Electric Potential

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 ra torb x2 In general we have to specify how we get from a to b, e.g., for friction force However, for a conservative force such as gravity we remember x1 where  is the potential energy

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

Electric potential energy Can we find a function U =U (r) such that is the force exerted by a point charge q on a test charge q0 ? We expect the answer to be yes, due to the similarity between Coulomb force and gravitational force Let’s try In fact we see Potential Potential energy simple because of radial symmetry where U(r)=U(r)

We conclude Electric potential energy of electrostatically interacting point charges q and q0 U U qq0<0 attractive potential qq0>0 repulsive potential r r As always, potential defined only up to an arbitrary constant. Expression above uses U(r)=0 as reference point

We know already the superposition principle for electric fields and forces, can we find a net potential energy for q0 interacting with several point charges? Force exerted on q0 by charge q3 at r3 Force exerted on q0 by charge q2 at r2 Force exerted on q0 by charge q1 at r1 Net force q0 experiences q1 y r1 r1-r r3-r q3 q0 r r3 r2-r x r2 q2 Note: textbook on p. 785 defines I prefer to keep r-dependence explicitly visible

The last expression answered the question about the potential energy of the charge q0 due to interaction with the other point charges q1, q2, …, Those point charges q1, q2, …, interact as well. Each charge with all other charges q1 y r1 r1-r r3-r q3 q0 r r3 If we ask for the total potential energy of the collection of charges we obtain r2-r x r2 q2 makes sure that we count each pair only once This is the energy it takes to bring the charges from infinite separation to their respective fixed positions ri

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

Goal: Making the potential energy a specific, test charge independent quantity Electric potential 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) .

Point a Meaning of a potential difference Wa->b work done by electric force during displacement of charge q0 from a to b. Point b Voltage of the battery Alternatively we can ask: What is the work an external force, F, has to do to move charge q0 from b to a This force is opposite to the electric force, Fel, above. Hence:

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

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

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

An important application of our “potential of a conducting sphere”- problem According to our considerations above we find at the surface of the conducting sphere: R There is a dielectric breakdown field strength, Em, for all insulating materials including air For E>Em 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 http://en.wikipedia.org/wiki/File:Plasma_wheel_2_med_DSIR2018.jpg

Demonstration: Surface Charge Density

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

Electric Potential