Electric Potential

# Electric Potential - Chapter 3 Electric Potential 3.1...

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Chapter 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 3-1

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Electric 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: 2 ˆ g Mm G r =− Fr G (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 2 6.67 10 N m /kg G ˆ r g G , defined as the gravitational force per unit mass, is given by 2 ˆ g GM mr == F g r G G (3.1.2) Notice that g 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 2 B B A A r r gg r BA r GMm r Wd d r G M m rr r ⎛⎞ =⋅= = = ⎜⎟ ⎝⎠ ⎡⎤ ∫∫ ⎢⎥ ⎣⎦ Fs G G (3.1.3) The result shows that g W is independent of the path taken; it depends only on the endpoints A and B . It is important to draw distinction between , g W the work done by the 3-2
field and , the work done by an external agent such as you. They simply differ by a negative sign: . ext W ext g WW =− Near Earth’s surface, the gravitational field g 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 . 8 m / E gG Mr =≈ s E r A y to (Figure 3.1.2) is B y cos cos ( ) B A BBy g g BA AAy W d mg ds mg ds mg dy mg y y θφ =⋅ = = = = ∫∫ Fs G G (3.1.4) Figure 3.1.2 Moving a mass m from A to B .

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Electric Potential - Chapter 3 Electric Potential 3.1...

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