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Lecture 2-1 COULOMB FORCE q1q2 F1,2 k 2 rˆ1,2 r1,2 “Test Charge” q2 Charge q1 Question: How can q1 sense q2 without direct contact? r “Is action at a distance possible?” • NO! just as air around us, there’re some medium around a charge, the electric field that transfers the force. Lecture 2-2 READING QUIZ 1 HOW DOES THE MAGNITUDE OF THE ELECTRIC FIELD E FROM A POINT CHARGE Q DEPEND ON THE MAGNITUDE OF Q AND THE DISTANCE R TO THE FIELD POINT? A) E = k Q Q / R2 A) E = k Q / R B) E = k Q / R2 C) E = k Q Q / R A) E = k Q / R3 Lecture 2-3 Electric Field Define electric field, which is independent of the test charge, q2 , and depends only on position in space: F E q Electric Field due to a Point Charge Q F 1 Q E rˆ 2 q 4 0 r Lecture 2-4 Electric Field Lines • A visualization tool to illustrate the geometry of an electric field. • Electric field lines originate from positive charges and terminates at negative charges. (or at infinity) • The direction of the electric field at any location is tangential to the field line there. • The magnitude of the electric field at any location is proportional to the density of the lines there. # lines/area ~ 1/r2 ~ |E| Lecture 2-5 Electric Field With this concept, we can “map” the electric field anywhere in space A Vector field 73 77 72 71 82 84 83 88 75 68 64 80 73 57 56 55 66 88 75 80 90 83 92 91 77 Lecture 2-6 Demonstration • Vandegraaff Generator Lecture 2-7 Electric field lines of two charges of equal magnitude dipole Far from the charges, electric field lines resemble those of a point charge. Field lines do NOT cross They do NOT come out of nowhere or vanish into a point. Lecture 2-8 Two charges of the same magnitude • Both charges > 0 • One is > 0, the other < 0 -q d E q electric dipole of dipole moment: p qd if |qR|>|qL| 2kp x3 if |qL|>|qR| Lecture 2-9 Electric field lines of two unequal charges Far from charges, the field lines are as if they are due to a point charge of +2q-q=+q # lines proportional to the magnitude of charge http://www.its.caltech.edu/~phys1/java/phys1/EField/EField.html http://www.falstad.com/vector3de/ Lecture 2-10 ©2008 by W.H. Freeman and Company Lecture 2-11 Electric Field due to Multiple Point Charges The force on a test charge is then given by F F1 F2 F3 so the electric field is, by definition, given by F F1 F2 F3 E q0 q0 q0 q0 E1 E2 E3 Principle of Superposition! Lecture 2-12 Lecture 2-13 Lecture 2-14 ( Sum symbol Σ ) Σ Fx = 3.97 x 10-7 N E = F/qO Σ Fy = - 2.77 x 10-7 N F2 = Fx2 + Fy2 = (4.84 x 10-7 N)2 Tan Θ = Fy /Fx = - 34.9 O Lecture 2-15 Warm-up quiz 2 What is the direction of the electric field at the center of the two point charges? (Take Q > 0.) a) b) c) d) e) toward the lower right toward the upper left upward to the right none of the above Q -Q Lecture 2-16 Point Charge in an external Electric Field • Electrostatic force on the charge q due to E: F qE F is parallel to E if the charge is positive and anti-parallel if the charge is negative • E acting on the charge is produced by the other charges (external field). The charge is not affected by its own electric field. • Dimension of E = [force]/[charge] and its SI units = N/C • Work done by E on q in moving the charge: WAB : AB qE dl W is path( )-independent => Conservative force Lecture 2-17 Demonstration • Dynamics of a charged Mass in an Electric field Lecture 2-18 Dynamics of a Charged Mass in Electric Field For -Q<0 in uniform E downward: F ma ( Q ) E QE a ay j j (E E j) m 1 2 y (t ) a y t , x (t ) v x t 2 -Q 2 1 x QEx 2 y ay 2 vy2 = at = qE/m t vx >>0 v 2 mv x x QEt v (t ) v v y (t ) v m tan y 2 xv 2 2 x 2 • Oscilloscope • Ink-Jet Printing • Oil drop experiment vx http://canu.ucalgary.ca/map/content/force/elcrmagn/simulate/electric_single_particle/applet.html Lecture 2-19 The Electric Dipole Along the + z-axis Where p=qd is the magnitude of the dipole moment. The electric dipole is a vector in the direction from –q to +q. Let d 0 keep qd constant.This a point dipole Lecture 2-20 Dipole in a uniform external electric field • No net force. The electrostatic forces on the constituent point charges are of the same magnitude but along opposite directions. So, there is no net force on the dipole and thus its center of mass should not accelerate. Clockwise about the center of mass • Net torque! There is clearly a net torque acting on the dipole with respect to its center of mass, since the forces are not aligned. http://qbx6.ltu.edu/s_schneider/physlets/main/dipole_torque.shtml Lecture 2-21 Torque on the Dipole F The magnitude of the torque: Fx sin F (d x) sin Fd sin qEd sin pE sin The torque points into the screen. p E x CM F Lecture 2-22 Electric Dipoles • Typical dipole consists of positive and negative charges slightly displaced. • General definition of dipole moments exists: p ( r )rd 3r • Water molecule can be thought of as consisting of 2 standard dipoles at an angle to each other. Net neutral molecules can have electrical dipole moments Permanent dipole moment (polar) vs. induced dipole moment Lecture 2-23 Physics 241 – 10:30 Quiz 3 – January 14, 2010 Four point charges are arranged at the corners of a square as shown. What is the direction of the electric field at the center of the square? (Take Q > 0.) a) b) c) d) e) toward the lower right toward the upper left upward to the right none of the above -Q -Q Q Q Lecture 2-24 Physics 241 – 11:30 Quiz 3 – January 14, 2010 Four point charges are arranged at the corners of a square as shown. What is the direction of the electric field at the center of the square? (Take Q > 0.) a) b) c) d) e) toward the lower right toward the upper left upward to the right none of the above -Q Q Q Q Lecture 2-25 Physics 241 – Sample Quiz C – January 9, 2008 Three point charges are arranged at three of the corners of a square as shown. What is the direction of the electric field at the center of the square? (Take Q > 0.) a) b) c) d) e) toward the lower right toward the upper left upward to the right none of the above Q Q Q