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# Lecture 13 - Purdue University

August 2018

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Chapter 19 A Microscopic View of Electric Circuits E and Drift Speed In steady state current is the same everywhere in a series circuit. Ethin Ethick i i What is the drift speed? i = nAv nAthin vthin = nAthick vthick vthin Athick = vthick Athin Note: density of electrons n cannot change if same metal What is E? v = uE uEthin Athick = uEthick Athin Ethin Athick = Ethick Athin Exercise i = nAv 0.1 mm 1 mm vthin = ? vthick = 410-5 m/s Ethin= 10-1 N/C Ethick= ? vthin = 40010-5 m/s Ethick= 10-3 N/C Direction of Electric Field in a Wire E must be parallel to the wire E is the same along the wire Does current fill the wire? Is E uniform across the wire? B C D A A B C D DVABCDA = - ò E1 ×dl - ò E3 ×dl - ò E2 ×dl - ò E3 ×dl = 0 VAB 0 E1 = E2 VCD 0 Electric Field in a Wire What charges make the electric field in the wires? E Bulb filament and wires are metals – there cannot be excess charges in the interior Are excess charges on the battery? E E i = nAuE A Mechanical Battery Van de Graaff generator Electron Current Field due to the Battery E Blue = vdrift Ebends In the steady state there must be some other charges somewhere that contribute to the net electric field in such a way that the electric field points upstream everywhere. Field due to the Battery i = nAuE Surface charge arranges itself in such a way as to produce a pattern of electric field that follows the direction of the wire and has such a magnitude that current is the same along the wire. Field due to Battery E Smooth transition from + surface charge to – to provide constant E. The amount of surface charge is proportional to the voltage. Amount of Surface Charge The average magnitude of E in a closed circuit can vary from ~.01 V/m in copper wire to more than 100 V/m in Nichrome wire - due to a much different electron mobility. What is easy: to draw E and i What is complex: to draw surface charge distribution Connecting a Circuit When making the final connection in a circuit, feedback forces a rapid rearrangement of the surface charges leading to the steady state. This period of adjustment before establishing the steady state is called the initial transient. The initial transient Connecting a Circuit The initial transient Egap faces is due to charges on gap faces Eother Enet Before the gap is closed, the net field in the wire must be zero, because the system is in static equilibrium. Connecting a Circuit The initial transient Speed of light: 30 cm/ns In just a few nanoseconds the rearrangement of the surface charges will extend all the way around the circuit. Connecting a Circuit 1. Static equilibrium: nothing moving (no current) 2. Initial transient: short-time process leading to the steady state 3. Steady state: constant current (nonzero) Surface Charge and Resistors Just after connection: E may be the same everywhere i = nAv = nAuE ithin = nAthin uE ithick = nAthick uE After steady state is reached: ithin = ithick ithin = nAthin uEthin ithick = nAthick uEthick Ethin Athick = Ethick Athin ithin Athin = ithick Athick Energy in a Circuit Vwire = EL Vbattery = ? Energy conservation (the Kirchhoff loop rule [2nd law]): V1 + V2 + V3 + … = 0 along any closed path in a circuit V= U/q  energy per unit charge Potential Difference Across the Battery Coulomb force on each e FC non-Coulomb force on each e FC 1. FC =eEC EC = e 2. FC =FNC EC DVbatt FC s FNC s = EC s = = e e Fully charged battery. Energy input per unit charge emf – electromotive force The function of a battery is to produce and maintain a charge separation. The emf is measured in Volts, but it is not a potential difference! The emf is the energy input per unit charge. chemical, nuclear, gravitational… Analysis of Circuits The current node rule (Charge conservation) Kirchhoff node or junction rule [1st law]: In the steady state, the electron current entering a node in a circuit is equal to the electron current leaving that node Conventional current: I = |q|nAuE The loop rule (Energy conservation) Kirchhoff loop rule [2nd law]: V1 + V2 + V3 + … = 0 along any closed path in a circuit V= U/q  energy per unit charge Field and Current in a Simple Circuit Round-trip potential difference: DVbatt + DVwire = 0 emf + ( - EL) = 0 E= emf L emf I = enAuE = enAu L We will neglect the battery’s internal resistance for the time being. Field and Current in a Simple Circuit Round-trip potential difference: Path 1 DVbatt + DV1 + DV3 = 0 emf + ( - E1L1 ) + (- E3L3 ) = 0 Path 2 E1 L1 = E2 L2 DVbatt + DV2 + DV3 = 0 emf + ( - E2 L2 ) + (- E3L3 ) = 0 Question: Twice the Length 1 2 i1 i2 Nichrome wire (resistive) A) i1 = i2 B) i1 = 2*i2 C) i1 = ½ i2 Twice the Length Nichrome wire (resistive) DV i = nAuE = nAu L i2 L = 1 iL 2 Current is halved when increasing the length of the wire by a factor of 2. Doubling the Cross-Sectional Area Doubling the cross-sectional area of the wire will A) not change electron current B) increase electron current by 2 C) decrease electron current by 2 Nichrome wire Doubling the Cross-Sectional Area i = nAuE = nAu DV L Loop: emf - EL = 0 Electron current in the wire increases by a factor of two if the crosssectional area of the wire doubles. Can we achieve infinitely large current using very thick wire? Nichrome wire Internal Resistance of a Battery FNC = FC = eEC fully charged DV (emf ) i = nAuE = nAu = nAu L L Increase mobility – current increases Real battery cannot provide  current Internal battery resistance – limits maximum current fixed EC = FC / e Drift speed in battery: vb = ub Enet = ub (FNC / e - EC ) We will neglect the battery’s internal resistance for the time being. Approximate Vbatt = emf V Across Connecting Wires The number or length of the connecting wires has little effect on the amount of current in the circuit. DV L DV = i L nAu + DVbattery = 0 i = nAuE = nAu DVwires + DVfilament DVbattery » emf uwires >> ufilament  DVwires << DV filament DV filament » (emf ) Work done by a battery goes mostly into energy dissipation in the bulb (heat).