MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 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 nal Applications of div and curl to physics Recall curl of velocity eld 2 angular velocity vector of the rotation component of motion E g for uniform rotation about z axis v y xj and v 2 k Curl singles out the rotation component of motion while div singles out the stretching compo nent Interpretation of curl for force elds If we have a solid in a force eld or rather an acceleration eld F such that the force exerted on m at x y z is F x y z m recall the torque of the force about the origin is de ned as r F for the entire solid take dV and measures how F imparts rotation motion Force d For translation motion acceleration velocity Mass dt Torque d For rotation e ects angular acceleration angular velocity Moment of inertia dt Force Torque Hence curl 2 Mass Moment of inertia Consequence 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 a ect Earth s rotation not strictly true actually Earth is deformable similarly friction and tidal e ects due to Earth s gravitational attraction explain why the Moon s rotation and revolution around Earth are synchronous Div and curl of electrical eld part of Maxwell s equations for electromagnetic elds charge density 0 physical constant 1 Gauss Coulomb law E 0 n dS dV Q where Q By divergence theorem can reformulate as E E 0 S D total charge inside the closed surface S This formula tells how charges in uence the electric eld e g it governs the relation between voltage between the two plates of a capacitor and its electric charge B B magnetic eld 2 Faraday s law E t is no longer conservative if we have a closed loop So in presence of a varying magnetic eld E d r d n dS of wire we get a non zero voltage induction e ect By Stokes E B dt S C 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 eld the ux of this magnetic eld through another output wire loop creates an output voltage between its ends B 0 and B There are two more Maxwell equations governing div and curl of B E 0 J 0 0 where J current density t 1
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