W02D1 Electric Dipoles and Continuous Charge Distributions 1 Announcements Math Review Tuesday Tues Feb 14 from 9-11 pm in 32-082 PS 1 due Tuesday Tues Feb 14 at 9 pm in boxes outside 32-082 or 26-152 W02D2 Reading Assignment Course Notes: Chapter Course Notes: Sections 4.1-4.2, 4.7 Make sure your clicker is registered 2 Outline Electric Dipoles Force and Torque on Dipole Continuous Charge Distributions 3 Nature Likes to Make Dipoles http://youtu.be/EMj10YIjkaY 4 Demonstration: Dipole in a Van de Graaff Generator D22 5 Concept Question: Dipole in NonUniform Field E A dipole sits in a non-uniform electric field E Due to the electric field this dipole will feel: 1. 2. 3. 4. force but no torque no force but a torque both a force and a torque neither a force nor a torque 6 Concept Question Answer: NonUniform Field E Answer: 3. both force and torque Because the field is non-uniform, the forces on the two equal but opposite point charges do not cancel. As always, the dipole wants to rotate to align with the field – there is a torque on the dipole as well 7 Continuous Charge Distributions 8 Continuous Charge Distributions Break distribution into parts: V Q = å Dqi ® i òòò dq V E field at P due to Dq Dq dq DE = ke 2 rˆ ® dE = ke 2 rˆ r r Superposition: EP ? ò E = å DE ® dE 9 Continuous Sources: Charge Density R dQ dV Volume = V = L w Area = A = wL L Length L L p R2 L r= Q V dQ dA Q s= A dQ dL Q l= L 10 Group Problem: Charge Densities A solid cylinder, of length L and radius R, is uniformly charged with total charge Q. (a)What is the volume charge density ρ? (b)What is the linear charge density λ? (c)What is the relationship between these two densities ρ and λ? 11 Examples of Continuous Sources: Finite Line of Charge Length L L dQ dL Q L E field on perpendicular bisector 12 Examples of Continuous Sources: Finite Line of Charge Length L L dQ dL Q L E field off axis 13 Examples of Continuous Sources: Finite Line of Charge Length L L dQ dL Q L Grass seeds of total E field 14 Concept Question Electric Field of a Rod A rod of length L lies along the x-axis with its left end at the origin. The rod has a uniform charge density λ. Which of the following expressions best describes the electric field at the point P l dx ˆ 1. E(P) = - ò i 3 (x - d) x=0 x=L l dx ˆ i x=L 2. E(P) = ò x=0 (x - d) x=L 3. E(P) = - ò x=0 4. E(P) = 5. E(P) = - 3 l dx ˆ i (x - d) l dx ˆ òx=0 (x - d)2 i x=L 2 6. E(P) = lL (d - x) lL (d - x) 2 2 ˆi 7. E(P) = ˆi lL ˆ i d lL ˆ 8. E(P) = 2 i d 9. None of the above. 2 15 Concept Question Electric Field of a Rod: Answer A rod of length L lies along the x-axis with its left end at the origin. The rod has a uniform charge density λ. Which of the following expressions best describes the electric field at the point P 5. E(P) = - lL (d - x) 2 ˆi 16 Group Problem: Line of Charge Point P lies on perpendicular bisector of uniformly charged line of length L, a distance s away. The charge on the line is Q. Find an integral expression for the direction and magnitude of the electric field at P. 17 Hint on Line of Charge Group Problem ˆj rˆ ˆi P r s x 2 2 s L 2 x dq dx dx L 2 Typically give the integration variable (x’) a “primed” variable name. ALSO: Difficult integral (trig. sub.) 18 E Field from Line of Charge Answer: E = ke Q s(s + L / 4) 2 2 1/2 ˆj Limits: s >> L (far away) and s << L (close) Qˆ lim E ® ke 2 j s>> L s Q ˆ lˆ lim E ® 2ke j = 2ke j s<<L Ls s Looks like the E field of a point charge if we are far away Looks like E field of an infinite charged line if we are close 19 Examples of Continuous Sources: Ring of Charge dQ dL Q 2 R E field on the axis of the ring of charge 20 Examples of Continuous Sources: Ring of Charge dQ dL Q 2 R E field off axis and grass seeds plot 21 Concept Question Electric Field of a Ring A uniformly charged ring of radius a has total charge Q. Which of the following expressions best describes the electric field at the point P located at the center of the ring? 1. E(P) = - q =2p òq =0 2. E(P) = q =2p òq =0 l adq ˆ a3 i l adq ˆ a Qˆ 3. E(P) = - 2 i a 3 i Qˆ 4. E(P) = + 2 i a 5. E(P) = 0 22 Concept Question Electric Field of a Ring: Answer A uniformly charged ring of radius a has total charge Q. Which of the following expressions best describes the electric field at the point P located at the center of the ring? 5. E(P) = 0 23 Demonstration Problem: Ring of Charge A ring of radius a is uniformly charged with total charge Q. Find the direction and magnitude of the electric field at the point P lying a distance x from the center of the ring along the axis of symmetry of the ring. 24 Ring of Charge 1) Think about it E = 0 Symmetry! ^ 2) Define Variables ( dq = l dl = l a dj r a x 2 ) 2 25 Ring of Charge 3) Write Equation rˆ dE = ke dq 2 r r dE = ke dq 3 r x dEx = ke dq 3 r dq = l a dj r a x 2 2 26 Ring of Charge dq = l a dj 4) Integrate x Ex = ò dEx = ò ke dq 3 r x = ke 3 ò dq r r a x 2 2 This particular problem is a very special case because everything except dq is constant, and dq ò = ò 2p 0 2p l a dj = l a ò dj = l × a2p Q 0 27 Ring of Charge 5) Clean Up x E x = ke Q 3 r x E x = ke Q 2 2 3/ 2 (a + x ) 6) Check Limit x ˆi E = ke Q 2 2 3/2 (a + x ) a®0 k eQ x Ex ® keQ 2 3/ 2 = 2 (x ) x 28 Group Problem: Uniformly Charged Disk (x > 0) P on axis of disk of charge, x from center Radius R, charge density . Find E at P 29 Disk: Two Important Limits Answer: E d isk é s ê x = 12 2 2e o ê x +R êë ( ) 1/2 ù ú ˆi ú úû Limits: x >> R (far) and x << R (close) lim E disk x>> R Qˆ ® i 2 4pe o x lim E disk x<<R 1 s ˆ ® i 2e o Looks like E of a point charge far away Looks like E field of an infinite charged plane close up 30 Scaling: E for Plane is Constant 1) 2) 3) 4) Dipole: E falls off like 1/r3 Point charge: E falls off like 1/r2 Line of charge: E falls off like 1/r Plane of charge: E constant 31

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