 1b

Upon reflection, ray α undergoes a phase shift of π radians. Ray β does not undergo any phase shift upon reflection. 

Write an expression for the path difference between ray α and ray β.

Justify your answer.

Thin-film interference occurs when reflected light from two different boundaries interfere. 

With reference to the path difference, describe the conditions for constructive and destructive thin-film interference of ray α and β in Figure 1b.

For a given angle of incidence θ, discuss what will be observed above the surface of the bubble for different colours of light.

Anti-reflective coatings use thin-film interference effects to make it appear that light is not reflected from the surface and instead passes straight through it. A simplified version of anti-reflective coated glass is shown in Figure 2:

The coating is designed such that there is a π phase shift at the first boundary (between the air and the coating) as well as at the second boundary (between the coating and the glass).         

By considering the conditions for constructive or destructive interference, discuss the limitations of this design.

A narrow beam of monochromatic blue light is directed at a double slit arrangement with slits S and T shown in Figure 1. Parallel blue and dark fringes are seen on the screen.

Figure 1

Explain briefly the role of diffraction in producing blue and dark fringes on the screen. 

You may draw a sketch to support your explanation if you wish.

Explain the formation of the fringes seen on the screen.

The blue light has a wavelength of 460 nm. 

The distance from S to X is 0.16 µm and from T to X is 0.85 µm. 

Determine whether point X in Figure 1 is a blue fringe or a dark fringe.

Describe how the appearance of the fringes would differ if red light had been used instead of blue light. Explain why this occurs.

Figure 1 shows the maxima of a two slit interference pattern produced on a screen when a laser was used as a monochromatic light source.

Figure 1

The wavelength of the monochromatic light = 640 nm

The distance from the slits to the screen = 7.5 m

 Use Figure 1 to calculate the slit spacing of the two slits that produced the pattern.

Calculate the angle between the middle of the central fringe and the middle of the third bright fringe.

The monochromatic light source is a red laser.

Describe and explain the change in the pattern seen on the screen when the red laser is replaced by a green laser. Assume the brightness of the central maximum is the same for both lasers.

Explain why less fringes would be seen if each of the slits was made wider, assuming that no other changes are made.

Figure 1 shows a diagram of apparatus used to demonstrate the formation of interference fringes using a white light source in a darkened room. Light from the source passes through a single slit and then through two narrow slits S₁ and S₂ with separation s.

Describe the interference pattern that is seen on the white screen. 

A filter transmits only blue light of wavelength  and yellow light of wavelength 1.3 . This filter is placed between the light source and the single slit. 

Describe the interference pattern now seen on the white screen. 

Use a calculation to support your answer.

A physicist carries out the Young double-slit experiment using the arrangement in Figure 1 using a laser that emits orange light of wavelength 610 nm. The separation of the slits, s, is 47 µm. 

Using a metre rule, the physicist measures the separation of two adjacent bright fringes in the central region of the pattern to be 7 mm. 

Calculate the distance D.

The physicist uses the same arrangement of apparatus to measure the wavelength of visible electromagnetic radiation emitted by another laser.

Describe what variables they could change in the apparatus in order to obtain an accurate value for the wavelength.

A laboratory ultrasound transmitter emits ultrasonic waves of wavelength 0.8 cm through two slits as shown in Figure 1. A receiver, moving along line AB, parallel to the line of the slits, detects regular rises and falls in the strength of the signal. 

A student measures a distance of 0.42 m between the first and the fourth maxima in the signal when the receiver is 1.5 m from the slits.

Figure 1

The ultrasound transmitter is a coherent source. 

Explain what is meant by the term coherent source.

Explain why the receiver detects regular rises and falls in the strength of the signals as it moves along the line AB.

Calculate the distance between the two slits.

One of the slits is now covered. No other changes are made to the experiment. 

State the difference between the observations made as the receiver is moved along AB before and after this change. Explain the changes that you mention.