January 19, 09Physics 2102Physics 2102Lecture: 04 WED 21 JANLecture: 04 WED 21 JANElectric Fields IIElectric Fields IIPhysics 2102Jonathan DowlingVersion: 1/19/09Michael Faraday (1791-1867)Electric Charges and FieldsElectric Charges and FieldsFirst: Given Electric Charges, WeCalculate the Electric Field UsingE=kqr/r3.Example: the Electric FieldProduced By a Single Charge, or bya Dipole:ChargeProduces E-FieldE-Field ThenProduces Forceon AnotherChargeSecond: Given an Electric Field,We Calculate the Forces onOther Charges Using F=qEExamples: Forces on a SingleCharge When Immersed in theField of a Dipole, Torque on aDipole When Immersed in anUniform Electric Field.Continuous Charge DistributionContinuous Charge Distribution• Thus Far, We Have Only DealtWith Discrete, Point Charges.• Imagine Instead That a Chargeq Is Smeared Out Over A:– LINE– AREA– VOLUME• How to Compute the ElectricField E? Calculus!!!qqqqCharge DensityCharge Density• Useful idea: charge density• Line of charge:charge per unit length = λ• Sheet of charge:charge per unit area = σ• Volume of charge:charge per unit volume = ρλ = q/Lσ = q/Aρ = q/VComputing Electric FieldComputing Electric Field of Continuous Charge Distribution of Continuous Charge Distribution• Approach: Divide the ContinuousCharge Distribution IntoInfinitesimally Small DifferentialElements• Treat Each Element As a POINTCharge & Compute Its ElectricField• Sum (Integrate) Over AllElements• Always Look for Symmetry toSimplify Calculation!dqdq = λ dLdq = σ dSdq = ρ dVDifferential Form of CoulombDifferential Form of Coulomb’’s Laws Lawq2 ! ! E 12P1P2E-FieldatPointdq2 ! d! E 12P1DifferentialdE-Field atPoint ! d! E 12=k dq2r122 ! ! E 12=k q2r122Field on Bisector of Charged RodField on Bisector of Charged Rod• Uniform line of charge+q spread over length L• What is the direction ofthe electric field at apoint P on theperpendicular bisector?(a) Field is 0.(b) Along +y(c) Along +x• Choose symmetricallylocated elements oflength dq!= λdx• x components of E cancelqLaPoyxdx ! d! E dx ! d! E ! " ELine of Charge: QuantitativeLine of Charge: Quantitative• Uniform line of charge,length L, total charge q• Compute explicitly themagnitude of E at pointP on perpendicularbisector• Showed earlier thatthe net field at P is inthe y direction — let’snow compute this!qLaPoyxLine Of Charge: Field on bisectorLine Of Charge: Field on bisector Distance hypotenuse: ! d = a2+ x2( )1/ 22)(ddqkdE =Lq=!Charge per unit length:2/322)()(cosxaadxkdEdEy+==!"qLaPoxdE!dxd2/122)(cosxaa+=!AdjacentOverHypotenuseLine Of Charge: Field on bisectorLine Of Charge: Field on bisector!"+=2/2/2/322)(LLyxadxakE#Point Charge Limit: L << a! =2k"La 4a2+ L22/2/222LLaxaxak!"#$%&'+=(! Ey=2k"La 4a2+ L2#2k"aLine Charge Limit: L >> a! Ey=2k"La 4a2+ L2#k"La2=kqa2Integrate: Trig Substitution!Coulomb’sLaw!! Nm2C1mCm" # $ % & ' =Nm" # $ % & ' Units Check!Binomial Approximation from Taylor Series:x<<1! 2k"La 4a2+ L2=k"La21+L2a# $ % & ' ( 2) * + , - . /1/ 20k"La21/12L2a# $ % & ' ( 2) * + , - . 0k"La2 ; (L << a)2k"La 4a2+ L2=2k"LaL1+2aL# $ % & ' ( 2) * + , - . /1/ 202k"a1/122aL# $ % & ' ( 2) * + , - . 02k"a ; (L >> a)! 1± x( )n" 1 ± nxExample Example —— Arc of Charge: Quantitative Arc of Charge: Quantitative• Figure shows a uniformlycharged rod of charge -Qbent into a circular arc ofradius R, centered at (0,0).• Compute the direction &magnitude of E at the origin.yx450xyθdQ = λRdθdθ!!coscos2RkdQdEdEx==!!==2/02/02coscos)(""##$##$dRkRRdkExRkEx!=RkEy!=RkE!2=λ = 2Q/(πR)E–QExample : Field on Axis of Charged DiskExample : Field on Axis of Charged Disk• A uniformly charged circulardisk (with positive charge)• What is the direction of E atpoint P on the axis?(a) Field is 0(b) Along +z(c) Somewhere in the x-y planePyxzExample : Arc of ChargeExample : Arc of Charge• Figure shows a uniformlycharged rod of charge –Qbent into a circular arc ofradius R, centered at (0,0).• What is the direction of theelectric field at the origin?(a) Field is 0.(b) Along +y(c) Along -yyx• Choose symmetric elements• x components cancelSummarySummary• The electric field produced by a system ofcharges at any point in space is the force perunit charge they produce at that point.• We can draw field lines to visualize the electricfield produced by electric charges.• Electric field of a point charge: E=kq/r2• Electric field of a dipole: E~kp/r3• An electric dipole in an electric field rotates toalign itself with the field.• Use CALCULUS to find E-field from a continuouscharge
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