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Methods of Math. Physics Dr. E.J. Zita, The Evergreen State College, 6 Jan.2011 Lab II Rm 2272, [email protected] Winter wk 1 Thursday: Electromagnetism * Overview of E&M * Review of basic E&M: prep for charge/mass ratio workshop * Griffiths Ch.1: Div, Grad, Curl, and d Introduction to Electromagnetism • • • • • • • • • 4 realms of physics 4 fundamental forces 4 laws of EM statics and dynamics conservation laws EM waves potentials Ch.1: Vector analysis Ch.2: Electrostatics 4 realms of physics, 4 fundamental forces Classical Mechanics Quantum Mechanics (big and slow: everyday experience) (small: particles, waves) Special relativity Quantum field theory (fast: light, fast particles) (small and fast: quarks) Four laws of electromagnetism Electric Magnetic Gauss' Law Gauss' Law Charges → E fields No magnetic monopoles Ampere's Law Faraday's Law Currents → B fields (and changing E→ B fields) Changing B → E fields Electrostatics • Charges → E fields and forces • charges → scalar potential differences dV • E can be found from V • Electric forces move charges • Electric fields store energy (capacitance) Magnetostatics • Currents → B fields • currents make magnetic vector potential A • B can be found from A • Magnetic forces move charges and currents • Magnetic fields store energy (inductance) Electrodynamics • Changing E(t) → B(x) • Changing B(t) → E(x) • Wave equations for E and B • Electromagnetic waves • Motors and generators • Dynamic Sun Some advanced topics • Conservation laws • Radiation • waves in plasmas, magnetohydrodynamics • Potentials and Fields • Special relativity Ch.1: Vector Analysis A = A x x̂ A y ŷ A z ẑ, B = B x x B y y Bz z Dot product: A.B = Ax Bx + Ay By + Az Bz = A B cos q Cross product: |AxB| = A B sin q = x Ax Bx y Ay By z Az Bz Examples of vector products Dot product: work done by variable force W = F dl = F cos q dl Cross product: angular momentum L = r x mv Differential operator “del” Del differentiates each component of a vector. = x yˆ zˆ x y y Gradient of a scalar function = slope in each direction f f f f = x yˆ zˆ x y y Divergence of vector = dot product = outflow Vy Vz Vx ˆ ˆ V = x y z x y y Curl of vector = cross product = circulation x V = x Vx y y Vy z z Vz = x yˆ zˆ Practice: 1.15: Calculate the divergence and curl of v = x2 x + 3xz2 y - 2xz z (3xz2 ) (2 xz) x 2 ˆ V = x y zˆ = .. . x y y x V = x x2 y z y z xz2 2 xz = x yˆ zˆ Ex: If v = E, then div E ≈ charge. If v = B, then curl B ≈ current. Prob.1.16 p.18 Develop intuition about fields Look at fields on p.17 and 18. Which diverge? Which curl? Separation vector vs. position vector: Position vector = location of a point with respect to the origin. r = x xˆ y yˆ z zˆ r = x2 y2 z 2 Separation vector: from SOURCE (e.g. a charge at position r’) TO POINT of interest (e.g. the place where you want to find the field, at r). r = r r ' = ( x x ') xˆ ( y y ') yˆ ( z z ') zˆ r = r r ' = ( x x ') ( y y ') ( z z ') 2 2 2 Source (e.g. a charge or current element) r (separation vector) Point of interest, or Field point Origin See Griffiths Figs. 1.13, 1.14, p.9 Fundamental theorems For divergence: Gauss’s Theorem v d = volume v da = flux surface For curl: Stokes’ Theorem v da = surface boundary v dl = circulation Dirac Delta Function rˆ f= 2 r This should diverge. Calculate it using (1.71), or refer to Prob.1.16. How can div(f)=0? Apply Stokes: different results on L ≠ R sides! How to deal with the singularity at r = 0? Consider 0 if x 0 d ( x) = if x = 0 and show (p.47) that f ( x) d ( x a) dx = f (a) Ch.2: Electrostatics: charges make electric fields • Charges make E fields and forces • charges make scalar potential differences dV • E can be found from V • Electric forces move charges • Electric fields store energy (capacitance) Gauss’ Law practice: What surface charge density does it take to make Earth’s field of 100V/m? (RE=6.4 x 106 m) 2.12 (p.75) Find (and sketch) the electric field E(r) inside a uniformly charged sphere of charge density r. 2.21 (p.82) Find the potential V(r) inside and outside this sphere with total radius R and total charge q. Use infinity as your reference point. Compute the gradient of V in each region, and check that it yields the correct field. Sketch V(r).