Let s review the four equations we now call Maxwell s equations. (Gauss s law for magnetism) (Faraday s law)

Transcription

1 Electromagnetic Waves Let s review the four equations we now call Maxwell s equations. E da= B d A= Q encl ε E B d l = ( ic + ε ) encl (Gauss s law) (Gauss s law for magnetism) dφ µ (Ampere s law) dt dφ B d E dl = = B da (Faraday s law) dt dt Recall that in our study of travelling waves, we saw that the harmonic wave functions for waves on a string obey a partial differential equation called the wave equation. y( x, x = 1 v y( x, t PHYS 153 8W 1

2 Electromagnetic waves arise from accelerated charges. Changing electric fields give rise to magnetic fields and changing magnetic fields give rise to electric fields. The wave equation information is contained in the last two of Maxwell s equations, Ampere s law (with conduction current being zero), and Faraday s law. First of all, think of a plane wave. The light reaching us from a distant star will arrive as a plane wave at Earth. E and B fields travelling in a plane wave are functions of time and one space coordinate only, which we take to be the x coordinate. A plane wave travelling in the x direction will have E and B constant across any plane perpendicular to the x axis. From Ampere s and Faraday s law, we can derive the following equation. E y x ( x, = ε µ E y t ( x, r 1 o r Think of the tiny circle at the centre being a distance star. Electromagnetic waves propagate outward in all directions. As the radius of curvature becomes >>r, the spherical wave becomes a plane wave. PHYS 153 8W

3 This is the equation of a wave. In this case, it is an electromagnetic wave in a vacuum. For waves in a non-vacuum, substitute for A similar equation holds for B. Bz( x, x = ε µ Bz( x, t εµ ε µ These are the equations for a plane E.M. wave travelling in the x direction (in a vacuum). y Note the E and B are perpendicular to each other and both are perpendicular to the direction of travel. z By comparing the general wave equation to the E.M. wave equation, we can see that 1 v= c ε µ B E E and B fields of an E.M. wave, at an instant of time travelling in the x direction. PHYS 153 8W 3 x

4 Sinusoidal E.M. waves are directly analogous to sinusoidal transverse mechanical waves on a rope, which we have studied. But E.M. waves are not mechanical. They do not require a medium through which to travel. Using sinusoidal functions, we can write E( x, jemaxcos( kx ω = B( x, kbmaxcos( kx ω = Fig shows these waves propagating along the positive x axis at velocity c (at a snapshot in time). At any point, the amplitudes are related by Fig E max = cb max The direction of travel of an E.M. wave is given by B Ex PHYS 153 8W 4

5 Summary of main ideas 1. An E.M. wave travels in a vacuum, with a constant speed c.. The wave is transverse. E B Both and are perpendicular to each other, and to the direction of propagation. The direction of propagation is the direction of the vector ExB product. 3. The ratio of the values E = B c 4. Unlike mechanical waves which require a medium to propagate, E.M. require no medium. In mechanical wave, the particles of the medium are doing the oscillating. In an E.M. wave, the E and B fields, are doing the oscillating. The visible part of the E.M. spectrum is only a small part of the whole spectrum. Fig. 3.4 PHYS 153 8W 5

6 Nature of Light, reflection and refraction Some fundamental ideas Fig We distiinguish between rays and wavefronts, for spherical and plane waves. Angle of reflection is equal to the angle of incidence and n a sin θ = a n b θ = sinθ r θ a b (Snell s law) Fig PHYS 153 8W 6

7 The wavelength of light in a vacuum λ and the wavelength of light In a material with index of refraction are related by λ= λ n v= fλ n> 1 Thus is less in a material with n λ Geometric Optics Reflection and refraction at a plane surface. Note that the image point is closer to the surface than the object point in a more dense medium. A pool of water, or a river, looks to be shallower than it is. Fig PHYS 153 8W 7

8 Image formed by a plane mirror. The eye thinks that the light is coming from P. But the light rays do not actually pass through P, so the image is virtual. Sign rules 1. Object distance. When the object is on the same side of the reflecting or refracting surface as the incoming light, the object distance s is positive; otherwise it is negative.. Image distance. When the image is on the same side of the reflecting or refracting surface as the outgoing light, the image distance s is positive; otherwise it is negative. 3. Radius of curvature of a spherical surface. when the centre of curvature C is on the same side as the outgoing light, the radius of curvature is positive; otherwise it is negative. PHYS 153 8W 8

9 Image of an extended object: Plane mirror P and P are on the axis, equal distances from the mirror. Note the rays which locate Q, the image of Q. The lateral magnification m is the ratio of image height to object height. y' m= y Fig Concave and convex mirrors Point object Curved mirrors can enlarge or reduce the image size, and can also produce a real image, all things that a plane mirror cannot do. Here is a brief review of the properties of concave and convex mirrors. PHYS 153 8W 9

10 Fig Fig A concave mirror has the centre of curvature C on the real side. Light can actually pass through it. A convex mirror has C on the virtual side. Light can not pass through it. An object at P (with the distance PV >R) will form a sharp image at P, for paraxial rays ( α small). For a concave mirror, the image at P will be a real image. for a convex mirror, it will be a virtual image. All paraxial rays from P will go through, or appear to go through P. Signs For the concave mirror, s, s, and R are positive. For the convex mirror, s is positive, and s, and R are negative. PHYS 153 8W 1

11 Using trigonometry and small angle approximations, it can be shown that 1 s 1 + = (spherical mirror) s' R Focal point and focal length Fig Fig PHYS 153 8W 11

12 = As the point P, the incoming rays become parallel. s Then and s ' = R The point at which the incident parallel rays converge is called the focal point F, and the image distance becomes the focal length f. s ' = f Parallel rays converge only for a paraboloid, but the inner section of a spherical surface (which is what we are dealing with for paraxial rays) differs by a negligible amount from a paraboloid. f = R ( (focal length of a spherical mirror) s= f s' = The rays can also be reversed. If, then R Substituting for, 1 s = (spherical mirror) s' f Parabolic mirrors are used in telescopes, because astronomers want an exact focal point for the whole mirror surface. PHYS 153 8W 1

13 f F is real and Is positive for a concave mirror. F is virtual, and is negative for a convex mirror. f Extended objects Fig Note that 4 different rays can be used to locate the image of an extended object. These are called principal rays. Only are needed. So far for concave mirrors, we have had objects lie outside the focal point. However, if, the image will be virtual. s< f PHYS 153 8W 13

14 Concave mirror examples. Note the changes in the position and size of the image as the object is brought from out-side the centre of curvature, to the centre of curvature, to the focal point, and then inside the focal point. Fig. 34. PHYS 153 8W 14