lectur15 - 15. Lecture 15 15.1 Demo: Sound transmission...

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Unformatted text preview: 15. Lecture 15 15.1 Demo: Sound transmission with (laser) light Consider the circuit in figure 85 where and audio source drives the primary of a transformer. The secondary of the transformer is in series with a battery and a laser or LED light. Since the secondary of the transformer acts as a battery whose voltage fluctuates following the original sound, the intensity of the light will also fluctuate accordingly. This fluctuations are small and too fast to be seen by the eye, so the laser pointer seems to be working as usual but in fact contains the information about the sound. This can be detected by using a solar cell. The solar cell generates a voltage that is proportional to the intensity of the light and that therefore fluctuates as the original voltage. If we connect to any device (in this case a children’s toy) that has an input for a microphone, it will be the sound manifest either on a loudspeaker (as we do) or by recording it. The idea for this interesting circuit can be found in several websites: http://www.i-hacked.com/index.php?option=com content&task=view&id=162&Itemid=44, http://www.wikihow.com/Transmit-Audio-With-a-Laser-Pen, etc. It is always interesting to look around and try to see how the ideas we learn can be used in simple ways to produce interesting results. 15.2 Electromagnetic waves A varying electric field produces a magnetic field, a (varying) magnetic field. A varying magnetic field creates a (varying) electric field. This gives rise to a periodic wave that propagates in space. In vacuum they propagate at speed c ￿ 3 × 108 m . This is the speed s of light which is an example of an electromagnetic wave. The speed of propagation can be written as 1 c= √ (15.1) µ0 ￿ 0 Generically a wave is profile that propagates (see fig.86). The intensity of the field oscillates in time and also changes in space. The distance between to crests is called the wave-length λ. At a given point the intensity oscillates in time and the interval between two maxima is called the period. It is clear that, if the profile propagate at velocity c, the two maxima separated by a distance λ will arrive at a time interval of T = λ . The inverse of the period is called the frequency c f= 1 c = T λ (15.2) and is measured in Hertz, where 1Hz = 1 1 a unit of frequency we already saw for s AC current. As a simple example consider a FM radio station transmitting at f = – 98 – Figure 85: Demo. A transformer is used to modulate, that is to change the intensity, of a laser following an audio input. The variations in the intensity of the light are detected by a solar cell (or photoelectric panel) and converted into sound by an amplifier. The same principle works if we use a LED but the laser beam stay narrow at much longer distances and transmits the signal more effectively. 1 100M Hz = 108 Hz . The period is T = f = 10−8 s. In that time the wave travels λ = cT = 3m which is the wavelength of that particular radio station. In the case of electromagnetic waves the fields that oscillate are the electric and magnetic fields. For that reason we need to know, not only their intensity but also their orientation. Both from theory and experiment it appears that electromagnetic waves are such that the electric and magnetic field are perpendicular to each other and perpendicular to ˆ the direction of propagation. The direction of propagation k is given by the right ˆ￿ ￿ hand rule: k ￿ E × B . In fig. 87 we see a depiction of an e.m. wave. The electric field can point in any direction contained in the plane perpendicular to the direction – 99 – Figure 86: Generic periodic wave. A disturbance propagates with velocity c. The distance between to peaks is called the wave-length λ. At a fixed point the time interval between the passage of two peaks is called the period T = λ . c of propagation of light. If it points always in the same direction then it is said that the wave is linearly polarized and the direction of polarization is the direction of the electric field. In general any wave can be consider as a superposition of two waves with orthogonal polarizations. That is, suppose that the wave propagates in direction z , ˆ then the electric field can point in the direction of x or y or can be a superposition of ˆ two waves, one polarized along x and the other along y . For example, an interesting ˆ ˆ case is circular polarization where the electric field rotates in the (xy ) plane. This can be considered as a superposition of two oscillations, one in x and the other in y which ￿ are out of phase (this is just from projecting E in its two components along x and y ). ˆ ˆ A simple demo show how this works (see fig.??). The e.m. wave is detected by putting two metallic rods connected by a light-bulb. When a electric field is present it moves the electrons in the metal and generates a current that lights up the electric bulb. The current has a maximum when the rods are parallel to the electric field which allows us to find its direction. Other important facts about e.m. waves are that modulus of electric and magnetic fields are related by: ￿ ￿ | E | = c| B | (15.3) and that the intensity of the wave is given by 1 ￿ I = ￿0 c| E | 2 2 – 100 – (15.4) W Is is measured in m2 = mJ s and determines how much energy crosses a given area per 2 unit time. For example for a solar cell this gives the power generated by square meter of solar cell (up to the efficiency factor since part of the energy is converted into heat). Figure 87: Electromagnetic wave. The electric and magnetic field are perpendicular to each other and to the direction of propagation. Although electromagnetic waves of different wave-length represent the same physical phenomenon, they interact (namely are generated and absorbed) with matter in a different way since they tend to interact with objects of the size of order the wavelength and/or with processes whose timescale is similar to the period of the wave. For that reason they are known with different names. For example γ -rays interact with the atomic nucleus and therefor have wave-length of order λ ∼ 10−15 m or shorter. X-rays and light interact with atoms and therefore have wave-length in the order of 10−10 m up to 10−6 m. Radio waves are generated by oscillating circuits and antennas and have wave-length in the meters or even kilometers. Fig.88 gives a more precise classification. 15.3 Light as an electromagnetic wave In the previous section we saw that light are electromagnetic in a particular narrow region of wave-length. It happens to be the range to which our eyes are sensitive to. The main reason seems to be that water is more transparent in that region allowing us to see under water and also in the atmosphere (which has large amounts of water vapor). In the case of light the polarization can be detected by a polarizer, a medium with a prefer direction which allows only the passage of waves polarized in that direction. If the wave is polarized in a different direction, only the component of the electric field parallel to the preferred direction goes through. That is (see fig.89), if the angle between the electric field and the preferred direction is θ, the magnitude of the electric ￿ ￿ field after the wave goes through the polarizer if |Eout | = |Eout | cos θ. the intensity is – 101 – Figure 88: Electromagnetic spectrum from wikipedia. Electromagnetic waves of different wave-length interact with matter differently and also have different applications. proportional to the square of the electric field (see eq.(15.4)) and therefore: Iout = cos2 θ Iin (15.5) 15.3.1 Index of refraction An important fact about light propagating in a medium is that its speed cm is slower that c the speed of light in vacuum. The ratio is called the index of refraction: n= c cm (15.6) The change in speed gives rise to a phenomenon known as refraction. A wave crossing a medium interfaces is bent. The change in angle is given by Snell’s law: n1 sin θ1 = n2 sin θ2 (15.7) where n1,2 are the index of refraction of the two media and the angles are defined in figure 90. This is nicely illustrated in a demo where we can also see that for a given angle the light emerges from the liquid parallel to the interface and for larger angles it does not emerge at all, a phenomenon known as total reflection. – 102 – Figure 89: A polarizer lets through only the component of the electric field parallel to a preferred direction (indicated with red). 15.3.2 Fermat’s principle A problem that works in a similar way to the refraction of light is the following. Looking at fig. 92 suppose you need to reach a buoy some distance from the shore in the shortest possible time. If you are in the water you swim straight to it since a straight line is the shortest path. If you are in-land however, you first need to run to shore and since you run faster than swim it is better to take a somewhat longest path on land if it cuts your swimming leg. It turns out that, as pointed out by Fermat already in the sixteen hundreds, the path of least time is such that Snell’s law is obeyed. Therefore, if we assume that a ray of light takes always the path of least time then we can derive Snell’s law as suggested by Fermat. On small correction is that in certain cases a ray of light would take actually take also the path of largest time. These are generically called extremal paths. A more sever objection is that such principle seems more appropriate to particles rather than waves. We will see later on how to derive Snell’s law using that light is a wave. Nevertheless Fermat’s principle was a great idea and had a profound impact physics. In fact, most of modern physics is based in similar extremal principles. That is for a certain physical system one finds a quantity called the action such that the system always follows the paths of extremal action. – 103 – Figure 90: Refraction of light in the interface between two media. Snell’s law determines that n1 sin θ1 = n2 sin θ2 . Part of the light is also reflected and the angle of reflection is the same as the angle of incidence θ1 . Figure 91: Demo: Refraction of light in the interface between two media. Illustrates Snell’s law. – 104 – Figure 92: Fermat’s principle can be illustrated with a simple consideration. If a person needs to get to a buoy in the water and part of the path is on land where he/she can run a fast and part in the water which slows them down, which is the optimal path?. It turns out that the optimal path is given by Snell’s law!. – 105 – ...
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This note was uploaded on 12/07/2011 for the course PHY 219 taught by Professor Na during the Fall '11 term at Purdue University-West Lafayette.

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