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JS 3010 Electromagnetism I Dr. Charles Patterson 2.48 Lloyd Building

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Course Outline Course texts: Electromagnetism, 2nd Edn. Grant and Phillips (Wiley) Electromagnetic Fields and Waves, 2nd Edn. Lorrain and Corson (Freeman) Online at: 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

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

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Electric field strength E(r,t) Volts m -1 or NC -1 Vector field of position and time Field at field point r due to single point charge at source point r’ (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 Electric Fields O r r’ r-r’

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Magnetic Fields Magnetic Induction (Magnetic flux density) B(r,t) Tesla (T) Vector field of position and time Field at field point r due to current element at source point r’ 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’ O r r’ r-r’ dB(r)dB(r) dl’

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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

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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)

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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

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

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

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

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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 ab c d area A contour C x y

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Green’s Theorem on plane Integral of derivative over A Integral around contour C ab c d Area A contour C x y

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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 = Contributions from boundaries cancel cancellation No cancellation on boundary C A

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

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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 V.n dS .V dv

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

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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. ( x V).n dS n outward normal dS local value of x V local value of V drdr V. drV. dr S C

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

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