29 30 1 SJP phys1120 Electric Potentials We ve been talking about electric forces and the related quantity E F q the E field or force per unit charge In 1110 after talking about forces we moved on to work and energy Quick Review of work and energy The work done by a constant force F moving something r r through a displacement d is W F d F d Fd cos F More formally if F varies as you follow some path r r W F dr d E g if you an external force lift a book at constant speed up a distance d Newton II says F net ma i e F ext F g 0 F ext because remember if speed is constant a 0 or F ext mg You do work W ext F ext d mgd The sign is because is 0 degrees your force is UP and so is the displacement vector The gravity field does W field F g d mgd The minus sign is because is 180 degrees the force of gravity points DOWN while the displacement vector is UP F g mg The NET work done by all forces is W ext W field 0 that s just the workenergy principle which says W net KE 0 here You did work Where did it go NOT into KE it got stored up it turned into potential energy PE In other words F ext did work which went into increased gravitational potential energy For gravity we defined this potential energy to be PE mgy so PE mg y final y initial mgd W ext The change in PE is all we ever cared about in real problems Summary If you do work on an object in a conservative force field r r dr PE W by you W by the field F field Knight generally uses the symbol U for Potential Energy by the way 29 30 2 SJP phys1120 Now let s drop the book and see what happens There is no more external force touching the book like me in the previous example only gravity acts Neglect friction i F g mg d Energy conservation says 1 i e mgd 0 0 mv 2f 2 PEi KEi PE f KE f f This formula gives a quick and easy way to find v f The concept of energy and energy conservation is very useful Another way of rewriting that equation is PE PE KE KE 0 f i f i i e PE KE 0 or Etot 0 If only conservative forces act U i e PE is independent of the path taken End of quick review of work and energy There is an electric analogue of the above examples Consider 2 charged parallel metal plates called a capacitor a fixed distance d apart Between the plates E is uniform constant and points from the towards the plate f E d q i Imagine a charge q initially located near the bottom plate The force on that charge is F E qE down do you see why Let s totally neglect gravity here Now LIFT q from the bottom to the top at constant speed r r You do work W ext Fext dr Fext d qEd r r The Electric field does W field FE dr FE d cos 180 qEd 1 Do you understand those signs Think about them Just like the previous case you did work but where did it go As before it didn t turn into KE it turned into potential energy We say the charge s electrical potential energy has increased U U final U initial qE y final y initial qE d W ext where y is the distance above the negative plate We lifted the charge from a region of LOW PE near the plate to a region of HIGH PE near the plate Note up and down are irrelevant here you could turn the picture on its side or even upside down It s not gravity in this story it s 100 electrical energy 29 30 3 SJP phys1120 Just like we defined E F q dividing out q gives force unit charge let s now define something we call electrical potential or just potential V PE q Calling this quantity potential is really a VERY bad name because this potential is DIFFERENT from potential energy Potential has units of energy charge Joules Coulomb J C We call 1 J C 1 Volt 1V People use the symbol V for the unit volt as well as for the quantity itself Another bad choice but we have to live with it Voltage and potential are basically synonyms A change in potential is called a potential difference U Br r AB E VAB VB VA dr q A and from this the change in potential energy U PE q V Potential is a number a scalar You assign this number to places points Potential difference is defined even if there are no charges moving around The sign of potential differences is meaningful Example A car battery maintains 12 V between the terminals If the headlights contain a 36 W bulb how much charge is the battery moving through the bulb each second And how many electrons is that Answer 36 W 36 Watt 36 J s Each second 36 Joules of energy are dissipated in a bulb This energy all comes from the loss of potential energy as charges flow from one terminal through the bulb to the other terminal If a charge q drops 12V the energy lost is PE q V or q 12V Each second 36 J are lost i e 36 J q 12 V or q 36 J 12 V 36 J 12 J C 3 C That s a lot of electric charge being moved by a car battery The number of electrons going through the bulb each second is 3C 1 6E 19 C electron 2E19 electrons A heck of a lot I was a little sneaky about signs the charge of an electron is negative just think about it Here s a related question for you given that it s negative electrons that flow out of a battery which way do they go from the terminal through the bulb to the or the other way The answer is from to Electrons are repelled from and attracted towards 29 30 4 SJP phys1120 For a parallel plate capacitor we just found two pages ago PE qE d which means V PE q qEd q E d Here s another sketch of a capacitor b V is high here DeltaV V b V a Ed 0 d a V is low here With gravity you can choose to call zero potential energy wherever you want You might choose sea level or the tabletop or the ground It s the same story with electricity you can pick any spot you want and call the electrical potential energy 0 there We usually call this point the ground Let s call point a in the diagram the ground or 0 potential Now put a charge q at the point …
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