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Chapter 3 Electromagnetic Waves & Maxwell's Equations

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Chapter 3 Electromagnetic Waves & Maxwell's Equations Chapter 3 Electromagnetic Waves & Maxwell’s Equations Part I Maxwell’s Equations Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist. Famous equations published in 1861 Maxwell’s Equations: Integral Form Gauss's law Gauss's law for magnetism Faraday's law of induction (Maxwell–Faraday equation) Ampère's law (with Maxwell's addition) Maxwell’s Equations Relation of the speed of light and electric and magnetic vacuum constants 1 c = 2.99792458 108 [m/s] 0 0 permittivity of free space, also called the electric constant permeability of free space, also called the magnetic constant Gauss’s Law For any closed surface enclosing total charge Qin, the net electric flux through the surface is This result for the electric flux is known as Gauss’s Law. Magnetic Gauss’s Law The net magnetic flux through any closed surface is equal to zero: As of today there is no evidence of magnetic monopoles See: Phys.Rev.Lett.85:5292,2000 Ampère's Law The magnetic field in space around an electric current is proportional to the electric current which serves as its source: B ds 0I I is the total current inside the loop. ds B i1 Direction of integration i 3 i2 Faraday’s Law The change of magnetic flux in a loop will induce emf, i.e., electric field B E ds dA A t Lenz's Law Claim: Direction of induced current must be so as to oppose the change; otherwise conservation of energy would be violated. Problem with Ampère's Law Maxwell realized that Ampere’s law is not valid when the current is discontinuous. When charge a parallel plate capacitor: For path L: B ds 0I For path R: B ds 0 Displacement Current He concluded that when the charge within an enclosed surface is changing, it is necessary to add to Ampere’s law another current called the Displacement current: ID Then the Ampere’s law is changed to: B ds ( I I ) 0 D E wikimedia.org 0I 0 0 dA A t 11 Maxwell’s Equations: Integral Form Q E dA Gauss's law 0 B dA 0 Gauss's law for magnetism B Faraday's law of induction E ds A dA (Maxwell–Faraday equation) t E Ampère's law (with B ds 0I 00 dA A t Maxwell's addition) Differential Operators A Ay A the divergence operator A x z Div (“del”) x y z xˆ y ˆ zˆ the curl operator A curl, rot x y z Ax A y A z the partial derivative with respect to time t Other notation used x x Transition from Integral to Differential Form Gauss’ theorem for a vector field F dA V FdV Volume V, surrounded by surface A Stokes' theorem for a vector field A F ds A F dA L Surface A, surrounded by contour L Maxwell’s Equations: Differential Form Divergence of electric field is a function E of charge density (Gauss Law) 0 B A closed loop of E field lines will exist when E the magnetic field varies with time t (Faraday’s Law) B 0 Divergence of magnetic field =0 (closed loops) (Gauss Law) E A closed loop of B field lines will exist in B 0 j 00 The presence of a current and/or t time varying electric field (Ampere’s Law) where is the charge density, and s current density Part II Electromagnetic Wave Maxwell’s Equation in Free Space Since a changing electric field produces a magnetic field, and a changing magnetic field produces an electric field, once sinusoidal fields are created they can propagate on their own. These propagating fields are called electromagnetic waves. Oscillating Electric Dipole Consider static electric field produced by an electric dipole (a) Positive (negative) charge at the top (bottom) (b) Negative (positive) charge at the top (bottom) Now imagine these two charge are moving up and down and exchange their position at every half-period. Then between the two cases there is a situation like as shown in figure below: What is the electric field in the blank area? Static electric fields Oscillating Electric Dipole Since we don’t assume that change propagate instantly once new position is reached the blank represents what has to happen to the fields in meantime. We learned that E field lines can’t cross and they need to be continuous except at charges. Therefore a plausible guess is as shown in the right figure. Oscillating Electric Dipole What actually happens to the fields based on a precise calculate is shown in Fig. Magnetic fields are also formed. When there is electric current, magnetic field is produced. If the current is in a straight wire circular magnetic field is generated. Its magnitude is inversely proportional to the distance from the current. Maxwell’s Equation in Free Space Since a changing electric field produces a magnetic field, and a changing magnetic field produces an electric field, once sinusoidal fields are created they can propagate on their own. These propagating fields are called electromagnetic waves. Maxwell’s Equation in Free Space In free space = 0 and = 0, thus B E 0 E t E B 0 B 00 t In Cartesian coordiantes, E Ey E x z 0 E 0 x y z B By B B 0 x z 0 x y z Maxwell’s Equation in Free Space E Ey B z x y z t B E E By E x z t z x t Ey E B x z x y t Bz By Ex 00 y z t E Bx Bz Ey B 00 00 t z x t By Bx Ez 00 x y t Maxwell’s Equation in Free Space Apply the curl identity ( ) ( ) 2 to Ampere’s law B B E ( E ) t t ( E ) 2E ( B ) t E Since E 0 and B 0 0 , one has t 2 2 2 2 2 E 2 E 00 t 2 x2 y2 z2 Maxwell’s Equation in Free Space Similarly, apply the curl identity ( ) ( ) 2 to Faraday’s law, one has 2 2 B B 00 t 2 These two equations are plane wave equations 2 2 2 2 Ex Ex Ex Ex 00 x2 y2 z2 t 2 2 2 2 2 2 2 E Ey Ey Ey Ey E 00 00 t 2 x2 y2 z2 t 2 2 2 2 2 Ez Ez Ez Ez 00 x2 y2 z2 t 2 Maxwell’s Equation in Free Space The general equation is a wave equation 2 2 2 1 2 x2 y2 z2 v2 t 2 And v is the speed of the wave. Thus, for the EM wave obtained from Maxwell’s equations, one has 1 v 00 Speed of light! Part III Plane Wave Plane Transverse Wave If there is only one component of E-field: One component has: Ey B z x t E The induced E-field and B- field perpendicular to each other. The wave equation B becomes 2 2 Ey Ey 00 x2 t 2 Plane Transverse Wave The solution for the wave equation is a plane wave 2 2 Ey Ey 00 x2 t 2 and c where 2f Angular Frequency Wavelength f 2 T 1/ f 2 / Period k Wavenumber c Speed k Energy Density Carried by the Plane Wave Electric energy density Magnetic energy density 2 B 2 0Ey u z uE B 2 20 2 Ey c 00 2 1 2 uB Ey 0Ey 20 20 2 For an EM wave, the instantaneous electric and magnetic energies are equal. 2 1 2 Bz uE uB 0Ey 2 20 Total energy density of an EM Wave 2 2 Bz u uE uB 0 Ey 0 2 2 0 E0 cos (t kx ) Energy Density Carried by the Plane Wave Since the light frequency is high (~ 1014 Hz), when measure the energy density, the detector usually obtain its time average: 2 u 0 Ey T T For a harmonic function f(t), / / T is the period of the harmonic function t / 2 2 u t / 0E0 cos (t kx )dt T 2 2 1 2 B0 0E0 2 20 Energy Carried by the Plane Wave Poynting vector : the flow of electromagnetic energy along the wave propagation direction, i.e., the transport of energy per unit time across a unit area EM wave speed c, in t time across area A, l ct uctA S uc tA A thus 1 S Ey Bz 0 Considering the energy flows in x-direction, S should be a vector 1 S E B 0 Energy Carried by the Plane Wave The intensity of the EM wave, or the light is the time average of the magnitude of the Poynting vector c 0 2 I S E0 T 2 or I S T cuav The intensity is c times the total average energy density Spherical Wave Another type of transversal wave is called spherical wave A spherical wave is a constant- frequency wave whose wavefronts (surfaces of constant phase) are parallel concentric spheres of constant amplitude normal to the phase velocity vector. When the distance from the source is very large, a spherical wave can be locally approximated as a plane wave. Spherical Wave The wave amplitude is , as it propagates further from the source, the amplitude decreases inversely with respect to r. The irradiance (W/m2) (intensity) of the wave is proportional to , That is a familiar inverse square law of propagation for spherical wave disturbance. At r = 0, wave is not valid but rather describes a point source.
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