Wolski-1 MAXWELLS EQUATION.pdf - THE CERN ACCELERATOR SCHOOL Theory of Electromagnetic Fields Part I Maxwell’s Equations Andy Wolski The Cockcroft

Wolski-1 MAXWELLS EQUATION.pdf - THE CERN ACCELERATOR...

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THE CERN ACCELERATOR SCHOOL Theory of Electromagnetic Fields Part I: Maxwell’s Equations Andy Wolski The Cockcroft Institute, and the University of Liverpool, UK CAS Specialised Course on RF for Accelerators Ebeltoft, Denmark, June 2010
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Theory of Electromagnetic Fields In these lectures, we shall discuss the theory of electromagnetic fields, with an emphasis on aspects relevant to RF systems in accelerators: 1. Maxwell’s equations Maxwell’s equations and their physical significance Electromagnetic potentials Electromagnetic waves and their generation Electromagnetic energy 2. Standing Waves Boundary conditions on electromagnetic fields Modes in rectangular and cylindrical cavities Energy stored in a cavity 3. Travelling Waves Rectangular waveguides Transmission lines Theory of EM Fields 1 Part I: Maxwell’s Equations
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Theory of Electromagnetic Fields I shall assume some familiarity with the following topics: vector calculus in Cartesian and polar coordinate systems; Stokes’ and Gauss’ theorems; Maxwell’s equations and their physical significance; types of cavities and waveguides commonly used in accelerators. The fundamental physics and mathematics is presented in many textbooks; for example: I.S. Grant and W.R. Phillips, “Electromagnetism,” 2nd Edition (1990), Wiley. Theory of EM Fields 2 Part I: Maxwell’s Equations
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Summary of relations in vector calculus In cartesian coordinates: grad f ≡ ∇ f ∂f ∂x , ∂f ∂y , ∂f ∂z ! (1) div ~ A ≡ ∇ · ~ A ∂A x ∂x + ∂A y ∂y + ∂A z ∂z (2) curl ~ A ≡ ∇ × ~ A ˆ x ˆ y ˆ z ∂x ∂y ∂z A x A y A z (3) 2 f 2 f ∂x 2 + 2 f ∂y 2 + 2 f ∂z 2 (4) Note that ˆ x , ˆ y and ˆ z are unit vectors parallel to the x , y and z axes, respectively. Theory of EM Fields 3 Part I: Maxwell’s Equations
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Summary of relations in vector calculus Gauss’ theorem: Z V ∇ · ~ A dV = I S ~ A · d ~ S, (5) for any smooth vector field ~ A , where the closed surface S bounds the volume V . Stokes’ theorem: Z S ∇ × ~ A · d ~ S = I C ~ A · d ~ ‘, (6) for any smooth vector field ~ A , where the closed loop C bounds the surface S . A useful identity: ∇ × ∇ × ~ A ≡ ∇ ( ∇ · ~ A ) - ∇ 2 ~ A. (7) Theory of EM Fields 4 Part I: Maxwell’s Equations
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Maxwell’s equations ∇ · ~ D = ρ ∇ · ~ B = 0 ∇ × ~ E = - ~ B ∂t ∇ × ~ H = ~ J + ~ D ∂t James Clerk Maxwell 1831 – 1879 Note that ρ is the electric charge density; and ~ J is the current density. The constitutive relations are: ~ D = ε ~ E, ~ B = μ ~ H, (8) where ε is the permittivity, and μ is the permeability of the material in which the fields exist. Theory of EM Fields 5 Part I: Maxwell’s Equations
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Physical interpretation of ∇ · ~ B = 0 Gauss’ theorem tells us that for any smooth vector field ~ B : Z V ∇ · ~ B dV = I S ~ B · d ~ S, (9) where the closed surface S bounds the region V .
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