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Electric Potential We introduced the concept of potential energy in mechanics Let’s remind to this concept and apply it to introduce electric potential energy We start by revisit the work done on a particle of mass m -by a force in general -by a conservative force such as the gravitational force In general, work done by a force F moving a particle from point r a to r b In general we have to specify how we get from a to b, e.g., for friction force x2x2 x1x1 However, for a conservative force such as gravity we remember where is the potential energy

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Independent of the path between and With this we obtain Gravity as an example Gravitational force derived from hh Pot. energy depends on h, not how to get there.

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Can we find a function U =U (r) such that is the force exerted by a point charge q on a test charge q 0 ? We expect the answer to be yes, due to the similarity between Coulomb force and gravitational force Potential energy Potential Let’s try In fact we see simple because of radial symmetry where U(r)=U(r)

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We conclude Electric potential energy of electrostatically interacting point charges q and q 0 r U qq 0 <0 attractive potential r U qq 0 >0 repulsive potential As always, potential defined only up to an arbitrary constant. Expression above uses U(r )=0 as reference point

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We know already the superposition principle for electric fields and forces, can we find a net potential energy for q 0 interacting with several point charges? Net force q 0 experiences Force exerted on q 0 by charge q 1 at r 1 Force exerted on q 0 by charge q 2 at r 2 Force exerted on q 0 by charge q 3 at r 3 r x y q0q0 r1r1 q1q1 r1-rr1-r r3-rr3-r q2q2 r2r2 r2-rr2-r q3q3 r3r3 Note: textbook on p. 785 defines I prefer to keep r-dependence explicitly visible

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The last expression answered the question about the potential energy of the charge q 0 due to interaction with the other point charges q1, q2, …, r x y q0q0 r1r1 q1q1 r1-rr1-r r3-rr3-r q2q2 r2r2 r2-rr2-r q3q3 r3r3 Those point charges q1, q2, …, interact as well. Each charge with all other charges If we ask for the total potential energy of the collection of charges we obtain This is the energy it takes to bring the charges from infinite separation to their respective fixed positions r i makes sure that we count each pair only once

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What is the speed of charge q after moving in the field E from the positive to the negative plate. Neglect gravity. Clicker question d 1) 2) 3) 4) 5) None of the above

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Goal: Making the potential energy a specific, test charge independent quantity We are familiar by now with the concept of creating specific quantities, e.g., Force on a test charge Electric field: test charge independent, specific quantity Gravitational potential energy test mass independent, specific potential Electric potential V Specific, test charge independent potential energy. The SI unit of the potential is volt (V).

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Meaning of a potential difference Point b Point a W a->b work done by electric force during displacement of charge q 0 from a to b. Voltage of the battery Alternatively we can ask: What is the work an external force, F, has to do to move charge q 0 from b to a This force is opposite to the electric force, F el, above. Hence:

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We know these two alternative interpretations already from mechanics z a b F g =-mg a b F=mg To slowly ( without adding kinetic energy ) move mass from b to a we need an external force acting against gravity

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From and We obtain the potential difference (voltage) from the path independent line integral taken between points a and b

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point a becomes variable point in distance r Let’s calculate the potential of a charged conducting sphere by integrating the E-field R r We start from point b becomes reference point at r for r>R :=0 For r

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An important application of our “potential of a conducting sphere”- problem R According to our considerations above we find at the surface of the conducting sphere: There is a dielectric breakdown field strength, E m, for all insulating materials including air For E>E m air becomes conducting due to discharge max potential of a sphere before discharge in air sets in depends on radius From Wartenberg pinwheel charged to a very high voltage Note, that discharge sets in at regions of small R

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Demonstration: Surface Charge Density

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How do we actually measure the charge on the proof plane ?

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