28 1 SJP Phys 1120 Electric Currents and Resistance So far we ve considered electrostatics charges which pretty much stay put In the demos of sparking Van de Graafs or discharging capacitors we ve seen the important effects of charges moving which leads us to discuss the flow of charges electric currents Electric Currents Whenever charges are free to move e g in conductors if you apply an E field they will move After all F qE acceleration Imagine a wire pick some spot and ask yourself q How much charge passes by that spot each second That s the current Mathematically current is called I Q amount of charge sec t Often written dQ dt or even more sloppily Q t for short The units of current are Coulombs sec C s 1 Ampere 1 A So if I 1 A that means 1 Coulomb flows by each second A LOT Your choice of that little spot is quite arbitrary If you moved it to the left or the right there s no difference The SAME NUMBER of charges pass by each second Evn if you tilted and stretched that area making it bigger the total current flowing past it would still be the same Think about that convince yourself We re just counting charges flowing by Current has a direction If the current is to the right there s a net flow of charge to the right This could occur in one of two ways It could mean s physically moving to the right OR it could mean s physically moving to the left There s almost no difference in terms of flow of charge Think about this it s an important point Negatives moving left are in most ways equivalent to positives moving right The flow of charge is the same in either case 28 2 SJP Phys 1120 Here s another way to think about this Start with two neutral plates Now you could EITHER move some charges down OR move some charges up but either way the final situation is the same Neutral plates OR Ends up in the same final situation Our convention is always to define current I as the flow of imaginary charges Even though in reality it s really negatives going the other way In most conductors it really is negative electrons flowing opposite the conventional current What makes currents flow Generally electric fields make charges move You can also think of it as arising from changes in electric potential energy a change in potential energy means you can convert potential energy into kinetic energy motion like a hill makes water flow down it We ll talk more about this next chapter introducing batteries and the more generic term EMF IMPORTANT Current is NOT the same as voltage not even close Current is the flow of electric charges Voltage is the energy per charge Totally different Your primary intellectual task for the next 2 chapters is to create a clear mental model that distinguishes these things for yourself Some important concepts to be aware of 1 In ideal wires electrons are free to roam around In good perfectly conducting metal it takes zero work to move electrons around Metals like to be at an equipotential throughout if they can There is no voltage drop along ideal wires 2 Charge is conserved In steady state circuits that means there is no buildup of charge anywhere Whatever charge comes in to some point must go right on by and out the other side 28 3 SJP Phys 1120 This leads us to the question of how to represent current in diagrams We generally just draw curvy lines with arrows to indicate the direction of conventional current flow The arrows follow wires usually It s a little funny though mathematically current as we defined it is NOT a vector it s a number positive means current is flowing in the direction shown Negative means current is actually flowing the other way This can be confusing at first Let s do some examples just remember the bottom line of current conservation is whatever current goes in must come out Think about why I2 I1 a I3 At this junction a which might be part of a bigger circuit I1 I2 I3 Current Entering Current Exiting Just be careful to watch the arrows You will be drawing arrows for currents and they might point either way E g in this picture I2 I1 a I3 I1 I3 I2 Current Entering Current Exiting Note the direction of I3 s arrow Fancier example We haven t learned all the symbols in this diagram but never mind it shows a battery some wires and resisters Focus your attention at the junction labeled a Here I in I in I1 I2 I3 I enters the other three exit a What goes in must go out V I2 I1 I out b I3 At the bottom of this circuit I1 I2 and I3 all flow back together at node b where we thus learn that the current flowing back into the bottom of the battery must be I out I1 I2 I3 That s the exact same as was flowing in at the top Current is NEVER eaten up it just flows around circuits 28 4 SJP Phys 1120 Understanding current microscopically current density Let s picture a chunk of metal a wire with charges drifting along in it q The chunk has cross section A and the charges q each drift to the right with on average constant drift velocity to the right vd vd A Since I q t we must estimate how many charges pass through the area A in a given time Think about this for a second in time t ANY charge q which started out in the dashed volume shown that extends back a distance L vd t will make it past the area A dL vd t Do you see that They re all drifting right We want to know how A q MANY will pass A in time t vd The ones that started a distance vd t back will JUST make it All the other ones that started inside that dashed volume certainly pass too Let s define n number of charge carries per volume the volume density of free charges That dashed cube contains volume volume n dL A charges As we argued they ALL make it past area A in time t Thus the total current I q t charges passing by the charge on each one t I n dL A q t n vd t A q t n vd A q We define the current density J current per unit area I A so J n vd q Recap current density J tells you the of charges flowing past some area each second DIVIDED by the area being crossed It s a current DENSITY not the current itself Most books define J as a vector J n q vd This is useful especially if current is not confined in a wire For us we won t worry much about the vector aspect 28 5 SJP Phys 1120 …
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