3Potential and Potential EnergyFigure 3.1.1Electric Potential in a Uniform FieldElectric Potential due to Point Charges3.3.1 Potential Energy in a System of ChargesContinuous Charge DistributionDeriving Electric Field from the Electric Potential3.5.1 Gradient and EquipotentialsExample 3.1: Uniformly Charged RodExample 3.2: Uniformly Charged RingExample 3.3: Uniformly Charged DiskExample 3.4: Calculating Electric Field from Electric PotentSummaryProblem-Solving Strategy: Calculating Electric PotentialSolved Problems3.8.1 Electric Potential Due to a System of Two Charges3.8.2 Electric Dipole Potential3.8.3 Electric Potential of an Annulus3.8.4 Charge Moving Near a Charged WireConceptual QuestionsAdditional Problems3.10.1 Cube3.10.2 Three Charges3.10.3 Work Done on Charges3.10.4 Calculating E from V3.10.5 Electric Potential of a Rod3.10.6 Electric Potential3.10.7 Calculating Electric Field from the Electric Potentia3.10.8 Electric Potential and Electric Potential Energy3.10.9. Electric Field, Potential and EnergyChapter 3 Electric Potential 3.1 Potential and Potential Energy.............................................................................. 3-2 3.2 Electric Potential in a Uniform Field.................................................................... 3-5 3.3 Electric Potential due to Point Charges ................................................................ 3-6 3.3.1 Potential Energy in a System of Charges....................................................... 3-8 3.4 Continuous Charge Distribution ........................................................................... 3-9 3.5 Deriving Electric Field from the Electric Potential ............................................ 3-10 3.5.1 Gradient and Equipotentials......................................................................... 3-11 Example 3.1: Uniformly Charged Rod ................................................................. 3-13 Example 3.2: Uniformly Charged Ring................................................................ 3-15 Example 3.3: Uniformly Charged Disk ................................................................ 3-16 Example 3.4: Calculating Electric Field from Electric Potential.......................... 3-18 3.6 Summary............................................................................................................. 3-18 3.7 Problem-Solving Strategy: Calculating Electric Potential.................................. 3-20 3.8 Solved Problems ................................................................................................. 3-22 3.8.1 Electric Potential Due to a System of Two Charges.................................... 3-22 3.8.2 Electric Dipole Potential..............................................................................3-23 3.8.3 Electric Potential of an Annulus ..................................................................3-24 3.8.4 Charge Moving Near a Charged Wire ......................................................... 3-25 3.9 Conceptual Questions ......................................................................................... 3-26 3.10 Additional Problems ......................................................................................... 3-27 3.10.1 Cube ........................................................................................................... 3-27 3.10.2 Three Charges ............................................................................................ 3-27 3.10.3 Work Done on Charges.............................................................................. 3-27 3.10.4 Calculating E from V ................................................................................. 3-28 3.10.5 Electric Potential of a Rod ......................................................................... 3-28 3.10.6 Electric Potential........................................................................................ 3-29 3.10.7 Calculating Electric Field from the Electric Potential ............................... 3-29 3.10.8 Electric Potential and Electric Potential Energy........................................ 3-30 3.10.9. Electric Field, Potential and Energy .......................................................... 3-30 3-1Electric Potential 3.1 Potential and Potential Energy In the introductory mechanics course, we have seen that gravitational force from the Earth on a particle of mass m located at a distance r from Earth’s center has an inverse-square form: 2ˆgMmGr=−Fr (3.1.1) where is the gravitational constant and is a unit vector pointing radially outward. The Earth is assumed to be a uniform sphere of mass M. The corresponding gravitational field 11 2 26.67 10 N m /kgG−=× ⋅ˆrg, defined as the gravitational force per unit mass, is given by 2ˆgGMmr==−Fgr (3.1.2) Notice that g only depends on M, the mass which creates the field, and r, the distance from M. Figure 3.1.1 Consider moving a particle of mass m under the influence of gravity (Figure 3.1.1). The work done by gravity in moving from A to B is m 211BBAArrggrBArGMmrWd dr GMmrrGMmr−⎛⎞⎛⎞=⋅= = = −⎜⎟⎜⎝⎠⎝⎠⎡⎤∫∫⎢⎥⎣⎦Fs⎟ (3.1.3) The result shows that gW is independent of the path taken; it depends only on the endpoints A and B. It is important to draw distinction between ,gW the work done by the 3-2field and , the work done by an external agent such as you. They simply differ by a negative sign: . extWextgWW=− Near Earth’s surface, the gravitational field g is approximately constant, with a magnitude , where is the radius of Earth. The work done by gravity in moving an object from height 22/9.8m/EgGMr=≈sErAy to (Figure 3.1.2) is By cos cos ( )BABBygg BAAAyW d mg ds mg ds mg dy mg y yθφ=⋅= =− =− =− −∫∫ ∫ ∫Fs (3.1.4) Figure 3.1.2 Moving a mass m from A to B. The result again is independent of the path, and is only a function of the change in vertical height . BAyy− In the examples above, if the path forms a closed loop, so that the object moves around and then returns to where it starts off, the net work done by the gravitational field would be zero, and we say that the gravitational force is conservative. More generally, a force F is said to be conservative if its line integral around a closed loop vanishes: 0d⋅=∫FsGGv (3.1.5) When dealing with a conservative
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