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From Last Time: Electric field of a single charge Electric field from multiple charges Superposition Ex: calculate electric field of dipole Today: Continuous charge distributions Electric field of line, ring, plane Conductors : electrostatic equilibrium Electrostatic potential Oct. 1, 2009 Physics 208 Lecture 9 1 Exam 1 results Physics 208 Exam 1 25 Mean = 66% B Average = 66% Curve Ave at B/BC boundary Contributes to course grade as curved score. 20 BC AB C 15 Count Exams returned in Monday discussion A 10 D F 5 0 20 30 40 50 60 70 80 90 100 Exam % Score Oct. 1, 2009 Physics 208 Lecture 9 2 From Last Time: Electric field of a single charge Electric field from multiple charges Superposition Ex: calculate electric field of dipole Today: Continuous charge distributions Electric field of line, ring, plane Conductors : electrostatic equilibrium Electrostatic potential Oct. 1, 2009 Physics 208 Lecture 9 3 Charge Densities Volume charge density: when a charge is distributed evenly throughout a volume = Q / V dq = dV Surface charge density: when a charge is distributed evenly over a surface area = Q / A dq = dA Linear charge density: when a charge is distributed along a line = Q / dq = d Oct. 1, 2009 Physics 208 Lecture 9 4 Infinite line of charge An infinite line of charge has a uniform charge density Coulombs / meter. What direction is the electric field at point x? B. C. x A. D. E. + + + + + + + + + + + + + + + + + + + Oct. 1, 2009 Physics 208 Lecture 9 5 E-field: infinite line of charge dE y cosdE y dE 2 r 2 x dQ dx r 1 4 o dQ r2 x 2 dQ 4 o r 2 x 2 2 1 1 cos r r2 x 2 rdx 4 o r 2 x 2 3 / 2 + + + + + + + + + + + + + + + + + + + + x Charge density Coulombs/meter Oct. 1, 2009 Physics 208 Lecture 9 6 Add all these up Ey 2 4 rdx 1 0 o r 2 x 2 3/2 xdx 2k 1/ 2 2 2 2o rr x r 2o r 0 Units? Oct. 1, 2009 [k] N m 2 /C 2 [ ] C /m [r] m Physics 208 Lecture 9 7 Non-infinite (finite) line of charge What direction is the E-field above the end of the line charge? C B A D E + + + + + + + + + + + + + + + + + + + + Oct. 1, 2009 Physics 208 Lecture 9 8 A) Ring of uniform positive charge z B) y Which is the graph of E z on the z-axis? x C) D) z Ez k E) Oct. 1, 2009 Physics 208 Lecture 9 zQ z 2 R 2 3/2 9 Quick quiz We have an infinite sheet of charge of uniform charge density . The electric field A. increases with distance from plane B. decreases with distance from plane C. is independent of distance from plane D. changes direction with distance from plane Oct. 1, 2009 Physics 208 Lecture 9 10 E-field of plane of charge Text does this as adding up rings of charge. E plane 2o Surface charge density Oct. 1, 2009 Physics 208 Lecture 9 11 Electric fields and forces Original definite of E-field was (Coulomb Force) / charge E-field produced force on charged particle. F qE Oct. 1, 2009 Physics 208 Lecture 9 12 Force on charged particle Electric field produces force qE on charged particle Force produces an acceleration a = FE / m Uniform E-field (direction & magnitude) produces constant acceleration Positive charge accelerates in direction of the field Negative charge accelerates in direction opposite the electric field Oct. 1, 2009 Physics 208 Lecture 9 13 Motion of charged particle If no other forces, positive charge accelerates in direction of E-field. But many systems have drag forces (e.g. molecules in a liquid, etc) Drag force is complex, but usually depends on velocity. Particle reaches terminal velocity, determined by force balance Oct. 1, 2009 Physics 208 Lecture 9 14 Application: Gel Electrophoresis Charged macromolecules in ‘gel’ with applied E-field Electric force: FE = qE Steady state: F +F =0 E D Drag force: FD ~ -cv v = qE / c Speed depends on charge and drag (molecule size) Sometimes too complex to interpret Protein electrophoresis: soak in detergent to give proteins all the same charge density. Result, small proteins move faster Oct. 1, 2009 Physics 208 Lecture 9 15 Conductors: electrostatic equilibrium Because charges are mobile in conductor : E-field inside conductor = 0 Local net charge resides at surface Non-zero E-field causes charges to move Stop at surface, generate canceling E-field Repulsive force between charges pushes them away from each other – stop at surface E-field at locally perpendicular to surface A parallel component would cause charge motion Has magnitude E surf /o Surface charge density Oct. 1, 2009 16 Charge distribution on conductor What charge is induced on sphere to make zero electric field? Oct. 1, 2009 + 17 Physics 208 Lecture 9 Electrogenic fish Dipole + nearby conducting object Some fish generate charge separation - electric dipole. Dipole is induced in nearby (conducting) fish Small changes detected by fish. 18 Summary of conductors E 0 everywhere inside a conductor Charge in conductor is only on the surface E surface of conductor --- Oct. 1, 2009 ++ + + ++ 19 Electric forces, work, and energy Consider positive particle charge q, mass m at rest in uniform electric field E Force on particle from field Opposite force on particle from hand Let particle go - it moves a distance d How much work was done on particle? W Fd qEd 1 2 K.E. mv W qEd How fast is particle moving? 2 v=0 Oct. 1, 2009 + v>0 + 20 Work and kinetic energy Work-energy theorem: Change in kinetic energy of isolated particle = work done dW F ds Fdscos Total work end K end dW F ds start start In our case, F qE Oct. 1, 2009 21 Electric forces, work, and energy Same particle, but don’t let go F qE Move particle distance d, keep speed ~0 How much work is done by hand on particle? W Fd qEd What is change in K.E. of particle? K.E . 0 How much force does hand apply? Conservation of energy? W stored in field as potential energy + Oct. 1, 2009 + 22 Work, KE, and potential energy If particle is not isolated, Wexternal K U Work done on system Change in kinetic energy Change in electric potential energy Works for constant electric field if U qE r Only electric potential energy difference Sometimes a reference point is chosen E.g. U r 0 at r (0,0,0) Then U r qE r for uniform electric field Oct. 1, 2009 23 Electric potential V Electric potential difference V is the electric potential energy / unit charge = U/q For uniform electric field, U r qE r V r E r q q This is only valid for a uniform electric field Oct. 1, 2009 24 Quick Quiz Two points in space A and B have electric potential VA=20 volts and VB=100 volts. How much work does it take to move a +100µC charge from A to B? A. +2 mJ B. -20 mJ C. +8 mJ D. +100 mJ E. -100 mJ Oct. 1, 2009 25 Check for uniform E-field Push particle against E-field, or across E-field Which requires work? + Increasing electric potential in this direction Oct. 1, 2009 Constant electric potential in this direction + Decreasing electric potential in this direction 26 Potential from electric field dV E d Potential changes largest in direction of V V d o E-field. Smallest (zero) E perpendicular to E-field d V Vo E d d V=Vo V Vo E d Oct. 1, 2009 27 Electric potential: general U F Coulomb ds qE ds q E ds Electric potential energy difference U proportional to charge q that work is done on U /q V Electric potential difference E ds Depends only on charges that create E-fields Electric field usually created by some charge distribution. V(r) is electric potential of that charge distribution V has units of Joules / Coulomb = Volts Oct. 1, 2009 28 Electric potential of point charge kQ Electric field from point charge Q is E 2 rˆ r What is the electric potential difference? V end r final E ds start k Define V r 0 Oct. 1, 2009 Q k dx 2 rinitial r r final Q Q Q k k r rinitial rinital rfinal Then Q V r k for point charge r 29 Electric Field and equipotential lines for + and - point charges The E lines are directed away from the source charge Oct. 1, 2009 A positive test charge would be repelled away from the positive source charge The E lines are directed toward the source charge A positive test charge would be attracted toward the negative source charge Blue dashed lines are equipotential 30