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Current and Resistance Putting electrons in motion Electron movement through conductors Resistivity and Resistance – Ohm’s Law Electrical Power Current Current : Flow rate of charges through a surface area. The direction of the current is the direction of the flow of positive charges. ΔQ I av = Δt dQ I (t ) = dt (C/s=Amperes) Microscopic Model ΔQ = ( nV ) q ΔQ = ( nAΔx ) q ΔQ = ( nAvd Δt ) q ΔQ I av = = nqvd A Δt vd : Drift Velocity Drift Speed In Copper Wire A = 3.31 x 10-6 m2 I = 10 A ρCu = 8.92 g/cm3 ΔQ I av = = nqvd A Δt 10 A = nvd 1.6 ×10 −19 3.31× 10 −6 ( V= m = 63.5 g/mol NA = 6.02 x 1023 at/mol 1 electron per Cu atom m ρ = )( ) 63.5 = 7.12cm3 / mol 8.92 N A 6.02 × 10 23 = = 8.46 × 10 22 el / cm3 = 8.46 × 10 28 el / m 3 n= V 7.12 10 −4 vd = = 2 . 23 × 10 m/s 28 −19 −6 1.6 × 10 3.31× 10 8.46 ×10 ( )( )( ) Ohm’s Law Current Density: I J = = nqvd A J = nqv d (for a uniform current perpendicular to the cross-section) General vector definition In some materials, the field applied and the current density are proportional. Ohm’s Law: J = σE Most metals obey Ohm’s Law σ: conductivity Resistance ⎛ l ⎞ ΔV = El = l = ⎜ ⎟ I σ ⎝ σA ⎠ J l ΔV R= ≡ σA I Metal Semiconductor diode (V/A=Ω) Resistivity Resistivity is the inverse of conductivity. Both are intrinsic properties of the material. Resistance is a also a function of the shape and size of the device. 1 l ρ= R=ρ σ A Material ρ (Ω.m) Copper 1.7x10-8 Gold 2.44x10-8 Aluminum 2.82x10-8 Silicon 640 Rubber ~1013 Resistance of a Conductor A=2x10-4 m2 ρAl = 2.82x10-8 Ω.m 10 cm RAl = ρ Al ρGlass = 3x1010 Ω.m l = 2.82 ×10 −8 A ) ( ) RGlass = ρ Glass ( ⎛ 0.1 ⎞ −5 1 . 41 10 = × Ω ⎜ ⎟ −4 ⎝ 2 ×10 ⎠ l ⎛ 0.1 ⎞ 13 = 1 . 5 × 10 Ω = 3 ×1010 ⎜ ⎟ −4 A ⎝ 2 ×10 ⎠ Resistance of a Coaxial Cable ρ ⎛b⎞ R= ln⎜ ⎟ 2πL ⎝ a ⎠ A Model for Electrical Conduction: The Drude Model Atoms Electrons A regular array of atoms surrounded by a “cloud” of free electrons Random movement under zero field Random movement modified by a field field Drude Model F = qE = mea qE a= me J = σE qE v f = v i + at = v i + t me Now take the average over all times. Then vi=0 (random movement), v f = vd = nq 2 E τ J = nqvd = me qE qE τ t= me me , where t is the mean time between collisions nq 2τ σ= me me ρ= 2 nq τ l = τ ⋅ vd is the mean free path Resistivity and Temperature Resistivity in metals is linear with temperature over a limited range ρ = ρ 0 [1 + α (T − T0 )] R = R0 [1 + α (T − T0 )] α : temperature coefficient of resistivity 1 Δρ α= ρ 0 ΔT (C-1) Electrical Power ΔU battery = ΔQΔV I ΔU resistor (heat ) = ΔU battery + - ΔV R ΔU ΔQΔV = = IΔV Electrical Power Δt Δt V.A = Watts(W) P = IΔV ( ΔV ) R= 2 Symbol of a Resistor P =I 2 R Power Dissipated on a Resistor Rank the Currents Ia=Ib Ic=Id Ie=If Ia=Ic+Ie P60W > P30W Ic>Ie Ia=Ib> Ic=Id>Ie=If Summary Current is the net rate of charge flow. Electrons move at the drift velocity. Resistance is the ratio of voltage applied to current. The ratio is linear for most metals. Resistivity is a material property. Electrical energy will be converted to thermal energy on a resistor. The rate of conversion is the power. For Next Class Reading Assignment Chapter 28 – Direct Current Circuits WebAssign: Assignment 5