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JS 3010 Electromagnetism I

JS 3010 Electromagnetism I. Dr. Charles Patterson 2.48 Lloyd Building. Course Outline. Course texts: Electromagnetism , 2nd Edn. Grant and Phillips (Wiley) Electromagnetic Fields and Waves , 2nd Edn. Lorrain and Corson (Freeman)

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JS 3010 Electromagnetism I

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  1. JS 3010 Electromagnetism I Dr. Charles Patterson 2.48 Lloyd Building

  2. Course Outline Course texts: • Electromagnetism, 2nd Edn. Grant and Phillips (Wiley) • Electromagnetic Fields and Waves, 2nd Edn. Lorrain and Corson (Freeman) Online at: www.tcd.ie/Physics/People/Charles.Patterson/Teaching/JS/JS3010/ Topics: • 0). Overview • 1). Vector Operators and Vector Analysis • 2). Gauss’ Law and applications • 3). Electrostatic and Dielectric Phenomena • 4). Ampere’s Law and Applications • 5). Magnetostatic and Magnetic Phenomena • 6). Maxwell’s Equations and Electromagnetic Radiation

  3. k v B i j F i j k F E Force on charge due to electric and magnetic fields • Lorentz force on single charge q FB = q v x BB magnetic induction (Tesla, T) FE = q E E electric field strength (Volts/m) i x j = k Sense of F depends on sign of q

  4. r-r’ r r’ O Electric Fields • Electric field strength E(r,t) Volts m-1 or NC-1 Vector field of position and time • Field at field pointr due to single point charge at source pointr’ (electric monopole) Note r-r’ vector directed away from source point when q is positive. Electric field lines point away from (towards) a positive (negative) charge

  5. I dB(r) r-r’ r O r’ dl’ Magnetic Fields • Magnetic Induction (Magnetic flux density) B(r,t) Tesla (T) Vector field of position and time • Field at field pointr due to current element at source pointr’ is given by Biot-Savart Law • Note dB(r) is the contribution to the circulating magnetic field which surrounds this infinite wire from the current element dl’

  6. Maxwell’s Equations • Expressed in integral or differential forms • Simplest to derive integral form from physical principle • Equations easier to use in differential form • Forms related by vector field identities (Stokes’ Theorem, Gauss’ Divergence Theorem) • Time-independent problems electrostatics, magnetostatics • Time-dependent problems electromagnetic waves Vacuum Matter

  7. 1). Vector Operators and Analysis • Div, Grad, Curl (and all that) • Del or nabla operator • In Cartesian coordinates • Combining vectors in 3 ways • Scalar (inner) product a.b = c (scalar) • Cross (vector) product axb = c (vector) • Outer product (dyad) ab = c (tensor)

  8. Scalar Product - Divergence • r is a Cartesian position vector r=(x,y,z) • A is vector function of position r • Div A = • Scalar product of del with A • Scalar function of position

  9. k j i Cross Product - Curl • Curl A = • Cross product of del with A • Vector function of position

  10. Gradient • f(x,y,z)is a scalar function of position • Grad f = f = • Operation of del on scalar function • Vector function of position f =const. f

  11. Div Grad – the Laplacian • Inner product Del squared • Operates on a scalar function to produce a scalar function • Outer product

  12. y d area A contour C c x a b Green’s Theorem on plane • Leads to Divergence Theorem and Stokes’ Theorem • Fundamental theorem of calculus • Green’s Theorem P(x,y), Q(x,y) functions with continuous partial derivatives

  13. y d Area A contour C c x a b Green’s Theorem on plane • Integral of derivative over A • Integral around contour C

  14. = cancellation Green’s Theorem on plane • Similarly • Green’s Theorem relates an integral along a closed contour C to an area integral over the enclosed area A • QED for a rectangular area (previous slide) • Consider two rectangles and then arbitrary planar surface • Green’s Theorem applies to arbitrary, bounded surfaces A C Contributions from boundaries cancel No cancellation on boundary

  15. V = (Vx,Vy) dx dy dr nds dx dy j i Divergence Theorem • Tangent dr = i dx + j dy • Outward normal n ds = i dy – j dx • n unit vector along outward normal • ds = (dx2+dy2)1/2 • P(x,y) = -Vy Q(x,y) = Vx Cartesian components of the same vector field V • Pdx + Qdy = -Vydx + Vxdy • (i Vx + j Vy).(i dy – j dx) = -Vy dx + Vx dy = V.n ds

  16. .V dv V.n dS Divergence Theorem 2-D 3-D • Apply Green’s Theorem • In words - Integral of V.n ds over surface contour equals integral of div V over surface area • In 3-D • Integral of V.n dS over bounding surface S equals integral of div V dv within volume enclosed by surface S

  17. local value of  x V k V = (Vx,Vy) j dx i dy dr Curl and Stokes’ Theorem • For divergence theorem P(x,y) = -Vy Q(x,y) = Vx • Instead choose P(x,y) = Vx Q(x,y) = Vy • Pdx + Qdy = Vx dx + Vy dy • V = i Vx + j Vy + 0 k A C

  18. local value of  x V n outward normal ( x V) .n dS dS dr V.dr local value of V Stokes’ Theorem 3-D • In words - Integral of ( x V) .n dS over surface S equals integral of V.dr over bounding contour C • It doesn’t matter which surface (blue or hatched). Direction of dr determined by right hand rule. S C

  19. local value of  x V n outward normal .A dv ( x V) .n dS dS V.n dS dr V.dr local value of V Summary • Green’s Theorem • Divergence theorem • Stokes’ Theorem • Continuity equation surface S volume v S C

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