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!t = amount of charge/sec. 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. 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. OR Neutral plates Ends up in the samefinal situation: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!) 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: 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) = I1 + I2 + I3 (I enters, the other three exit) What goes in must go out. 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! I1 a I2 I3 I1 I3I2 I(in) V a b I(out) I1 a I2 I328-4 (SJP, Phys 1120) Understanding current microscopically: current density: Let's picture a chunk of metal (a wire) with charges drifting along in it. 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 . Since I = Δq/Δt, we must estimate how many charges pass through the area "A" in a given time. Think about this for a second: in time "Δt", ANY charge q which started out in the dashed volume shown, that extends back a distance ΔL = vd*Δt will make it past the area "A". Do you see that? They're all drifting right. We want to know how MANY will pass "A" in time Δt. The ones that started a distance vd *Δt 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 Δt Thus, the total current I = Δq/Δt = (# charges passing by)*(the charge on each one)/Δt I = (n dL A)*(q) / Δt = (n vd Δt A) (q) / Δt = 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
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