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Quiz August 30, 2010 From Mechanics (PHYS 220) There is a certain net flux Φο through a Gaussian sphere of radius r enclosing an isolated charged particle. Suppose the enclosing Gaussian surface is changed to • Energy Kinetic Energy: associated with the state of motion 1 A Gaussian cube with edge length r 2 A larger Gaussian sphere of radius 2r 3 A Gaussian cube with edge length 2r Potential Energy: associated with the configuration of the system Rank the net flux for each Gaussian surface: • Conservative Forces: (A) Φο = Φ1 < Φ2 = Φ3 (B) Φ2 = Φο > Φ1 = Φ3 (C) Φο = Φ1 > Φ2 = Φ3 (D) Φο = Φ1 = Φ2 = Φ3 (Ε) Φ1 = Φ3 > Φ2 = Φο 8/31/10 Work done by a conservative force is independent of path. 1 Potential Energy m1m2 r2 PE = −G m1m2 r 2 Potential Energy of Two Point Gravitational Potential Energy Fg = G 8/31/10 U=0 at r=∞ Electric Potential Energy Fe = k q1q2 r2 PE = k q1q2 r U g = −G m1m2 r PEelec = k q1q2 r Potential energy: can be positive or negative Field forces push object into area with lower potential energy: Potential energy decreases, work is done on object – for example, kinetic energy of the object may increase (speed) Gravity: always attraction. U is more negative (“lower”) when objects are closer q1q2 < 0 : attraction. U is more negative (“lower”) when objects are closer q1q2 > 0 : repulsion. U is lower when objects are farther apart Total energy is conserved 8/31/10 3 8/31/10 4 1 Potential Energy of Several Point Charges: Electric Potential q5 An Example with 3 charges q4 Suppose a test charge q moves through the area with many fixed charges around q1 Total potential energy PE: +q1 -q3 PE = PE12 + PE23 + PE13 q6 q q3 q2 +q2 ⎛ qq qq ⎞ qq qq PEelec = k ⎜ 1 + 2 + ... 1 2 + 1 3 + ...⎟ r2 r12 r13 ⎝ r1 ⎠ The changes in potential energy are proportional to that charge Electric potential V: electric potential energy per unit charge: Potential energy of many charges is a scalar sum of potential energies of all the pairs of charges in the system. Note: each pair should be counted only once. 8/31/10 *Electric potential energy PEelec ≠ electric potential (or potential) V 5 Electric Potential versus Electrical Potential Energy 8/31/10 6 Potential & Electric Fields The electric field points in the direction in which the potential decreases most rapidly. Electric Potential is a property of an electric field and is measured in J/C or V Electric Potential Energy is an energy of system consisting of the charged object and the external electric field, and is measured in Joules. 8/31/10 7 8/31/10 8 2 Potential Difference Potential Difference *independent of path *independent of test charge To find ΔV, find work done on a positive charge qo by the electric field as it moves from i to f. The potential difference between two points is the negative of the work done by the electrostatic force to move a unit charge from one point to another. At any point, the differential work done on the particle by electrostatic force: Work: Potential Difference: 8/31/10 9 Potential due to a point charge: 8/31/10 10 Potential due to a Group of Point Charges V in space around a charged particle relative to the zero potential at infinity: Find the Potential at the center of the square. N V (r) = ∑ Vn (r) = For a point charge n =1 V(r) to ∞ V= + q 4πε 0 r q1 = +12 nC r q2 = -24 nC + - + + q3 =+31 nC 1 N qn ∑ 4πε o n =1 rn q3 =+17 nC V (r = 0) = ∞ 8/31/10 11 8/31/10 12 3 Equipotentials Definition: locus of points with the same potential. • Calculate Electric Potential from the Electric Field • General Property: The electric field is always perpendicular to an equipotential surface. ΔV = −EΔx • Calculate Electric Field from the Electric Potential E=− ΔV Δx 8/31/10 13 Equipotentials: Examples Point charge infinite positive charge sheet 8/31/10 14 E due to an infinite line charge electric dipole Corona discharge around a high voltage power line, which roughly indicates the electric field lines. 8/31/10 15 8/31/10 16 4 Equipotentials Demo Get energy out ΔPE = q ΔV charge flow + + V= + R + + + + + + + 17 ΔV = V (r1 ) − V (r2 ) + across fluorescent light bulb 8/31/10 18 Supplemental: Potential Energy: Gravitational & Electric Potential inside & outside a conducting sphere Gravitational potential energy Vref = 0 ΔPE = -Wgrav.field at r = ∞. Common mistake: thinking that potential must be zero inside because electric field is zero inside. 8/31/10 r2 r1 Also try an elongated neon bulb. 8/31/10 kQ R 19 8/31/10 W = F.d.cos(θ) mm Fg = G 1 2 2 r mm PE = −G 1 2 r Electric potential energy ΔPE = -Welec.field U=0 at r=∞ W = F.d.cos(θ) qq Fe = k 1 2 2 r qq PE = k 1 2 r 20 5