I-1 Electric Currents and Simple Circuits Electrons can flow along inside a metal wire if there is an E-field present to push them along (). The flow of electrons in a wire is similar to the flow of water in a pipe. FqE=GG Definition: electric current QIt∆=∆ = rate of flow of charge units [ 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. I In electrostatic problems, inside a metal, but if I ≠ 0, then the situation is not static, the E-field is not zero. E=G0EnElectrons flow in metals, not protons, so (–) charges are moving when there is a current. The electron feel a force Fed goes "upstream" against the E-field. = −GGaE The flow of (–) charge in one direction is electrically equivalent to the flow of (+) charge in the opposite direction: neutral plates (−) or (+) ++++either way, get: −−− −Last update: 9/25/2009 Dubson Phys2020 Notes, ©University of ColoradoI-2 By convention, we define current I as the flow of imaginary (+) charges, when it is really (–) charges flowing the other way: I (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 19QNe NI 1A 1C/sI5t t t e 1.6 10 C 1.6 10 C−−−∆∆⋅ ∆== ⇒ == = = ×∆∆ ∆ ××.610s∝∆ 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∝∆∝GG lo V hi V E I ∆V From now on, we usually follow the (bad) convention and write "V" when we really mean "∆V". VI V ( really I V) constantI∝∝∆⇒= 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 V or I, is called Ohm's Law : VR constantI==, usually written V = I R (R constant) Units: [R] = volt / ampere = ohm (Ω) ["Ω" is Greek letter omega] Last update: 9/25/2009 Dubson Phys2020 Notes, ©University of ColoradoI-3 Ohm'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 is approximately 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 do not obey Ohm's Law. The average speed of electrons in a current-carrying wire results from a competition between two effects: (1) the E-field, which causes an acceleration according to , 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. FqEm==GGGa 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 Ω, Rhuman ≈ 105 Ω 4535V10VI10A (R10V 100VI 10 A (painful!)R10−−== =Ω== =Ωharmless) ⇒ 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∝, area A L so big L means big R, big A means small R Last update: 9/25/2009 Dubson Phys2020 Notes, ©University of ColoradoI-4 LRAρ= ρ (Greek letter "rho") = resistivity ( We show were this comes in the next section below.) Resistivity ρ depends on the material composition, not on the shape. ρ is a measure of the scattering of electrons in that material, like viscosity of fluid in a pipe. Big ρ means lots of scattering (friction), big resistance to flow. Units: Nlength lengthlengthARL×ρ = ⋅ [ ρ ] = [R] × length = Ω⋅m material ρ use Cu 1.7 × 10-8 Ω⋅m house wiring Al 2.7 × 10-8 Ω⋅m power lines W (tungsten) 5.3 × 10-8 Ω⋅m (cool) 60 × 10-8 Ω⋅m (hot) light bulb filaments Fe 9.8 × 10-8 Ω⋅m not used in wiring glass 10+10 Ω⋅m electrical insulator Microscopic 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: 9/25/2009 Dubson Phys2020 Notes, ©University of ColoradoI-5 area A volume = A ⋅ ∆x J The charge ∆Q in the length of wire ∆x is ∆x = vdrift ∆t number of carriers chargeQ volume n q A xvolume carrier∆ =××=∆ So the current density is driftIQ1nqAxJnAtAAt∆∆== ⋅ ==∆∆qv Done. In ohmic materials, the current density J is proportional to the electric field E JEσ=KK , where the proportionality constant σ is called the conductivity. The resistivity ρ is defined as 1ρσ= and so E = ρ J. JEσ= or E, where ρ = 1/σ = 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. JJρ= Start with Eρ= , now multiply both sides by L and write J as J = I / A. ⇒ IEL . Notice that ∆V = E L. So we have LAρ=LVIAρ∆, or ∆, where we have defined the resistance R as = R=VILRAρ=. We have just shown that EJρ= is the same as Ohm's Law ∆V = I R, where
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