PHYS 202 Notes, Week 8

Transcription

1 PHYS 202 Notes, Week 8 Greg Christian March 8 & 10, 2016 Last updated: 03/10/2016 at 12:30:44 This week we learn about electromagnetic waves and optics. Electromagnetic Waves So far, we ve learned about magnetic and electric fields as separate entities. However, as it turns out, they are very closely linked; two sides of the same coin, so to speak. We ve had hints of this already, as Faraday s law shows that time-varying magnetic fields are a source of an electric field (manifested as an emf). As it turns out, the reverse it true too: time varying electric fields are also a source of magnetic field. Fundamentally, electric and magnetic fields are described by four equations called Maxwell s equations. These use highly advanced mathematics, so we won t go into detail; we ll just examine the consequences. The main consequence is that time-varying electric and magnetic fields can travel, or propagate, through space in the form of waves. These are called Electromagnetic Waves (or EM waves for short). Important points Time-varying #» E and #» B fields propagate through space in waves. EM waves have both the electric and magnetic fields perpendicular to the direction of motion. EM wavelengths fall along a large spectrum, including visible light. Important equations EM wave speed ( speed of light): c = 1 / ɛ0 µ 0 i = m/s Figure 1 shows how electromagnetic waves propagate through space. As shown in the top panel, the electric and magnetic field directions are always perpendicular to the direction of motion. To get the field directions, we can use a variation of the right hand rule: Figure 1: Propagation of an electromagnetic wave. Point your thumb in the direction of the direction of motion. Point your fingers in the direction of the #» E field.

2 phys 202 notes, week 8 2 Rotate your fingers by 90 deg. They now point in the direction of the #» B field. Propagation Speed Electromagnetic always waves travel through empty space with a constant speed, c = 1 ɛ0 µ 0. (1) Numerically, this comes out to c = 1 [ C 2 / (N m 2 )] (4π 10 7 N/A 2 ) (2) = m/s. (3) As you may recognize, this is exactly the speed of light in a vacuum! This is because light is a specific form of electromagnetic wave. Another property of EM waves is the relationship between the magnitudes of #» E and #» E. They are always related by the speed of propagation c, E = cb. (4) Properties Summarized The following points summarize the properties of EM waves: 1. Both #» E and #» B are to the direction of propagation and to each other. 2. The magnitudes of #» E and #» B are always related by E = cb. 3. The wave travels in a vacuum with unchanging speed c = 1 / ɛ0 µ 0 = m/s. (5) 4. Electromagmetic fields do not need a medium to travel through. It is the #» E and #» B fields themselves that are doing the waving. The Electromagnetic Spectrum As mentioned, visible light is a particular form of electromagnetic wave. More generally, EM waves span a wide range of wavelengths called the electromagnetic spectrum. This is summarized in Figure 2. Electromagnetic waves span many orders of magnitude in wavelength, from 10 m on the high end to m on the low end. Equivalently, they span a wide range of frequencies given by f = c/λ. (6)

3 phys 202 notes, week 8 3 Figure 2: The electromagnetic spectrum. EM waves are roughly broken down into seven (overlapping) categories as shown in Figure 2. You are probably already familiar with all of these from your everyday life. At the low end of the spectrum are long wavelength, high frequency radio waves like the ones you pick up in an FM radio. At the high end are short wavelength, high frequency gamma rays, which are the result of nuclear de-excitations. Visible light lies roughly in the middle of these two extremes. EM Wave Properties Like all waves, EM wave propagation can be described by mathematical functions called wave functions. These describe the amplitude of the #» E and #» B fields as a function of time t and position x: where E = E max sin (ωt kx) (7) B = B max sin (ωt kx), (8) ω = 2π/T and k = 2π/λ T is the period of the wave and λ is the wavelength E max = cb m ax The negative sign means a wave traveling in the +x direction; the positive sign means a wave traveling in the x direction; 1 and When E is positive, B may be positive or negative; get the sign using the right hand rule as in Figure 1. Figure 1 shows what a propagating wave described by Eqns. (7) and (8) and traveling in the +x direction looks like. A picture of an EM wave traveling in the x direction is shown in Figure 3. Important points EM wave propagation is described by equations called wave functions. The power carried by EM waves can also be described mathematically. Important equations Wave functions Energy density E = E max sin (ωt kx) B = B max sin (ωt kx) u = ɛ 0 E 2 /2 + B 2 /(2µ 0 ) = ɛ 0 E 2 Average intensity I = ɛ 0 ce 2 max/2 1 Note the sign flip!!! = E max B max/(2µ 0 )

4 phys 202 notes, week 8 4 Figure 3: An EM wave traveling in the x direction. Energy in EM Waves As we ve already learned, energy can be stored in #» E and #» B fields. Recall that the equations for the respective energy densities are and u E = 1 2 ɛ 0E 2 (9) u B = B2 2µ 0. (10) In EM waves, the total energy density is the sum of these two, u = 1 2 ɛ 0E 2 + B2 2µ 0. (11) Relating E and B by B = E/c = ɛ 0 µ 0 E, we can express the energy density just in terms of the electric field strength: u = 1 2 ɛ 0E µ 0 ( ɛ 0 µ 0 E) 2 (12) = ɛ 0 E 2. (13) Note that the energy oscillates with time since E also oscillates. It is not a constant quantity. We can also talk about the power carried by EM waves. In particular, let s talk about the power per unit area, S. This is equal to the energy density times the velocity of the wave c, S = cɛ 0 E 2. (14) Again, this is an instantaneous quantity; it s constantly changing with time. If we want to talk about an average quantity, we can define the intensity I as the average power per unit area. This is found simply by

5 phys 202 notes, week 8 5 replacing E with E max /2 in Eq. (14), This can also be expressed using E max and B max : I = 1 2 ɛ 0cE 2 max. (15) I = 1 2 ɛ 0c 2 E max B max (16) = 1 ( ) 1 2 ɛ 0 E max B max (17) ɛ 0 µ 0 = 1 E max B max. (18) 2µ 0 Remember: this is an average quantity that does not vary sinusoidally. Optics The nature of light has been pondered for hundreds of years. Starting around the 1600s, there were competing theories: one treating light as a steady stream of particles (as proposed by Isaac Newton) and the other treating it as a wave. As it turns out, both of these theories are correct. Sometimes light behaves like a wave while other times it behaves like a particle. Only with the advent of quantum mechanics in the 20 th century can we fully resolve these two competing properties. For the time being, we ll focus on the wave nature of light. Recall that light is simply a specific form of EM wave. As such, it travels in wave fronts as represented in Figure 4. To model the propagation of light in media, we can represent light in the form of rays, or straight lines, rather than visible wave fronts. Figure 5 shows two different forms of light rays: the first (left panel) is the result of a spherical wave front resulting in light rays diverging from a point source; the second (right panel) is the result of a planar wave front, resulting in parallel light rays. Important points Light and other EM waves can be modeled as rays. Light rays undergo optical phenomena: refraction and reflection. Important equations Index of refraction Snell s law n = c/v n a sin θ a = n b sin θ b Total internal reflection sin θ crit = n b/n a Figure 4: A spherical wave front. Figure 5: Spherical (left) and planar (right) light rays.

6 phys 202 notes, week 8 6 Reflection and Refraction Although light travels at a constant speed c in a vacuum, it slows down when inside physical media. The amount by which light slows down defines something called the index of refraction, deonted by n. This is related to the speed of light in a vacuum c and the speed in the medium v by n = c/v. (19) Note that the speed of light in a medium is always less than c. This means that n is always greater than one. When traveling in a medium, the frequency of light stays the same as the vacuum frequency. However, the wavelength changes according to λ = λ 0 n. (20) When light encounters a medium boundary, it can either be reflected or refracted as shown in Figure 6. Let s talk about these two possibilities in a bit more detail. Reflection In reflection, the light bounces off the medium, staying in the same plane. In this case, the outgoing angle, θ r is always equal to the incoming angle θ a : θ r = θ a. (21) Figure 6: Reflection and refraction. Refraction In refraction, the light crosses the medium boundary, changing its direction to a new angle θ b. The relationship between θ a and θ b is given by Snell s Law, n a sin θ a = n b sin θ b. (22) Note that these angles are with respect to the normal to the interface. Furthermore, if the incident light is normal to the interface (θ a = 0), there is no bending. Total Internal Reflection As shown in Figure 7, there is always some critical angle θ crit above which refraction is impossible because above this angle the outgoing angle is greater than 90. The critical angle for total internal reflection is given by sin θ crit = n b n a. (23) Figure 7: Total internal reflection.

7 phys 202 notes, week 8 7 When the incident angle θ a is greater than this angle, refraction is impossible and all of the light will be reflected. This phenomena is referred to as total internal reflection. Polarization Recall (Figure 1) that EM waves are transverse waves, i.e. the directions of the #» E and B #» fields are perpendicular to the direction of propagation, and to each other. However, within these constraints, the displacement directions of the #» E and B #» fields can be in different orientations. We refer to the displacement direction of the electric field as the polarization direction. 2 Now let s pretend an EM wave has #» E field displacements in only one direction, for example the y direction. We then say that the wave is linearly polarized in the y direction. Figure 8 shows examples of waves polarized in the y and z directions, as well as how a slit can act as a polarizing device. To set the polarization of light, we can use a device called a polarizing filter, as demonstrated in Figure 9. This device serves to only transmit light with a specific polarization direction. However, we have to pay a price for this: the intensity of the light is reduced. Let s say we have light striking a polarizing filter at an angle φ to the direction of polarization. The intensity reduction is described by the equation, 2 The reason we choose the electric and not the magnetic field is that most EMwave detectors, such as your eyes, are sensitive to the electric part of the wave. I = I max cos 2 φ, (24) where I max is the maximum intensity of the light transmitted (at φ = 0, i.e. already polarized in the direction of the filter). Note that if unpolarized light strikes the filter, φ is randomized and it turns out that I = I max /2. Figure 8: Illustration of polarization along different axes. Figure 9: Example of a polarizing filter.

8 phys 202 notes, week 8 8 Mirrors Images The first concept we need to introduce is that of an image. This is basically the apparent source of light rays as seen by an observer. This is illustrated in Figure 10 (left side). Although the real source of the light rays is point P, when they reach the observer s eyes it looks like they came from point P. This is because the rays reflect off the surface of the mirror and reach the observer s eyes at an angle that makes them look like they came from P. Figure 10: An image due to a mirror (left) and refraction (right). Similarly, images can also form due to refraction, as shown in the right side of Figure 10. Here, the rays change angle when crossing the medium boundary. Thus they reach the observer s eyes at an angle that makes them look like they came from P. Figure 11 shows what s going on in more detail and defines a few terms. First, we have the distance s, the horizontal distance from P to the mirror plane. This is called the object distance. When the object is on the same side of the reflecting (or refracting) surface as the incoming light, we say that the object distance is positive. Otherwise, it s negative. We can also define the image distance, s in the figure. This is the horizontal distance from the mirror to the image point. When the image is on the same side of the reflecting (or refracting) surface as the outgoing light, we say that the image distance is positive. Otherwise, it s negative. The image shown in Figure 11 is a virtual image. This is because the light rays don t actually come from the image point P. They only look like they do. There is also a type of image called a real image where the

9 phys 202 notes, week 8 9 Figure 11: An image from a mirror, in more detail. rays really do pass through the image point. Magnification In the Figure 11, the object distance s and image distance s both have the same magnitude. Thus if we were to place an image of height y in front of the mirror, the apparent image would have height y y. In this case, there is no change in the size of the image; the magnification is unity. In general, the apparent height of an image is not always the same as the real height. We can define the lateral magnification of an image as m = y y. (25) Depending on the geometry of the problem, images can either appear in the same vertical orientation as the original, in which case they are upright. They can also appear flipped along the y-axis, in which case they are inverted. The orientation of 3d objects can also be changed by mirrors. This is represented in Figure 12, where the orientation of the hand changes from left-handed to right. We call this type of image reversed. 3 Flat plane mirrors always form images that are the same size as the original, but reversed. Figure 12: A reversed image. 3 Or colloquialy, people often use the term mirror image.

10 Example Problems phys 202 notes, week 8 10

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