Electric charge; Electric fields; Dipoles

1 P02 -Class 02: OutlineAnswer questionsHour 1:Review: Electric FieldsChargeDipolesHour 2:Continuous Charge Distributions2 P02 -Last Time: FieldsGravitational & Electric3 P02 -Gravitational & Electric FieldsMass MCharge q(±)2ˆMGr=−gr􀁇2ˆeqkr=Er􀁇CREATE:gm=Fg􀁇􀁇Eq=FE􀁇􀁇This is easiest way to picture fieldFEEL:4 P02 -PRS Questions:Electric Field5 P02 -Electric Field Lines1.Direction of field line at any point is tangent to field at that point2.Field lines point away from positive charges and terminate on negative charges3.Field lines never cross each other6 P02 -In-Class Problemdsq−q+PiˆjˆConsider two point charges of equal magnitude but opposite signs, separated by a distance d. Point Plies along the perpendicular bisector of the line joining the charges, a distance sabove that line. What is the E field at P? 7 P02 -Two PRS Questions:E Field of Finite Number of Point Charges8 P02 -Charging9 P02 -How Do You Charge Objects?•Friction•Transfer (touching)•InductionNeutral----+++++q10 P02 -Demonstrations:Instruments for Charging11 P02 -Electric DipolesA Special Charge Distribution12 P02 -Electric DipoleTwo equal but opposite charges +q and–q,separated by a distance 2aq-q2acharge×displacementˆˆ×22qaqa≡==pjj􀁇Dipole Momentp􀁇p􀁇points from negative to positive charge13 P02 -Why Dipoles?Nature Likes To Make Dipoles!http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/20-Molecules2d/20-mole2d320.html14 P02 -Dipoles makeFields15 P02 -Electric Field Created by DipoleThou shaltuse components!33xexxEkqrr+−⎛⎞∆∆=−⎜⎟⎝⎠33yeyyEkqrr+−+−⎛⎞∆∆=−⎜⎟⎝⎠3/23/22222()()exxkqxyaxya⎛⎞⎜⎟=−⎜⎟⎡⎤⎡⎤+−++⎣⎦⎣⎦⎝⎠3/23/22222()()eyayakqxyaxya⎛⎞−+⎜⎟=−⎜⎟⎡⎤⎡⎤+−++⎣⎦⎣⎦⎝⎠2333ˆˆˆxyrrrr∆∆==+rrij􀁇16 P02 -PRS Question:Dipole Fall-Off17 P02 -Point Dipole ApproximationTake thelimitra>>You can show…Finite Dipole303sincos4xpErθθπε→()2303cos14ypErθπε→−Point Dipole18 P02 -Shockwave for Dipolehttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/06-DipoleField3d/06-dipField320.html19 P02 -Dipoles feelFields20 P02 -Demonstration:Dipole in Field21 P02 -Dipole in Uniform FieldˆE=Ei􀁇ˆˆ2(cossin)qa=+pij􀁇θθ()0netqq+−=+=+−=FFFEE􀁇􀁇􀁇􀁇􀁇tends to align with the electric field p􀁇Total Net Force:Torque on Dipole:=×τrF􀁇􀁇􀁇()()2sin()aqEθ==×pE􀁇􀁇sin()rFτθ+=sin()pEθ=22 P02 -Torque on DipoleTotal Field (dipole + background) shows torque:http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/43-torqueondipolee/43-torqueondipolee320.html•Field lines transmit tension•Connection between dipole field and constant field “pulls” dipole into alignment23 P02 -PRS Question:Dipole in Non-Uniform Field24 P02 -Continuous Charge Distributions25 P02 -Continuous Charge Distributions()?P=E􀁇ViiQqBreak distribution into parts:=∆∑2ˆeqkr∆∆=Er􀁇E field at Pdue to ∆qSuperposition:=∆∑EE􀁇􀁇Vdq→∫d→∫E􀁇2ˆedqdkr→=Er􀁇26 P02 -Continuous Sources: Charge DensitydVdQρ=RL2VolumeVRLπ==QVρ=LQ=λQAσ=dAdQσ=wLAreaAwL==dLdQλ=LengthL=L27 P02 -Examples of Continuous Sources: Line of chargeLQ=λLengthL=LdLdQλ=http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/07-LineIntegration/07-LineInt320.html28 P02 -Examples of Continuous Sources: Line of chargeLQ=λLengthL=LdLdQλ=http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/08-LineField/08-LineField320.html29 P02 -Examples of Continuous Sources: Ring of Charge2QRλπ=dLdQλ=http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/09-RingIntegration/09-ringInt320.html30 P02 -Examples of Continuous Sources: Ring of Charge2QRλπ=dLdQλ=http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/10-RingField/10-ringField320.html31 P02 -Example: Ring of ChargePon axis of ring of charge, xfrom centerRadius a, charge density λ.Find Eat P32 P02 -Ring of ChargeSymmetry!0E⊥=1) Think about ithttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/09-RingIntegration/09-ringInt320.html2) Define Variablesdqdlλ=22xar+=()adλϕ=33 P02 -Ring of Charge3) Write Equationdqadλϕ=2ˆerdkdqr=E􀁇a) My way3xexdEkdqr=b) Another way22xar+=3erkdqr=􀁇cos()xdEdθ=E􀁇231eexxkdqkdqrrr=⋅=34 P02 -Ring of Charge4) Integrate3xxexEdEkdqr==∫∫22xar+=dqadλϕ=3exkdqr=∫Veryspecial case: everything except dqis constant2aλπ=dq∫2200adadππλϕλϕ==∫∫Q=35 P02 -Ring of Charge5) Clean Up3xexEkQr=()3/222xexEkQax=+0a→()3/222ˆexkQax=+Ei􀁇6) Check Limit()3/222exekQxEkQxx→=36 P02 -rˆ2L−2L+sPjˆiˆIn-Class: Line of ChargePoint Plies on perpendicular bisector of uniformly charged line of length L, a distance saway. The charge on the line is Q. What is Eat P?37 P02 -rˆθθ2L−2L+xd′x′xddq′=λs22xsr′+=PjˆiˆHint:http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/07-LineIntegration/07-LineInt320.htmlTypically give the integration variable (x’) a “primed” variable name.38 P02 -E Field from Line of Charge221/2ˆ(/4)eQkssL=+Ej􀁇Limits:2ˆlimesLQks>>→Ej􀁇Point chargeˆˆ22limeesLQkkLssλ<<→=Ejj􀁇Infinite charged line39 P02 -In-Class: Uniformly Charged Disk Pon axis of disk of charge, xfrom centerRadius R, charge density σ.Find Eat P(0)x>40 P02 -Disk: Two Important Limits()1/222ˆ12diskoxxR⎡⎤⎢⎥=−⎢⎥+⎣⎦Ei􀁇σεLimits:21ˆlim4diskxRoQx>>→Ei􀁇πε***Point chargeˆlim2diskxRo<<→Ei􀁇σεInfinite charged plane41 P02 -E for Plane is Constant????1)Dipole: E falls off like 1/r32)Point charge:E falls off like 1/r23)Line of charge:E falls off like 1/r4)Plane of charge: E constant