ch27 - Currents and Ohm’s Law Chapter Outline ENGINEERING...

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Unformatted text preview: Currents and Ohm’s Law Chapter Outline ENGINEERING PHYSICS II Electric Current PHY 303L Resistance and Ohm’s Law Resistivity of Materials Chapter 27: Currents and Ohm’s Law Resistances in Combinations Maxim Tsoi Physics Department, The University of Texas at Austin http://www.ph.utexas.edu/~tsoi 303L: Current and Resistance (Ch.27) 303L: Current and Resistance (Ch.27) Electric Current Electric Current Flow of electric charges Microscopic model of conduction in a metal • Charges move perpendicular to a surface of • ∆Q is the amount of charge that passes through • The average current I= • If the carriers move with a speed vd their displacement ∆x = v d ∆t in ∆t is this area in a time interval ∆t • The instantaneous current • Consider conductor of cross-sectional area A with n charge carriers per unit volume area A ∆Q I av = ∆t dQ dt • The total charge through a given cross-section in ∆t is ∆Q = Nq = (nV )q = (nA∆x )q = (nAv d ∆t )q I av = • The average current in the conductor ∆Q = nqvd A ∆t • The SI unit of current is the ampere (1 A= 1 C/1 s) • The current direction is the direction of the flow of positive charge • vd (opposite the direction of flow of charge carriers - electrons - in conductors) is an average speed of charge carriers called the drift speed The current through a surface is the rate at which charge flows through the surface 303L: Current and Resistance (Ch.27) 303L: Current and Resistance (Ch.27) Resistance • Current density Resistance Ohm’s law of uniform block of material current per unit area: • If ∆V is maintained across a conductor j= I = nqv d A r r j = nqvd r r j = σE σ • ρ=1/σ • Ohmic conductivity of the conductor resistivity of the conductor obey Ohm’s law 303L: Current and Resistance (Ch.27) • Potential difference ∆V = El E and j are established in the conductor For many materials (including most metals), the ratio of the current density to the electric field is a constant σ that is independent of the electric field producing the current • • Straight wire of cross-sectional area A and length l • Ohm’s law: ∆V j = σE = σ l ⇒ E= ∆V l l ∆V = j = I = R I σ σA l • Resistance R= ∆V I • SI unit of resistance is Ohm (1 Ω = 1 V / 1 A) • Resistance of a uniform block of conductor: R= l σA = ρl A 303L: Current and Resistance (Ch.27) 1 A Model for Electrical Conduction Resistance and Temperature Drude model of conduction in metals Resistivity of conductor • Conductor atoms + • Over a limited temperature range, the resistivity of conduction electrons • E=0 • E≠0 ρ = ρ 0 [1 + α (T − T0 )] random + drift r r qE a= me • Temperature coefficient of r r rr r qE v f = vi + a t = vi + t me ⇒ • Average velocity at time τ (at which the next collision occurs) r vf a conductor varies ~linearly with T random motion r r r r qE qE τ t= = vd = vi + me me • Magnitude of the current density r nq 2 E τ j = nqvd = me resistivity: σ= nq 2τ me me nq 2τ l τ= v ρ= Example 3 303L: Current and Resistance (Ch.27) α= 1 ∆ρ ρ 0 ∆T • Because R ~ ρ we can write: R = R0 [1 + α (T − T0 )] 303L: Current and Resistance (Ch.27) Superconductors Resistors no resistance at low temperatures • Superconductors a class of metals and compounds whose resistance decreases to zero when they are below a certain temperature Tc • Tc in series • For a series combination of two resistors, the currents are the same in both resistors because the amount of charge that passes one resistor must also pass through the other one in the same time interval critical temperature I1 = I 2 = I • Discovered in 1911 by Kamerlingh-Onnes ∆V = IR1 + IR2 = = I ( R1 + R2 ) = IReq Req = R1 + R2 Req = R1 + R2 + R3 + ... The equivalent resistance of a series combination of resistors is the algebraic sum of the individual capacitances (is always greater than any individual resistance) 303L: Current and Resistance (Ch.27) 303L: Current and Resistance (Ch.27) Resistors Resistors in parallel • When resistors are connected in parallel, the potential differences across the resistors is the same application Christmas tree lights connected in series a failed bulb results in an open circuit all bulbs go out. I = I1 + I 2 1 1 ∆V ∆V ∆V I= + = ∆V + = R1 R2 R1 R2 Req 1 1 1 = + Req R1 R2 1 111 = + + + ... Req R1 R2 R3 If the filament breaks and the bulb fails, the bulb’s resistance increases dramatically most of the applied voltage appears across the failed bulb the high voltage burns the jumper insulation, causing the metal wire to make electrical contact with the supports thus providing a conducting path through the failed bulb. The inverse of the equivalent resistance of resistors connected in parallel is equal to the sum of the inverses of the individual resistances 303L: Current and Resistance (Ch.27) 303L: Current and Resistance (Ch.27) 2 Electrical Power SUMMARY rate of energy transfer Currents and Ohm’s Law • Follow a charge Q moving clockwise around the circuit • Electric current in a conductor: I= dQ dt • It is related to motion of charged carriers through: • From A to B electric potential energy of the system increases by Q∆V (chemical potential energy in the battery decreases) • From C to D the system loses this potential j= • Current density: • Ohm’s law: I av = nqvd A I = nqv d A j = σE • Resistance of a conductor: R= ∆V I R= ρl energy (transformed into internal energy) • Rate of energy loss: • Resistance of a uniform block of material: dU d dQ = (Q∆V ) = ∆ V = I∆ V dt dt dt • Drift velocity of current carriers in a uniform electric field: • Resistivity of a metal: A me ρ= 2 nq τ • Temperature dependence of resistivity of a conductor: • Power delivered to the resistor: • Joule heating P = I∆ V = I 2 R = ∆ V 2 R process of loosing power as internal energy in conductor of resistance R 303L: Current and Resistance (Ch.27) r qE vd = τ me ρ = ρ 0 [1 + α (T − T0 )] • Power (rate at which energy is supplied to an element): P = I∆V = I 2 R = ∆V 2 R 303L: Current and Resistance (Ch.27) 3 ...
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