MIDTERM 1 UTC 4.132 Thu-Sep 27, 7:00PM - 9:00PM Bring pencils, calculators (memory cleared) Course Summary Unit 1 Provided
Chapter 17 Electric Potential
Potential Energy To understand the dynamics of moving objects we used: forces, momenta, work, energy The concept of electric field Edeals with forces Electric potential –> for work and energy Electric potential:electric potential energy per unit charge Practical importance: • Reason about energy without having to worry about the details of some particular distribution of charges • Batteries: provide fixed potential difference • Predict possible pattern of E field
Energy of a Single Particle q1 Particle energy Kinetic energy Rest energy For v<<c: The energy of a single particle with charge q1 consists solely of its particle energy. Kinetic energy is associated with motion A single particlehas no (electric) potential energy The kinetic energy of a single particle can be changed if positive or negative work is done on the particle by its surroundings.
Electric Potential Energy of Two Particles Potential energy is associated with pairs of interacting objects Energy of the system: • Energy of particle q1 • Energy of particle q2 • Interaction energy Uel q2 r12 q1 Esystem = E1+E2+Uel To change the energy of particles we have to perform work. Wext – work done by forces exerted by other objects Wint – work done by electric forces between q1and q2 Q – thermal transfer of energy into the system
Electric Potential Energy of Two Particles if q2 r12 Uel -Wint q1 Total energy of the system can be changed (only) by external forces or by adding (thermal) energy. Work done by internal forces:
Electric Potential Energy of Two Particles Fint q2 r12 q1
Electric Potential Energy of Two Particles Fint q2 r12 q1 The potential energy of a pair of particles is:
Electric Potential Energy of Two Particles q2 q2 q1 q1 Uel > 0 for two like-sign charges (repulsion) Uel < 0 for two unlike-sign Charges (attraction)
Electric Potential Energy of Two Particles q2 q2 q1 q1 r12 Meaning of U0: Choose U0=0 – no potential energy if r12 (no interaction) Potential energy = amount of work the two charges can do on each other when moved away from each other to
Three Electric Charges Interaction betweenq1andq2is independent of q3 There are three interacting pairs: q1 q2 q2 q3 q3 q1 U12 U23 U31 U= U12+ U23+ U31
Multiple Electric Charges q1 q3 q6 q2 q5 q4 Each (i,j) pair interacts: potential energyUij Notation: i<j avoids double counting: ij, ji
Electric Potential Alessandro Volta (1745 - 1827) Electric potential electric potential energy per unit charge Units: J/C = V (Volt) Volts per meter = Newtons per Coulomb Electric potential – often called potential Electric potential difference – often called voltage
V due to One Particle q2 Single charge hasnoelectric potential energy Single charge haspotentialto interact with other charge – it creates electric potential probe charge J/C, or Volts Electric potential at B due to charge q1.
V due to Two Particles q3 Electric potential is scalar: Electric potential energy of the system: If we add one more charge at position C:
V at Infinity r, V=0 Positive charge Negative charge
Exercise What is the electrical potential at a location 1Å from a proton? 1Å What is the potential energy of an electron at a location 1Å from a proton?
Exercise 2Å 1Å What is the change in potential in going from 1Å to 2Å from the proton? What is the change in electric potential energy associated with moving an electron from 1Å to 2Å from the proton? Does the sign make sense?
Electric Potential Difference in a Uniform Field Electric potential electric potential energy per unit charge
Sign of the Potential Difference 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) 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.
Sign of the Potential Difference If freed, a positive charge will move to the area with a lower potential: Vf – Vi < 0 (no external forces) V1 < V2 Moving in the direction of E means that potential is decreasing
Sign of the Potential Difference To move a positive charge to the area with higher potential: Vf – Vi > 0 Need external force to perform work V1 < V2 Moving opposite to E means that potential is increasing
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: Is constant during the motion Decreases Increases Not enough information V1 < V2
Potential Difference in a Nonuniform Field C x Ato C: DV1 = -|E1x|(xC-xA) Cto B: DV2 = |E2x|(xB-xC); Ato B: DV= DV1+ DV2 = -|E1x|(xC-xA) + |E2x|(xB-xC)
Example: Different Paths near Point Charge 1. Along straight radial path: rf ri Origin at +q +q
Example: Different Paths near Point Charge + 2. Special case iA: AB: BC: Cf:
Example: Different Paths near Point Charge 3. Arbitrary path +