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CU-Boulder PHYS 1120 - Electric Currents and Simple Circuits

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I-1Electric Currents and Simple CircuitsElectrons can flow along inside a metal wire if there is an E-field present to push them along (F q E=r r). The flow of electrons in a wire is similar to the flow of water in a pipe.Definition: electric current QItD=D = rate of flow of chargeunits [ I ] = coulomb/second = 1 C / 1 s = 1 ampere (A) = "amp""It's not the voltage that kills you, it the amps." About 0.05 A is enough to kill you.If current I = 1 A in a wire, then 1 coulomb of charge flows past any point every second. In electrostatic problems, E 0=r inside a metal, but if I  0, then the situation is not static, the E-field is not zero.Electrons flow in metals, not protons, so (–) charges are moving when there is a current. The electron feel a force F e E=-r rand goes "upstream" against the E-field.The flow of (–) charge in one direction is electrically equivalent to the flow of (+) charge in the opposite direction:Last update: 1/13/2019 Dubson Phys2020 Notes, University of Colorado Ineutralplates() or (+)either way, get:+ + + +   EI-2By convention, we define current I as the flow of imaginary (+) charges, when it is really (–) charges flowing the other way: (Some texts refer to I as the "conventional current" to distinguish it from the "electron current".)Example: How many electrons flow past per second when the current is 1 A? 18 119 19Q N e N I 1 A 1 C / sI 5.6 10 st t t e 1.6 10 C 1.6 10 C-- -D D � D= = � = = = = �D D D � �About 0.01 A = 10 mA flowing though your heart is lethal, yet I could grab a wire carrying 1000A and be safe! Why? Because my body has a much higher electrical resistance than the metal. The electrons prefer to flow through the metal wire.For most materials, the current I is proportional to the voltage difference between the ends.I E (since F = q E) and V E, so I V� D � � Dr rFrom now on, we usually follow the (bad) convention and write "V" when we really mean "V".VI V ( really I V) constantI� � D � =Definition of resistance R (of a piece of wire or other material): VRI�The experimental fact that (for most materials) the ratio R = V / I is a constant, independent of Vor I, is called Ohm's Law : VR constantI= =, usually written V = I R (R constant)Units: [R] = volt / ampere = ohm () ["" is Greek letter omega]Last update: 1/13/2019 Dubson Phys2020 Notes, University of Colorado IE Ihi Vlo VVI-3Ohm's Law should be written V = I R, but the bad convention is to write V = I R."Ohm's Law" is not really a law, because it is not always true. For many materials, Ohm's Law isapproximately true, the resistance R is approximately constant, independent of V or I. Materials that obey Ohm's Law are called "ohmic materials". But some materials are "non-ohmic"; they donot obey Ohm's Law. The average speed of electrons in a current-carrying wire results from a competition between twoeffects: (1) the E-field, which causes an acceleration according to F q E m a= =r rr, making the electrons go faster and faster, and (2) the scattering of electrons due to impurities and thermal vibrations, which act like friction, making the electrons slow down.For typical currents in real wires, the average electron speed (often called the drift velocity) is actually quite slow, typically less than 0.1 mm/s. (Incidentally, the term drift velocity is incorrect,it should be called the drift speed.)A material with lots of electron scattering has a high resistance:Rwire << 1 , Rhuman  105  4535V 10 VI 10 A (harmless)R 10V 100 VI 10 A (painful!)R 10--= = =W= = =W  10 V safe, 100 V dangerous !The resistance R of a piece of material depends on its shape and composition.Shape: long and skinny  R big short and fat  R small — just like the flow of water through a pipe. Long skinny water pipes resist flow of water.Turns out that LRA�, so big L means big R, big A means small RLast update: 1/13/2019 Dubson Phys2020 Notes, University of Colorado area ALI-4LRAr= (Greek letter "rho") = resistivity( We show were this comes in the next section below.)Resistivity  depends on the material composition, not on the shape.  is a measure of the scattering of electrons in that material, like viscosity of fluid in a pipe. Big  means lots of scattering (friction), big resistance to flow.Units: {length lengthlengthARL�r = � [  ] = [R]  length = mmaterial  useCu 1.7  10-8 m house wiringAl 2.7  10-8 m power linesW (tungsten) 5.3  10-8 m (cool)60  10-8 m (hot)light bulb filamentsFe 9.8  10-8 m not used in wiringglass 10+10 m electrical insulatorMicroscopic view of Ohm's Law.Definition: current density IJA= (current per area in a conductor). We also define current density vector J where direction of J is the direction of the current.J is related to the average speed vdrift of the charge carriers (usually electrons) by J = n q vdrift ,n is the number of carrier per volume, q is the charge per carrier (usually q = e).Proof: Consider a wire with carrier density n (#/volume), cross-sectional area A, and drift speed vdrift. In a time t, all the charges move an average distance x = vdrift t.Last update: 1/13/2019 Dubson Phys2020 Notes, University of Colorado area AJx = vdrift tvolume = A  xI-5The charge Q in the length of wire x is number of carriers chargeQ volume n q A xvolume carrierD = � � = DSo the current density is driftI Q 1 n q A xJ n q vA t A A tD D= = � = =D D Done.In ohmic materials, the current density J is proportional to the electric field E J Es=v v , where the proportionality constant  is called the conductivity.The resistivity  is defined as 1rs= and so E =  J.J Es= or E Jr=, where  = 1/ = constant is the microscopic version of Ohm's Law. We now show that this is equivalent to V = I R : Consider a cylindrical section of conductor, length L, cross-sectional area A, with current density J, and E-field E.Start with E Jr= , now multiply both sides by L and write Jas J = I / A.  IE L LAr= . Notice that V = E L. So we have LV IArD =, or V I RD =, where we have defined the resistance R as LRAr=. We have just shown that E Jr= is the same as Ohm's Law V = I R, where LRAr=.Last update: 1/13/2019 Dubson Phys2020 Notes, University of Colorado ALJ, EI-6A


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CU-Boulder PHYS 1120 - Electric Currents and Simple Circuits

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