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Topic 3 (Electric Current, Power, & Resistance) Readings for Topic 3 (Knight,Jones,Field 4th): Ch.22 + Ch.23

Plan for Topic 3 Quiz on Tue.Feb.06 - its header will look like this:

Physics II Topic 3 Summary

Idea #7: conduction electrons get average velocity, from local E-field pulling between collision stops - - -

(recall from Topic 1) in matter , conduction electrons can move - - - - - - - - - - Some fraction of electrons in bulk material escape their "original host (+) ion core" . . . they are thermally mobile ... and move "freely" thru the surrounding neutral matter     * at room Temperature (295 K) an electron's thermal K   = 3/2 kB T = 6.1E-21 J = .0382 eV       so these electrons (.911E-30 kg) are moving fast: K = ½mv² => v ≈ 116 km/s (at room T),     * but their total Energy is still a few eV negative, trapped in the bulk material, due to + ions' V.       Neighboring atoms are close together, so the U hilltop between their wells is low       (typical Vmin ≈ +6 [V] => Umax ≈ −6 [eV])       so an electron's average U (along its path) is a bit lower yet (about −8 [eV]).

What fraction depends on the kind of atoms in the material (mostly) , and it increases with Temperature

. . . most metal atoms have essentially one electron able to conduct
      (gold, silver, copper: 97.1% at T=300Kelvin → 97.4% at 330[K] ... almost constant w/Temperature)
    * random directions for thermal motion implies that their average velocity vector is usually = 0
      (so their average momentum = 0, and their average current = 0 if there is no E-field applied )
    * they typically collide with lattice defects in a fairly short distance , and very short time
      (distance about 40 nano-meters, in copper ... takes Δt = d/v ≈ ¹/₃ pico-second between collisions )
    * WITH an Electric field applied, their acceleration (a=qE/m) drifts them a wee bit into the E-field before colliding.
      Δx=½ a Δt ² ; with a stiff (for inside a metal) E = 0.040 [V/m] , a= −7 Giga-m/s² => Δx= −0.39 femto-meter )
    => average velocity becomes non-zero, due to the small "drift velocity" : v_avg = Δx/Δt ≈ 1.2 [mm/s]
      (this example uses values for copper wire supplying a typical house lamp ... good conductor, so not much E .

. . . in semi-conductors, very few electron have enough Energy to conduct (so as T=300 → 330[K] the number might triple)
      (the fraction with Thermal Energy E is roughly exp(−E/k_BT) . . . k_BT is 39 [milli-eV] at 300[K] "room Temperature" ...
      (but they need typical "conduction band" Energy E_c≈½[eV] => exp(−.5/.039)= 2.7E−6 ; at 330[K] => 8.7E−6)
    * but it is fairly easy to get 8000 [V/m] in Ge or Si ... 0.8[V]/0.1[mm] to light an LED ... so the rare conduction electron accelerates 20,000× quicker.
      Their drift velocities are 20,000× as fast, but they are so sparse that their current density is only 1/20 as much as in metal.
      to handle the same current, semiconductors need a bigger cross-section Area.

. . . mobile charge is very dense in metals :
      ρ_q = Q_mobile/Vol = (Q_mobile/atom)(atoms/Vol) ≈ (−1)(8.5E28) [e/m³] = 14E9 [C/m³]
      so even a very weak E-field moves a lot of Coulombs rapidly (recall that a mole of electrons is 96,000 C)
=> we can usually ignore the potential difference (voltage) along a wire ... (unless very long, or poor conductor, or overloaded ...)

7.a: charge passing through an Area is Indicated current ; Indicated current is charge flow rate - - - - - - - - - - - - - -
. . . analogy with mass current in a water stream or in wind : more stuff crosses an Area if the material is denser
      more stuff crosses if material is moving faster . . . more stuff crosses a larger Area
=> I = Δq / Δt = ρ_q v · A   . . . where ρ_q is the charge density [C/m³] that has average velocity v .
      in a wire those are "mobile electron density" and the "drift velocity" detailed above.
. . . Indicated electric charge current is the total current density   j = ρ_q v   piercing (  |  ) the Area
      like momentum density, but (+) mass replaced by (±) charge.

In a complicated material, especially solutions (for electroplating, electrophoresis, chromatography ...):
I is the sum of all current densities for each type of (mobile) constituent ... times the same Area.
. . . different ions have different mass (mitigating acceleration) and different size (reducing velocity)
      so each ion species will have a different average velocity.
. . . different ions have different charge ! +/− will be pushed/pulled in opposite directions:
      negative charge density with negative velocity (say, electrons e⁻, F⁻, Cl⁻ ... moving leftward or down)
      and positive charge density with positive velocity (say, protons H⁺, Li⁺, Ca⁺⁺ ... moving rightward or up)
      both have positive sign for current density ... a vector rightward or up ... j points along the (+)'s velocity !
. . . since charge is additive, total charge flow rate is the sum from each type of ion;
      since it is not specific to any one ion "species", it is called Indicated current :
=> I = Σ (ρ_q v) ·A = Δq / Δt     . . .   this is what current is

7.b: electrons' average velocity is proportional to E , since random thermal motion dominates - - - - - -
Time from one collision to the next (τ) is short at high Temperature but is almost independent of E-field {except in gases}
      average velocity of electron is caused by E ; random thermal motions average to zero
. . . average velocity is v_avg = ½ (v_t=0 + v_τ) = ½ a τ . . . m a = q E , so
      v_avg = ½ e E τ / m   . . . (τ depends on material properties and Temperature) .
. . . I = ρ_q v·A = ½ (ρ_q τ / m) E·A ... but voltage is   δV = E·δx :
=> I = { ½ ρ_q τ / m } ( A/δx ) δV . . . { material properties } ( object's form ) imposed voltage
      this is "Ohm's Law" ... derived via the classical "Drude model" (minor temperature tweak needs Topic 9)

7.c: Lab-scale results depend on the object's material and geometry - - - - - -
properties of the material are collected - - - - - - - - - - - - - -
. . . into conductivity   σ = j / E
=> I = {σ} (A/ℓ) δV . . . object conductance G = σ A/ℓ
      big conductance allows a lot of current easily (at low voltage)
. . . or into resistivity   ρ_r = E / j = 1/σ     <= notice, this   ρ_r   is not charge density ρ_q !
=> δV = I {ρ_r} (ℓ/A) . . . object resistance R = ρ_r ℓ/A
      so high resistance allows little current thru, even withstanding high voltage .

Ge example:   Suppose I have a crystal of pure Germanium (ρ_r = 0.46 [Ω·m]) that is 20[mm]×20[mm] and 5[mm] thick.
. . . to push current thru the wide, short direction, its Area would be .020[m]×.020[m] = .0004m² that the E and v vectors pierce ;
      its length along E would be .005[m] , so   => R = 0.46 [Ω·m²/m] (0.005[m]) / 0.0004[m²] = 5.75 Ω
      ... to push 5 Amps thru that crystal would take   V_GE = I R = 5 A · 5.75 Ω = 28.75 V ... at room Temperature.
. . . most materials' resistivity (1/conductivity) does not vary much at different E-fields (until near breakdown)

=> look-up material resistivities in a table (Wikipedia's) - includes Temperature coefficients !

Layered devices like diodes have one layer "doped" with a few % of Positive ions,
      directly contacting a layer doped with a few % of Negative ions ... called a "PN junction" .
. . . this set up a very steep E-field across the junction , so current cannot flow from N to P
      but current can flow easily from P to N , if the voltage across the junction is more than 0.6 V (typical "turn-on")
=> diodes act essentially like 1-way valves (in water-pumps & blood veins) ... turn-on is higher voltage for LED's ... 3.2 V for violet

δVab = Va − Vb = V2 = I2 R2   is a statement about one device ... not a circuit! . . . Use subscripts to identify which device each statement is about ! => letter different locations in a circuit:     a is one end of a device, b is the other end (as high potential is relative to low potential); => number devices: B1 is battery #1, R2 is resistor #2, S4 is switch #4, C5 is capacitor #5 . . . this circuit is   1,2,(3||(3,4)) ... a is at R2's "high" end, b its "low" end       and a node where I splits into left branch & right branch (parallel)       ... c is between switch and capacitor, d is a node where the branches rejoin .

If the Temperature changes , the resistance for many devices will change in an "almost linear" way :
metals typically have resistance that is proportional to Temperature , in Kelvin
. . . so they have twice the resistance in a 500[K] oven than they do in a −20 °C chest freezer
      wires in a toaster ... their resistance goes to 4× R_o while they're glowing at 1100[K]
      . . . (so a toaster's current is very large only as it starts)
      and their resistance becomes very small near absolute zero (nearly superconductor).
. . . ΔR/R_o ≈ ΔT/T_o . . . T_o is traditionally 0[°C] = 273[K] , but might be 20°C
      describing this (almost-constant) slope a slightly different way,
=> ΔR/R_o = α ΔT   . . . where α for metals ≈ 1/273[K] = 0.0037 [1/K]

semi-conductors typically resist less at high Temperature   (ρ_q is higher).
. . . pretending that their resistance at their working Temperature is "part of a straight line"   ( | \    )
=> ΔR/R_o ≈ α ΔT . . . where for example,   Germanium's α ~ −0.048[1/K] , at 20°C = 293K
      . . . at that slope, α ~ −1/20[K] , would make R → 0 by T = 313[K] ... but R(T) curves, so it does not cross zero .

Ge example 1 (above): if ΔR = −R₂₀, then that R_final = R_initial + ΔR = 0 .
      so ΔR / R = −1 = α ΔT , so ΔT = −1/α = (+)1/0.048 = 20.8°   so T_final = T_initial + ΔT = 20°C + 20.8K = 40.8°C .
. . . Ge example 2: What would that big crystal's resistance be at liquid N2 Temperature (77K)?
      ΔR = R α ΔT = (5.75 Ω)(−0.048 1/K)(77K − 293K) = (+)59.6 Ω   so   R₇₇ = 5.75 Ω + 59.6 Ω = 65 Ω .
      ... disclamer: Ge's resistivity curves up so much that its resistance at 77K would be about 2500 Ω (has a maximum there)

designed devices like various thermistors are made with resistances that increase (or others that decrease) "sensitively"
      PTC's increasing to 50 R₀ with ΔT=25[K] ... you choose T₀ at 50°C, 75°C, 100°C, or 125°C
      (or NTC's dropping to ¼ R₀ with ΔT=25[K] ... smooth curve)

Circuits use Resistors and Conductors - - - - - -

Parallel : Currents in Parallel Add ... Conductance Areas in Parallel Add
Total Current going into a junction (or other device) is equal to the total current coming out of it
. . . this is true after a pico-second or so (first few femtosec, charge builds up on the surface of the solder spot...)

If the paths are parallel, they re-converge at the bottomG_|| = G₁ + G₂ + G₃ + ... .
=> 1/R_|| = 1/R₁ + 1/R₂ + 1/R₃ + ... .

Notice that Conductances carry charge "like" Capacitances ... but Resistances relate reciprocally.

Series : Voltages in Series Add ... Resistance Lengths in Series Add
Total of all the Voltages around any closed circuit (in same clockwise/ccw sense) is zero
. . . simply, the battery's Potential boost (+) is lost (along each passive route) going back to the P.S.(−)

R_series = R₁ + R₂ + R₃ + ... .

Current from High Potential to Low Potential Transforms Power - - - - - -

I = dQ/dt , so if the "landscape" δV remains constant along a path
. . . (due to a power supply continually replacing the charges that flow thru that path)
=> charge Q loses electric Potential Energy δU during Δt
      the mobile charges collide with substrate ion cores (after being pushed by E-field for time  τ)
      so they hit with more Kinetic Energy than the substrate ion has (only thermal E, since they don't move).
. . . these collisions transfer Energy from the mobile charges (Electric Potential Energy) to the substrate (Thermal Energy);
      a toaster wire gets hot when current flows thru it (from high Potential to low Potential).
=> = I · δV ... = I R ² ... = δV ² /R
      I is the current thru that device , and δV = V_R is the voltage across that device .

Often a variable Resistor is wired in series to control the amount of current (hence Power) that the load Resistor gets.
. . . current thru the load decreases as the variable Resistor's value increases (you turn down the loudness knob)
      the load's power is   = I_load V_load = V_PS / (R_vary + R_load) · V_PS / (R_vary + R_load) (R_load)
=> = V_PS² R_load / (R_vary + R_load)² . . . from V_PS²/R down to 0 as R_vary gets large
. . . the Energy efficiency η = Power to the load / Power from the supply
      if the input voltage is constant , η decreases from almost 100% down to roughly 0% ...

A transistor is a 3-layer device , with one PN junction and another N or P layer they start to conduct when the PN turns on
typical resistance varies from about 1000 kΩ when its "base" is at 0.5 V , down to 2.5 kΩ or lower when its base is raised to 0.8 V or above
. . . The Power-Supply adjustment knob raises or lowers the base voltage (gradually, by hand)
      another transistor can raise this one's base voltage very fast, so it acts as a voltage-controlled switch.

Resistor & Capacitor Circuits - - - - - - a Capacitor has voltage because charge has accumulated on its plates, . . . but a Resistor has voltage because charge traverses it. a Resistor in series with a Capacitor will slow the Capacitor's charging . . . current thru the R drops its voltage in R (not in C) . . . but the current's charge   ΔQ = I Δt   changes the Capacitor's voltage.       how fast? I0 = Vbattery/R   and Q∞ = V∞ C => looks like the current will fill the capacitor in time   Δt = Q/I = RC   a big capacitor will need more time "to fill" (or to empty) than a small capacitor . . . the more resistance to flow, the less current, so the longer it will take. R C is the "relaxation time" for the circuit to change ... abbreviated "τ" ← "lower-case greek "tau"       UNITS: [ Ω F ] = [ Volt/Amp · Coulomb/Volt ] = [ C / (C/s) ] = [s] . as current continues thru R, charge accumulates on C ... so the Resistor's voltage gets less steep . . . when a swich opens or closes, everything changes most rapidly at first, getting slower and slower , after while the Resistor's voltage goes as the time-slope of the Capacitor's voltage . . . the Capacitor's charge finally quits changing, only as the current (and VR) becomes zero.       the difference from how it will eventually be (after ∞ time) is an exponential decay. . . . with RC = τ   is the time needed to accomplish most of the change ... 63% of the change ... all but 37% of it => It = I0 exp(−t / τ)