MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable CalculusFall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.��� � 18.02 Lecture 33. – Thu, Dec 6, 2007 Handouts: PS12 solutions; exam 4 solutions; review sheet and practice final. Applications of div and curl to physics. Recall: curl of velo c ity field = 2 angular velocity vector (of the rotation component of motion). · E.g., for uniform rotation about z-axis, v = ω(−yˆı + xˆj), and � × v = 2ωkˆ. Curl singles out the rotation component of motion (while div singles out the stretching compo-nent). Interpretation of curl for force fields. If we have a solid in a force field (or rather an acceleration field!) F such that the force exerted on Δm at (x, y, z) is F(x, y, z) Δm: recall the torque of the force about the origin is defined as τ = �r × F (for the entire solid, take . . . δ dV ), and measures how F imparts rotation motion. Force dFor translation motion: = acceleration = (velo city). Mass dtTorque dFor rotation effects: = angular acceleration = (angular velocity).Moment of inertia dtForce Torque Hence: curl( ) = 2 .Mass Moment of inertiaConsequence: if F derives from a potential, then �×F = �×(�f) = 0, so F does not induce any rotation motion. E.g., gravitational attraction by itself does not affect Earth’s rotation. (not strictly true: actually Earth is deformable; similarly, friction and tidal effects due to Earth’s gravitational attraction explain why the Moon’s rotation and revolution around Earth are synchronous). Div and curl of electrical field. – part of Maxwell’s equations for electromagnetic fields. 1) Gauss-Coulomb law: � · E�= ρ (ρ = charge density, �0 = physical constant). �0 �� ��� By divergence theorem, can reformulate as: E�nˆdS = E dV = Q , where Q = · � · ��0S D total charge inside the closed surface S. This formula tells how charges influence the electric field; e.g., it governs the relation between voltage b etween the two plates of a capacitor and its electric charge. ∂B�2) Faraday’s law: � × E�= − ∂t (B�= magnetic field). So in presence of a varying magnetic field, E�is no longer conservative: if we have a closed loop �� of wire, we get a non-zero voltage (“induction” effect). By Stokes , C E�· d�r = −dtd S B�· nˆdS. This principle is used e.g. in transformers in power adapters: AC runs through a wire looped around a cylinder, which creates an alternating magnetic field; the flux of this magnetic field through another output wire loop creates an output voltage between its ends. There are two more Maxwell equations, governing div and curl of B�: � · B�= 0, and � × B�= ∂E�µ0J�+ �0µ0 (where J�= current density). ∂t
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