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Electromagnetism Electromagnetism Christopher R Prior Fellow and Tutor in Mathematics ASTeC Intense Beams Group Trinity College, Oxford Rutherford Appleton Laboratory Contents • Review of Maxwell’s equations and Lorentz Force Law • Motion of a charged particle under constant Electromagnetic fields • Relativistic transformations of fields • Electromagnetic waves – Waves in vacuo – Waves in conducting medium • Waves in a uniform conducting guide – Simple example TE01 mode – Propagation constant, cut-off frequency – Group velocity, phase velocity – Illustrations 2 Reading • J.D. Jackson: Classical Electrodynamics • H.D. Young and R.A. Freedman: University Physics (with Modern Physics) • P.C. Clemmow: Electromagnetic Theory • Feynmann Lectures on Physics • W.K.H. Panofsky and M.N. Phillips: Classical Electricity and Magnetism • G.L. Pollack and D.R. Stump: Electromagnetism 3 Basic Equations from Vector Calculus For a scalar function ϕ(x,y,z,t), ⎛ ∂ϕ ∂ϕ ∂ϕ ⎞ Gradient is normal to surfaces gradient : ∇ϕ = ⎜ , , ⎟ φ=constant ⎝ ∂x ∂y ∂z ⎠ For a vector F = (F1,F2,F3) ∂F ∂F ∂F divergence : ∇⋅ F = 1 + 2 + 3 ∂x ∂y ∂z ⎛ ∂F3 ∂F2 ∂F1 ∂F3 ∂F2 ∂F1⎞ curl : ∇ ∧ F = ⎜ − , − , − ⎟ ⎝ ∂y ∂z ∂z ∂x ∂x ∂y ⎠ 4 Basic Vector Calculus ∇ ⋅(F ∧ G) = G ⋅∇ ∧ F − F ⋅∇ ∧ G ∇ ∧ ∇ϕ = 0, ∇ ⋅∇ ∧ F = 0 ∇ ∧ (∇ ∧ F) = ∇(∇ ⋅ F) − ∇2F Stokes’ Theorem Divergence or Gauss’ Theorem ∫∫ ∇ ∧ F ⋅ dS = ∫ F ⋅ dr S C ∫∫∫ ∇⋅ F dV = ∫∫ F ⋅ dS n V S dS = n dS Closed surface S, volume V, Oriented outward pointing normal boundary C 5 What is Electromagnetism? • The study of Maxwell’s equations, devised in 1863 to represent the relationships between electric and magnetic fields in the presence of electric charges and currents, whether steady or rapidly fluctuating, in a vacuum or in matter. • The equations represent one of the most elegant and concise way to describe the fundamentals of electricity and magnetism. They pull together in a consistent way earlier results known from the work of Gauss, Faraday, Ampère, Biot, Savart and others. • Remarkably, Maxwell’s equations are perfectly consistent with the transformations of special relativity. Maxwell’s Equations Relate Electric and Magnetic fields generated by charge and current distributions. E = electric field D = electric displacement ∇ ⋅ D = ρ H = magnetic field B = magnetic flux density ∇ ⋅ B = 0 ρ= charge density ∂B j = current density ∇ ∧ E = − µ (permeability of free space) = 4π 10-7 ∂t 0 -12 ε0 (permittivity of free space) = 8.854 10 ∂D ∇ ∧ H = j + c (speed of light) = 2.99792458 108 m/s ∂t 2 In vacuum D = ε 0 E, B = µ0 H, ε 0µ0c =1 ρ ∇ ⋅ E = st Maxwell’s 1 Equation ε0 Equivalent to Gauss’ Flux Theorem: ρ 1 Q ∇ ⋅ E = ⇔ ∫∫∫∇ ⋅ E dV =∫∫ E ⋅ dS = ∫∫∫ ρ dV = ε0 V S ε0 V ε0 The flux of electric field out of a closed region is proportional to the total electric charge Q enclosed within the surface. A point charge q generates an electric field q E = 3 r 4πε0r q dS q E ⋅dS = = ∫∫ ∫∫ 2 sphere 4πε0 sphere r ε 0 Area integral gives a measure of the net charge enclosed; divergence of the electric field gives the density of the sources. 8 nd Maxwell’s 2 Equation ∇ ⋅ B = 0 Gauss’ law for magnetism: B! =0 B! dS! ∇ · ⇐⇒ · The net magnetic flux out!! of any closed surface is zero. Surround a magnetic dipole with a closed surface. The magnetic flux directed inward towards the south pole will equal the flux outward from the north pole. If there were a magnetic monopole source, this would give a non-zero integral. Gauss’ law for magnetism is then a statement that There are no magnetic monopoles ∂B rd ∇ ∧ E = − Maxwell’s 3 Equation ∂t Equivalent to Faraday’s Law of Induction: ∂B ∫∫ ∇ ∧ E ⋅ dS = −∫∫ ⋅ dS S S ∂t d dΦ ⇔ E ⋅ dl = − B ⋅ dS = − ∫ dt ∫∫ dt C S (for a fixed circuit C) N S The electromotive force round a circuit ε = ∫ E ⋅ d l is proportional to the rate of change of flux of magnetic field, Φ = ∫∫ B ⋅ d S through the circuit. Faraday’s Law is the basis for electric generators. It also forms the basis for inductors and transformers. 1 ∂E th ∇ ∧ B = µ0 j + 2 Maxwell’s 4 Equation c ∂t Originates from Ampère’s (Circuital) Law : ∇ ∧ B = µ0 j B ⋅ dl = ∇ ∧ B ⋅ dS = µ j ⋅ dS = µ I ∫ ∫∫ 0 ∫∫ 0 C S S Satisfied by the field for a steady line current (Biot-Savart Law, Ampère 1820): µ I dl ∧ r B = 0 4π ∫ r3 µ I For a straight line current B = 0 θ 2π r Biot Need for Displacement Current • Faraday: vary B-field, generate E-field • Maxwell: varying E-field should then produce a B-field, but not covered by Ampère’s Law. Apply Ampère to surface 1 (flat disk): line integral of B =μ0I Surface 1 Surface 2 Applied to surface 2, line integral is zero since no current penetrates the deformed Current I surface. Q dQ dE In capacitor, E = , so I = = ε A A dt 0 dt ε0 Closed loop ∂E Displacement current density is j = ε d 0 ∂t ∂E ∇ ∧ B = µ0 (j + jd )= µ0 j + µ0ε0 ∂t 12 Consistency with Charge Conservation Charge conservation: From Maxwell’s equations: Total current flowing out of a Take divergence of (modified) region equals the rate of decrease Ampère’s equation of charge within the volume. 1 ∂E ∇ ∧ B = µ j + 0 c 2 t d ∂ j ⋅ dS = − ρdV 1 ∂ ∫∫ ∫∫∫ ⇒ ∇⋅∇ ∧ B = µ ∇⋅ j + ∇⋅ E dt 0 c 2 ∂t ( ) ∂ρ ⇔ ∇⋅ j dV = − dV ∂ ⎛ ρ ⎞ ∫∫∫ ∫∫∫ ∂t ⇒ 0 = µ0∇⋅ j + ε0µ0 ⎜ ⎟ ∂t ⎝ ε0 ⎠ ∂ρ ⇔ ∇⋅ j + = 0 ∂ρ ∂t ⇒ 0 = ∇⋅ j + ∂t Charge conservation is implicit in Maxwell’s Equations 13 Maxwell’s Equations in Vacuum In vacuum Equivalent integral forms 1 (useful for simple geometries) D = ε0 E , B = µ0H , ε0µ0 = 2 c Source-free equations: 1 ∫∫ E ⋅ dS = ∫∫∫ ρdV ∇⋅ B = 0 ε0 ∂B B dS 0 ∇ ∧ E + = 0 ∫∫ ⋅ = ∂t d dΦ Source equations: ∫ E ⋅ dl = − ∫∫ B ⋅ dS = − ρ dt dt ∇⋅ E = 1 d ε0 B ⋅ dl = µ j ⋅dS + E ⋅ dS ∫ 0 ∫∫ 2 ∫∫ 1 ∂E c dt ∇ ∧ B − 2 = µ0 j c ∂t 14 Example: Calculate E from B d E ⋅dl = − B ⋅dS z ∫ dt ∫∫ d r < r 2π rE = − π r2B sinωt = −π r2B ω cosωt 0 θ dt 0 0 r 1 ⇒ Eθ = − B0ω r cos ωt ⎧B0 sinωt r < r0 2 Bz = ⎨ ⎩ 0 r > r0 d r > r 2π rE = − π r2B sinωt = −π r2B ω cosωt 0 θ dt 0 0 0 0 B 2 Also from ∂ ω r B ∇ ∧ E = − ⇒ E = − 0 0 cosωt ∂t θ 2r 1 ∂E ∇ ∧ B = µ j + then gives current density necessary 0 c2 dt to sustain the fields Lorentz Force Law • Thought of as a supplement to Maxwell’s equations but actually implicit in relativistic formulation, gives force on a charged particle moving in an electromagnetic field: f = q(E + v ∧ B) • For continuous distributions, use force density f d = ρE + j ∧ B • Relativistic equation of motion dP #v f# 1 dE dp# – 4-vector form: F = = γ · , f# = γ , dτ ⇒ c c dt dt ! " # $ – 3-vector component: Energy component: d dE 2 dγ (m0γ v ) = f = q(E + v ∧ B ) v ⋅ f = = m0c dt dt dt 16 Motion of charged particles in constant magnetic fields d m γ v = f = q E + v ∧ B = qv ∧ B dt ( 0 ) ( ) d 2 (m0γ c ) = v ⋅ f = qv ⋅v ∧ B = 0 dt 1. From energy equation, γ is constant No acceleration with a magnetic field 2. From momentum equation, d d B ⋅ (γ v ) = 0 = γ (B ⋅v ) ⇒ v // is constant dt dt !v constant and !v constant |=| !v also constant| !| ⇒ | ⊥| 17 Motion in Constant magnetic field d dv q (m0γ v ) = qv ∧ B ⇒ = v ∧ B dt dt m0γ Constant magnetic field 2 gives uniform spiral about B v⊥ q ⇒ = v⊥B with constant energy. ρ m0γ m γ v ⇒ circular motion with radius ρ = 0 ⊥ qB v⊥ qB at angular frequency ω = = (m = m0γ ) ρ m m γ v p Bρ = 0 = q q Magnetic rigidity Motion in Constant Electric Field d d (m0γ v ) = f = q(E + v ∧ B ) → (m0γ v ) = q E dt dt d q Solution of (γ v)= E dt m0 2 2 qE 2 ⎛ γ v⎞ ⎛ qE ⎞ is γ v = t ⇒ γ = 1+ ⇒ γ = 1+ ⎜ t⎟ m0 ⎝ c ⎠ ⎝ m0c ⎠ 2 dx v m c 2 ⎡ ⎛ qEt ⎞ ⎤ γ 0 ⎢ ⎥ = ⇒ x = x0 + 1+ ⎜ ⎟ −1 dt γ qE ⎢ ⎝ m c⎠ ⎥ ⎣ 0 ⎦ 1 qE 2 ≈ x0 + t for qE << m0c 2 m0 Energy gain is qEx Constant E-field gives uniform acceleration in straight line 19 Relativistic Transformations of E and B • According to observer O in frame F, particle has velocity v, fields are E and B and Lorentz force is f! = q E! + !v B! × ! " • In Frame F’, particle is at rest and force is f!! = q!E! ! • Assume measurements give same charge and force, so q! = q and E! ! = E! + !v B! × q !v !r • Point charge at rest in F: ! ! q E = ×3 , B =0 4π#0 r µ0q !v !r 1 • See a current in F’, giving a field B! ! = × = !v E! Rough idea− 4π r3 −c2 × 1 • Suggests B! ! = B! !v E! − c2 × Review of Waves ∂2u 1 ∂2u • 1D wave equation is 2 = 2 2 with general solution ∂x v ∂t u(x, t)=f(vt x)+g(vt + x) − • Simple plane wave: 1D : sin(ωt − k x) 3D : sin ωt − k ⋅ x ( ) 2π Wavelength is λ = #k | | ω Frequency is ν = 2π Phase and group velocities ∞ ∫ A(k)ei[ω(k )t−kx]dk −∞ Superposition of plane waves. While Plane wave sin( ω t kx ) has constant shape is relatively undistorted, pulse phase ω t kx = −1 π at peaks − 2 travels with the Group Velocity ω k∆x =0 dω − vg = dk ∆x ω v = = ⇐⇒ p ∆t k Wave packet structure • Phase velocities of individual plane waves making up the wave packet are different, • The wave packet will then disperse with time 23 Electromagnetic waves • Maxwell’s equations predict the existence of electromagnetic waves, later discovered by Hertz.
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