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• Exam 1 on Feb 17 (Tuesday), 8-9:30 PM, PHYS Rm 112. 1 Clicker Question 1 Which arrow best represents the field at the “X”? A) B) C) D) E) E=0 7 Question 2 Which one of these statements is false? A. The electric field of a very long uniformly charged rod has a 1/r distance dependence. B. The electric field of a capacitor at a location outside the capacitor is very small compared to the field inside the capacitor. C. The fringe field of a capacitor at a location far away from the capacitor looks like an electric field of a point charge. D. The electric field of a uniformly charged thin ring at the center of the ring is zero. 8 Infinitesimals and Integrals in Science Q Charge distribution is not continuous Charge distribution is not exactly uniform Mathematical idealization Works well for macroscopic systems Atomic force microscope: scans microscopic structure using variations in charge density on surface 9 Chapter 17 Electric Potential 10 Electric and Gravitational Potential Energy 1 q2 q1q2 F r̂ 2 4 0 r m1m2 F G 2 rˆ r q1 m2 m1 1 q1q2 U el 40 r m1m2 U grav G r 11 Electric Potential Energy of Two Particles r12 Potential energy is associated with pairs of interacting objects Energy of the system: 1. Energy of particle q1 q2 2. Energy of particle q2 3. Interaction energy Uel q1 E = E1+E2+Uel To change the energy of particles we have to perform work. E1 E2 Wext Wint Q Wext – work done by forces exerted by other objects Wint – work done by electric forces between q1 and q2 Q – thermal transfer of energy into the system 14 Electric Potential Energy of Two Particles E1 E2 Wext Wint Q q2 r12 E1 E2 Wint Wext Q Uel -Wint q1 if (mc ) 0 K system Uel Wext Q 2 Total energy of the system can be changed (only) by external forces. Work done by internal forces: f U el Wint Fint dr i 16 Electric Potential Energy of Two Particles Fint f U el Wint Fint dr i q2 f 1 q1q2 U el rˆ12 dr12 2 4 0 r12 i r12 f 1 q1q2 U el dr12 2 40 r12 i q1 F 1 q1q2 rˆ12 2 40 r12 1 f 1 U el q1q2 2 dr12 40 r i 12 f 1 U el q1q2 40 r12 i 1 17 Electric Potential Energy of Two Particles 1 q1q2 Uel 4 0 r12 Meaning of U0: r12 1 q1q2 U el U0 4 0 r12 U el U 0 Choose U0=0 – no potential energy if r12 (no interaction) q2 q1 Potential energy = amount of work the two charges can do when moved away from each other to q2 q1 18 Electric Potential Energy of Two Particles q2 1 q1q2 U el (joules) 40 r12 q1 Uel > 0 for two like-sign charges (repulsion) q2 q1 Uel < 0 for two unlike-sign Charges (attraction) 20 Electric and Gravitational Potential Energy 1 q2 q1q2 F r̂ 2 4 0 r m1m2 F G 2 rˆ r q1 m2 m1 1 q1q2 U el 40 r m1m2 U grav G r 21 Three Electric Charges Interaction between q1 and q2 is independent of q3 There are three interacting pairs: q1 q2 U12 q2 q3 U23 q3 q1 U31 U= U12+ U23+ U31 1 q1q2 1 q2 q3 1 q1q3 U el 4 0 r12 4 0 r23 4 0 r13 26 Multiple Electric Charges q1 q3 q6 q2 q4 Each (i,j) pair interacts: potential energy Uij q5 1 qi q j U el U ij i j i j 40 rij 27 Electric Potential Electric potential electric potential energy per unit charge U el V q Units: J/C = V (Volt) Volts per meter = Newtons per Coulomb Alessandro Volta (1745 - 1827) Electric potential – often called potential Electric potential difference – often called voltage 28 V due to One Particle U el V q q2 Single charge has no electric potential energy Single charge has potential to interact with other charge – it creates electric potential 1 q1q2 U el 40 r 1 q1 VB 40 r probe charge J/C, or Volts 29 V due to Two Particles Electric potential is scalar: VC VC ,1 VC , 2 1 q1 1 q2 40 r13 40 r23 Electric potential energy of the system: q3 U sys 1 q1q2 U12 40 r12 If we add one more charge at position C: U sys 1 q1q2 1 q1 1 q2 U12 VC q3 q3 4 0 r12 4 0 r13 4 0 r23 U sys 1 q1q2 1 q1q3 1 q2 q3 U12 U13 U 23 40 r12 40 r13 40 r23 31 V at Infinity 1 q1 V 40 r r, V=0 Positive charge Negative charge 32 Exercise What is the electrical potential at a location 1Å from a proton? 1Å 19 1.6 10 C 1 q1 9 Nm V 9 10 14.4 J/C 14.4 V 2 10 40 r C 10 m 2 What is the potential energy of an electron at a location 1Å from a proton? 19 18 U el Vq 14.4 J/C 1.6 10 C 2.3 10 J 33 Exercise 2Å 1Å What is the change in potential in going from 1Å to 2Å from the proton? o o V V 2 A V 1A 7.2 V What is the change in electric potential energy associated with moving an electron from 1Å to 2Å from the proton? o o o o Uel Uel 2 A Uel 1A qV 2 A qV 1A qV Uel 1.6 1019 C 7.2 J/C 1.15 1018 J Does the sign make sense? 34 Electric Potential Difference in a Uniform Field Electric potential electric potential energy per unit charge U electric U electric qV, V q Units: J/C = V (Volt) Volts per meter = Newtons per Coulomb Electric potential, V – often called potential Electric potential difference, V – often called voltage 40 V E l ( Ex x E y y Ez z ) If we multiply through by q, we recover the relation between the change in potential energy and work done on q by the 41 internal force. Example 300 V E l E l cos 100 (N/C) 2 (m) cos(30o ) 173 volts 42 Example x An electron traveling to the right enters capacitor through a small hole at A. Electric field strength is 2x103 N/C. What is the change in the electron’s potential energy in traveling from A to B? What is its change in kinetic energy? AB)= 4mm Uelectric F l (eEx )x eEx x = (1.6x10-19 C)(2x103 N/C)(0.004m) =1.3x10-18 J K Uelectric = -1.3x10-18 J 43 Sign of the Potential Difference U el qV The potential difference V can be positive or negative. The sign determines whether a particular charged particle will gain or lose energy in moving from one place to another. If qV < 0 – then potential energy decreases and K increases If qV > 0 – then potential energy increases and K decreases Path going in the direction of E: Potential is decreasing (V < 0) Path going opposite to E: Potential is increasing (V > 0) Path going perpendicular to E: Potential does not change (V = 0) 44 Question 1 A proton is free to move from right to left in the diagram shown. There are no other forces acting on the proton. As the proton moves from right to left, its potential energy: A) B) C) D) Is constant during the motion Decreases Increases Not enough information V1 < V2 Sign of the Potential Difference U el qV If freed, a positive charge will move to the area with a lower potential: Vf – Vi < 0 U el qV 0 K U el 0 (no external forces) V1 < V2 Moving in the direction of E means that potential is decreasing 46 Sign of the Potential Difference U el qV To move a positive charge to the area with higher potential: Vf – Vi > 0 U el qV 0 Need external force to perform work V1 < V2 Moving opposite to E means that potential is increasing 47