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Lecture 2 Maxwell's Equations in Free Space Co-Ordinate Systems And Lecture 2 Maxwell’s Equations in Free Space In this lecture you will learn: • Co-ordinate Systems and Course Notations • Maxwell’s Equations in Differential and Integral Forms • Electrostatics and Magnetostatics • Electroquasistatics and Magnetoquasistatics ECE 303 – Fall 2007 – Farhan Rana – Cornell University Co-ordinate Systems and Vectors Cartesian Coordinate System z (xo, yo ,zo ) zo y xo yo x Vectors in Cartesian Coordinate System r A = Ax xˆ + Ay yˆ + Az zˆ All mean exactly the same thing … r ˆ ˆ ˆ just a different notation for the unit A = Ax i + Ay j + Az k vectors r ˆ ˆ ˆ A = Ax ix + Ay iy + Az iz The first one will be used in this course ECE 303 – Fall 2007 – Farhan Rana – Cornell University 1 Co-ordinate Systems and Vectors Cylindrical Coordinate System z (ro,φo,zo ) y φo zo ro x Vectors in Cylindrical Coordinate System r A = A rˆ + A φˆ + A zˆ Both mean exactly the same thing … r φ z just a different notation for the unit vectors r ˆ ˆ ˆ A = Ar ir + Aφ iφ + Az iz The first one will be used in this course ECE 303 – Fall 2007 – Farhan Rana – Cornell University Co-ordinate Systems and Vectors Spherical Coordinate System z (r ,φ ,θ ) r o o o θo o y φo x Vectors in Spherical Coordinate System r ˆ ˆ Both mean exactly the same thing … A = Ar rˆ + Aφ φ + Aθ θ just a different notation for the unit vectors r ˆ ˆ ˆ A = Ar ir + Aφ iφ + Aθ iθ The first one will be used in this course ECE 303 – Fall 2007 – Farhan Rana – Cornell University 2 Vector Fields In layman terms, a vector field implies a vector associated with every point is space: Examples: r r Electric Field: E()x, y,z,t or E(rr,t ) r r Magnetic Field: H()x,y,z,t or H(rr,t ) ECE 303 – Fall 2007 – Farhan Rana – Cornell University Maxwell’s Equations in Free Space Integral Form Differential Form r r r (1) ∫∫ εo E . da = ∫∫∫ ρ dV Gauss’ Law ∇. εo E = ρ r r r (2) ∫∫ µo H . da = 0 Gauss’ Law ∇. µo H = 0 ∂ r r r r r r ∂ µoH (3) ∫ E . ds = − ∫∫ µoH. da Faraday’s Law ∇ × E = − ∂t ∂t r r r r r ∂ r r r r ∂ ε E (4) ∫ H . ds = ∫∫ J .da + ∫∫ εoE . da Ampere’s Law ∇ × H = J + o ∂t ∂t Lorentz Force Law r r r r Lorentz Law describes the effect of F = q ( E + v × µoH ) electromagnetic fields upon charges ECE 303 – Fall 2007 – Farhan Rana – Cornell University 3 Physical Quantities, Values, and SI Units Quantity Value/Units r E Electric Field Volts/m r H Magnetic Field Amps/m εo Permittivity of Free Space 8.85x10-12 Farads/m µo Permeability of Free Space 4πx10-7 Henry/m q Electronic Unit of Charge 1.6x10-19 Coulombs ρ Volume Charge Density Coulombs/m3 r J Current Density Amps/m2 r r 2 D = εoE Electric Flux Density Coulombs/m r r B = µoH Magnetic Flux Density Tesla ECE 303 – Fall 2007 – Farhan Rana – Cornell University Gauss’ Law – Integral Form r r r r ∫∫ εo E . da = ∫∫∫ ρ dV or ∫∫ D . da = ∫∫∫ ρ dV What is this law saying ……?? A closed surface of arbitrary shape surrounding a ρ(x,y,z) charge distribution Carl F. Gauss D (1777-1855) Gauss’ Law: The total electric flux coming out of a closed surface is equal to the total charge enclosed by that closed surface (irrespective of the shape of the closed surface) Points to Note: This law establishes charges as the “sources” or “sinks” of the electric field (i.e. charges produce or terminate electric field lines). If the total flux through a closed surface is positive, then the total charge enclosed is positive. If the total flux is negative, then the total charge enclosed is negative ECE 303 – Fall 2007 – Farhan Rana – Cornell University 4 Gauss’ Law – Differential Form Divergence Theorem: r r r For any vector field: ∫∫ A . da = ∫∫∫ ∇ . A dV The flux of a vector through a closed surface is equal to the integral of the divergence of the vector taken over the volume enclosed by that closed surface Using the Divergence Theorem with Gauss’ Law in Integral Form: r r ∫∫ D . da = ∫∫∫ ρ dV r D ⇒ ∫∫∫ ∇ .D dV = ∫∫∫ ρ dV r r ρ(x,y,z) ⇒ ∇ .D = ρ or ∇ .εo E = ρ ECE 303 – Fall 2007 – Farhan Rana – Cornell University Gauss’ Law for the Magnetic Field – Integral Form r r r r ∫∫ µo H . da = 0 or ∫∫ B . da = 0 What is this law saying ……?? A closed surface of arbitrary shape B Gauss’ Law for the Magnetic Fields: The total magnetic flux coming out of a closed surface is always zero. Points to Note: This law implies that there are no such things as “magnetic charges” that can emanate or terminate magnetic field lines. If magnetic field is non-zero, then the flux into any closed surface must equal the flux out of it - so that the net flux coming out is zero. ECE 303 – Fall 2007 – Farhan Rana – Cornell University 5 Gauss’ Law – Differential Form Divergence Theorem: r r r For any vector field: ∫∫ A . da = ∫∫∫ ∇ . A dV The flux of a vector through a closed surface is equal to the integral of the divergence of the vector taken over the volume enclosed by that closed surface Using the Divergence Theorem with Gauss’ Law for the Magnetic Field in Integral form: r r r r ∫∫ B . da = 0 (Remember B = µo H ) B r ⇒ ∫∫∫ ∇ .B dV = 0 r r ⇒ ∇ .B = 0 or ∇ .µo H = 0 ECE 303 – Fall 2007 – Farhan Rana – Cornell University Faraday’s Law – Integral Form r r ∂ r r r r ∂ r r ∫ E . ds = − ∫∫ µ H . da or ∫ E . ds = − ∫∫ B . da ∂t o ∂t What is this law saying ……?? B Michael Faraday A closed contour (1791-1867) Faraday’s Law: The line integral of electric field over a closed contour is equal to –ve of the time rate of change of the total magnetic flux that goes through any arbitrary surface that is bounded by the closed contour Points to Note: This law says that time changing magnetic fields can also generate electric fields The positive directions for the surface normal vector and of the contour are related by the right hand rule ECE 303 – Fall 2007 – Farhan Rana – Cornell University 6 Faraday’s Law – Differential Form Stokes Theorem: r r r For any vector field: ∫ A.ds = ∫∫ (∇ × A). da The line integral of a vector over a closed contour is equal to the surface integral of the curl of that vector over any arbitrary surface that is bounded by the closed contour Using the Stokes Theorem with Farady’s Law in Integral Form: r r ∂ r r ∫ E . ds = − ∫∫ B . da ∂t r r ∂ r r ⇒ ∫∫ ()∇ × E .da = − ∫∫ B . da ∂t r r r ∂ B r ∂ µ H ⇒ ∇ × E = − or ∇ × E = − o ∂t ∂t ECE 303 – Fall 2007 – Farhan Rana – Cornell University Ampere’s Law – Integral Form r r r r ∂ r r r r r r ∂ r r ∫ H . ds = ∫∫ J .da + ∫∫ ε E . da or ∫ H . ds = ∫∫ J .da + ∫∫ D.da ∂t o ∂t What is this law saying ……?? electric flux density J D electric current density A. M. Ampere (1775-1836) Ampere’s Law: The line integral of magnetic field over a closed contour is equal to the total current plus the time rate of change of the total electric flux that goes through any arbitrary surface that is bounded by the closed contour Points to Note: This law says that electrical currents and time changing electric fields can generate magnetic fields. Since there are no magnetic charges, this is the only known way to generate magnetic fields The positive directions for the surface normal vector and of the contour are related by the right hand rule ECE 303 – Fall 2007 – Farhan Rana – Cornell University 7 Ampere’s Law – Differential Form Stokes Theorem: r r r For any vector field: ∫ A.ds = ∫∫ (∇ × A). da The line integral of a vector over a closed contour is equal to the surface integral of the curl of that vector over any arbitrary surface that is bounded by the closed contour Using the Stokes Theorem with Ampere’s Law in Integral Form: r r r r ∂ r r ∫ H . ds = ∫∫ J .da + ∫∫ D.da ∂t r r r r ∂ r r ⇒ ∫∫ ()∇ × H .da = ∫∫ J .da + ∫∫ D.da ∂t r r r r ∂ D r r ∂ ε E ⇒ ∇ × H = J + or ∇ × H = J + o ∂t ∂t ECE 303 – Fall 2007 – Farhan Rana – Cornell University Maxwell’s Equations and Light – Coupling of E and H Fields r r ∂ µ H r ∇ × E = − o ∇. εo E = ρ ∂t r r ∇. µo H = 0 r r ∂ ε E ∇ × H = J + o ∂t Time varying electric and magnetic fields are coupled - this coupling is responsible for the propagation of electromagnetic waves Electromagnetic Wave Equation in Free Space: r Assume: ρ = J = 0 and take the curl of the Faraday’s Law on both sides: r r r r ⎛ ∂ µ H ⎞ ∂ µ ∇ × H ∂2 E ⎜ o ⎟ o ∇ × ()∇ × E = −∇ × ⎜ ⎟ = − = −µo εo 2 ⎝ ∂t ⎠ ∂t ∂t 2 r 1 r ∂ E c = ≈ 3 ×108 m/s ⇒ ∇ × ()∇ × E = −µo εo 2 ∂t εo µo 2 r r 1 ∂ E Equation for a wave traveling ⇒ ∇ × ()∇ × E = − c2 ∂t 2 at the speed c ECE 303 – Fall 2007 – Farhan Rana – Cornell University 8 Maxwell’s Equations and Light r r 1 ∂2 E Equation for a wave traveling ∇ × ()∇ × E = − 2 2 c ∂t at the speed c: 1 c = ≈ 3 ×108 m/s εo µo In 1865 Maxwell wrote: “This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws” ECE 303 – Fall 2007 – Farhan Rana – Cornell University Electrostatics and Magnetostatics Suppose we restrict ourselves to time-independent situations (i.e.
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