Department of physics and astronomy san francisco state university 1600 holloway avenue san francisco, california 942 u. Reflection coefficients for a glasstoair interface. The equation explaining amperes law which is the final maxwells equation is given below. If is time, then you get the equation above, which is the same as saying that the color going in is equal to the color going out. Calculate reflection and transmission coefficients, r and t, as a function of incident light polarisation and angle of incidence using em boundary conditions. Gauss electric law describes the sources of the electric displacement. Of course, the above expressions correspond to the law of reflection and snell s law of refraction, respectively. Fermats principle and the laws of reflection and refraction. Lecture 22 electromagnetic waves mit opencourseware. Reflection and snells law electromagnetic geophysics. When an electromagnetic wave strikes the surface of a dielectric, both.
Fresnels equations for reflection and transmission incident, transmitted, and reflected beams. Reflection and refraction 17 be scattered away from its original direction figure 2. Instead, the description of electromagnetics starts with maxwells equations which are written in terms of curls and divergences. Maxwell s equations in the latter half of the nineteenth century, the scottish physicist james clerk maxwell demonstrated that all previously established experimental facts regarding electric and magnetic fields could be summed up in just four equations. I would like to explain the continuity conditions for electrical and magnetic fields at such an interface with specific applications with deriving the law of reflection andor snells law of refraction from these conditions and with using maxwell equations. Feynmans proof of maxwells equations get a copy of the paper feynmans proof of maxwells equations by freeman dyson published in ajp 58 3 march 1990.
Maxwell was the first person to calculate the speed of propagation of electromagnetic waves which was same as the speed of light and came to the conclusion that em waves and visible light are similar these are the set of partial differential equations that form the foundation of classical electrodynamics, electric circuits and classical optics along with lorentz force law. The derivation of snell s law using fermats principle is straightforward. Snell s law of refraction can be derived from fermats principle that light travels paths that minimize the time using simple calculus. In the middle of twentieth century there were a few attempts to have a better understanding of the index of refraction. Clifford algebra derivation of the characteristic hypersurfaces of maxwells equations william m. Derivation of poyntings theorem 3 the frequency domain phasors and maxwells equations. Maxwells equations have a simpler form when written in this way, and may in consequence be easier to remember. Snell s law explained snell s law also known as snell descartes law and the law of refraction is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other wave s passing through a boundary between two different isotropic media, such as water, glass, or air. Geometrical optics is an interesting subject full of wonderful applications, a subject that can appear.
To get maxwell s equations well look at a special case. Fermats principle states that light travels in the shortest path that takes the least time. Snells law is a formula used to describe the relationship between the angles of incidence and. Maxwell s equations are a set of coupled partial differential equations that, together with the lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The relation between electric field and current density in ohms law is different in structure compared to the connection following from maxwell s equations between the same vector fields. This proof has many interesting points, not the least of which is that maxwells equations appear to be derivable from newtons second law and the basic. Weile department of electrical and computer engineering.
Their physical content is, of and course, unaltered by the introduction of the two new auxiliary fields d h. We also use the definition of the sine as opposite side over hypotenuse to relate the lengths to the angles of incidence and reflection. In addition well assume that e x doesnt depend on where in the xy plane it is measured. Assume that the electric and magnetic fields are constrained to the y and z directions, respectfully, and that they are both functions of only x and t. I suggest that you should explain, line by line, how we can obtain maxwell. These equations are called the fresnel equations for perpendicularly polarized spolarized light. Snell s law also known as snell descartes law and the law of refraction is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air. What is the direction of energy flux of the em wave. Explicit expressions are given for the velocity and acceleration components of the ray. Consider a charge free currentfree interface between materials with. Equation 5 is known as snell s law or the law of refraction and it gives us the relationship between \. We can use maxwells equations to derive the boundary conditions on the. Yet another way to derive snell s law is based on translation symmetry considerations.
Simple derivation of electromagnetic waves from maxwells. Where i is the angle of incidence and r is the angle of refraction. A ray is a line that has an origin but does not have an end. Yet another way to derive snells law is based on translation symmetry considerations. The laws of reflection and refraction the full treatment of reflection and refraction is dervived from the maxwells equations of electrodynamics. Maxwell s equations and boundary conditions at interfaces. Outline te and tm fields refraction and snells law. Notice that snells law shows that the index of refraction and the sine of the angle of refraction are inversely proportional that is, as the refractive index gets larger n 2 n 1 the sine of the refracted angle gets smaller sin. Nor is it the only logical approach, because studying lens systems teaches you nothing more about maxwells equations than you can learn by deriving snells law.
In this special situation we have an electric field with only an xcomponent e x the y and z components of e are set to be zero. Snell s law is the generalization of the above in that it does not require the medium to be the same everywhere. Reflection at a dielectric boundary university of texas at. An authoritative view of maxwell s equations that takes theory to practicemaxwell s equations is a practical guide to one of the most remarkable sets of equations ever devised.
Geometric derivation of snells law for refraction doc. Maxwells equations and boundary conditions at interfaces. The magnetic field created by an electric current is proportional to the size of that electric current with a constant of proportionality equal to the permeability of free space. I know for a fact that we need to know the lorentz force in order to derive a lagrangian for electromagnetism. Using noethers theorem to get a constant of motion get all possible constants of motion given an explicit hamiltonian number of angular momentum states 3 particles. Snells law is then used to characterize the refraction angle in terms of the. From te analysis from phase matching from fermats principle of least time total internal reflection and fibers fios. Introduction to partial di erential equations, math 4635. To evaluate the left hand side of the equation, lets use a small cylinder shown in the drawing below as the surface of integration.
Refraction and snells law reading shen and kong ch. Maxwells equations imply the existence of electromagnetic waves as ligh, x. Fresnel s equations for reflection and transmission incident, transmitted, and reflected beams boundary conditions. This scattering is caused by the interaction of light with small particles, even atoms or molecules, within the material. Opticsfermats principle wikibooks, open books for an. For the case of oblique incidence, we need to consider two independent wave polarizations separately. In 1865, he predicted the existence of electromagnetic waves. Its a really cool argument that comes down to this. An alternative, pedagogically simpler derivation of the allowed. Setup here, we will consider the reflection and refraction of a uniform, linearly polarized, homogeneous plane wave at a horizontal interface fig.
If the interface lies in a plane which is crucial to this argument, then can represent one of the spatial dimensions in the plane and you get the law of reflection and snells law. Jan 10, 2018 this physics video tutorial provides a basic introduction into maxwell s equations and electromagnetic waves. For example, a timevarying current j on a linear antenna generates a circulating and timevarying magnetic. Together with a vector identity due to stokes i c d. Because the rate of change of the flux of the electric field acts as a. Reflection above the critical angle phase shift is different polarization states linearly polarized circularly polarized elliptically polarized. When light finds a surface separating two media air and water, for. Nowadays, these equations are generally known as maxwell s equations.
This means that maxwell s equations will allow waves of any shape to propagate through the universe. Although upwind schemes are well established for firstorder hyperbolic systems, it was only recently shown by banks and henshaw how upwinding could be incorporated into the secondorder form of the wave equation. Amperes law definition, statement, examples, equation, video. Consider a chargefree currentfree interface between materials with. Geometric derivation of snell s law for refraction doc physics. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. The first polarization has all the wave electric fields perpendicular to the plane of incidence, whilst the second has all the. Next, well solve the wave equation with these boundary conditions, much like we solved the. The equations of motion for a ray in a snell s law medium with a varying index of refraction are derived.
Snells law from an elementary particle viewpoint d. First of all, we suppose that a plane monochromatic wave is incident on the planar surface separating two isotropic media with nearly equal permittivity. Professor paul huray presents techniques that show the reader how to obtain analytic solutions for maxwell s equations for ideal materials and boundary conditions. Physics 202, lecture 24 todays topics polarization of light polarization by absorption polarization by reflection polarization by scattering polarization by birefringence circular versus linear polarization derivation of laws of reflection and refraction using principles of light propagation. Oct 27, 2010 im wondering where to start for the proof of snell s law using maxwell s equations. Snell s law can be derived using elementary calculus and trigonometry. A ray of light beginning in the top medium travels through the interface into the bottom medium. Snells law is defined as the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant, for the light of a given colour and for the given pair of media. Start with maxwells equations in a dielectric medium no currents, no charges 0 0 1 0 0 1 b t b c xe d t d c xh along with the material equations. This law is more complicated than that for reflection, but an understanding of refraction will be necessary for our future discussion of lenses and their applications. One of the most important consequences of the maxwell equations is the equations for. Start with maxwell s equations in a dielectric medium no currents, no charges 0 0 1 0 0 1 b t b c xe d t d c xh along with the material equations.
Snells law formula is derived from fermat s principle. The normal on the surface is used to gauge the angles that the refracted ray creates at the contact point. Simple derivation of electromagnetic waves from maxwells equations by lynda williams, santa rosa junior college physics department. Introduction to partial di erential equations, math 4635, spring 2015 jens lorenz april 10, 2015 department of mathematics and statistics, unm, albuquerque, nm 871. Here, we will derive these angle by considering an electric field. Since snell s law only involves sines i wonder whether this minimum problem has a simple geometric solution.
Derivation of maxwells equations to get maxwells equations well look at a special case. Using that the speed of light is v c n we deduced that. Solving metamaterial maxwells equations via a vector wave. Light is an electromagnetic disturbance and, as such, is described using maxwells equations, which expresses the relationship between the electric. From maxwells equation through the wave equation to the eikonal equation up to the derivation of snells law. B h d e substituting the equation for b into the second maxwell s equation and. You use maxwells equations to get the relationships for field at a boundary. Snell s law like with reflection, refraction also involves the angles that the incident ray and the refracted ray make with the normal to the surface at the point of refraction. Using the pythagorean theorem, we know that equations 1 since light travels at a constant speed in each medium, we also know that equations 2 the total time that the light ray requires to travel between its predetermined starting and ending points can now be written as equation 3.
The question asks for the angle of the light beam in the diamond, relative to the surface. How do we derive snells law using maxwells equations a solve the wave equation from physics 95. Snells law is then used to characterize the refraction angle in terms of the complex wavenumber for both media. In this lesson we are going to look at a derivation of snell s law based on the principle of least time. A point of stationary phase moves with velocity jvj. The law of refraction is also known as snell s law, named for willobrord snell, who discovered the law in 1621. It is further shown that the propagation of a ray can be modeled in terms of newtonianlike equations of motion and that. Describe how reflection and refraction are accounted for by the wave theory. Another way to derive snells law involves an application of the general boundary conditions of maxwell equations for electromagnetic radiation.
Here, we derive the propagation angles of reflected and refracted waves at a. In the diagram shown above, two mediums are juxtapositioned one below the other. Equations of motion for rays in a snells law medium. Snells law is used to determine the direction of light rays through refractive. B h d e substituting the equation for b into the second maxwells equation and. But yes, there must be some trade off that allows you to get directions using fermats principle, but not amplitudes.
Derivation of the eikonal equation neglecting derivative of. From maxwell s equation through the wave equation to the eikonal equation up to the derivation of snell s law. This starts from observations due to faraday and ampere and a suppostion of maxwell. Derivation of the laws of reflection and refraction.
Questions you should be able to answer by the end of todays lecture. I love that derivation, because we show that the boundary conditions that maxwells equations enforce lead directly to three things. To minimize the time we set the derivative of the time with respect to x equal to zero. The question is then whether or not such a description in terms of curls and divergences is sufficient and.
Notice that snell s law shows that the index of refraction and the sine of the angle of refraction are inversely proportional that is, as the refractive index gets larger n 2 n 1 the sine of the refracted angle gets smaller sin. Pdf the equations of motion for a ray in a snells law medium with a varying index of refraction are derived. Fresnel equations consider reflection and transmission of light at dielectricdielectric boundary. Let us now discuss the amplitudes of the of the electric and magnetic. First, it says that any function of the form f zct satisfies the wave equation. Highorder accurate upwind approximations for the wave equation in secondorder form on overlapping grids are developed. Derivation from conservation of energy and momentum. This equation is mathematically the same as the harmonic. I want to know why is that students are not taughtor encouraged to go through this classic,why is that maxwell equations are just taught to be emperical when a lot of underlying theoretical work was done by maxwell,to arrive at the eqns. Another way to derive snell s law involves an application of the general boundary conditions of maxwell equations for electromagnetic radiation. Proof of snells law using maxwells equations physics.
How do we derive snells law using maxwells equations a solve. Deriving the fresnel equations 1 introduction the intensity of light re. One major contribution of this paper is that after some effort we find that the metamaterial maxwells equations can be beautifully reduced to a vector wave integrodifferential equation. Snells law lets use the following form of snells law. Snells law same independent of the polarization critical angle same reflection below critical angle different for the tmpolarization, there is the brewster angle at which r0. Pdf equations of motion for rays in a snells law medium. The forth law states that there are no magnetic currents.
Chapter maxwells equations and electromagnetic waves. Lect 16 electromagnetic induction, faradays law, lenz law, super demo. Light waves at the boundary of nonlinear media pdf. Snells law that relates the incident and refracted wave 3. Maxwells equations, electromagnetic waves, displacement. Proof of snells law using maxwells equations physics forums. Electromagnetic waves maxwell, a young admirer of faraday, believed that the closeness of these two numbers, speed of light and the inverse square root of. Geometrical optics textbook weizmann institute of science.
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