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Transcript
Chapter 21-part1 Current and Resistance 1 Electric Current Whenever electric charges move, an electric current is said to exist The current is the rate at which the charge flows through a certain crosssection For the current definition, we look at the charges flowing perpendicularly to a surface of area A Definition of the current: + - Charge in motion through an area A. The time rate of the charge flow through A defines the current (=charges per time): I=DQ/Dt Units: C/s=As/s=A SI unit of the current: Ampere Electric Current, cont The direction of current flow is the direction positive charge would flow This is known as conventional (technical) current flow, i.e., from plus (+) to minus (-) However, in a common conductor, such as copper, the current is due to the motion of the negatively charged electrons It is common to refer to a moving charge as a mobile charge carrier A charge carrier can be positive or negative 2 Current and Drift Speed Charged particles move through a conductor of crosssectional area A n is the number of charge carriers per unit volume V (=“concentration”) nADx=nV is the total number of charge carriers in V Current and Drift Speed, cont The total charge is the number of carriers times the charge per carrier, q (elementary charge) ΔQ = (nAΔx)q [unit: (1/m3)(m2 m)As=C] The drift speed, vd, is the speed at which the carriers move vd = Δx/Δt Δx Rewritten: ΔQ = (nAvdΔt)q Finally, current, I = ΔQ/Δt = nqvdA Current and Drift Speed, final If the conductor is isolated, the electrons undergo (thermal) random motion When an electric field is set up in the conductor, it creates an electric force on the electrons and hence a current Charge Carrier Motion in a Conductor The electric field force F imposes a drift on an electron’s random motion (106 m/s) in a conducting material. Without field the electron moves from P1 to P2. With an applied field the electron ends up at P2’; i.e., a distance vdDt from P2, where vd is the drift velocity (typically 10-4 m/s). Does the direction of the current depend on the sign of the charge? No! (a) Positive charges moving in the same direction of the field produce the same positive current as (b) negative charges moving in the direction opposite to the field. qvd E vd E vd (-q)(-vd) = qvd Current density: The current per unit cross-section is called the current density J: J=I/A= nqvdA/A=nqvd In general, a conductor may contain several different kinds of charged particles, concentrations, and drift velocities. Therefore, we can define a vector current density: J=n1q1vd1+n2q2vd2+… Since, the product qvd is for positive and negative charges in the direction of E, the vector current density J always points in the direction of the field E. Example: An 18-gauge copper wire (diameter 1.02 mm) carries a constant current of 1.67 A to a 200 W lamp. The density of free electrons is 8.51028 per cubic meter. Find the magnitudes of (a) the current density and (b) the drift velocity. Solution: (a) A=d2p/4=(0.00102 m)2p/4=8.210-7 m2 J=I/A=1.67 A/(8.210-7 m2)=2.0106 A/m2 (b) From J=I/A=nqvd, it follows: J 2.0 10 A / m vd nq (8.5 10 28 m 3 )(1.60 10 19 C) 6 vd=1.510-4 m/s=0.15 mm/s 2 3 Electrons in a Circuit The drift speed is much smaller than the average speed between collisions When a circuit is completed, the electric field travels with a speed close to the speed of light Although the drift speed is on the order of 10-4 m/s the effect of the electric field is felt on the order of 108 m/s Meters in a Circuit – Ammeter An ammeter is used to measure current In line with the bulb, all the charge passing through the bulb also must pass through the meter (in series!) Meters in a Circuit - Voltmeter A voltmeter is used to measure voltage (potential difference) Connects to the two ends of the bulb (parallel) QUICK QUIZ Look at the four “circuits” shown below and select those that will light the bulb. 4 Resistance and Ohm’s law In a homogeneous conductor, the current density is uniform over any cross section, and the electric field is constant along the length. b a V=Va-Vb=EL Resistance The ratio of the potential drop to the current is called resistance of the segment: V R Unit: V/A=W (ohm) I Resistance, cont Units of resistance are ohms (Ω) 1Ω=1V/A Resistance in a circuit arises due to collisions between the electrons carrying the current with the fixed atoms inside the conductor Ohm’s Law V I V=const.I V=RI Ohm’s Law is an empirical relationship that is valid only for certain materials Materials that obey Ohm’s Law are said to be ohmic I=V/R R, I0, open circuit; R0, I, short circuit Ohm’s Law, final Plots of V versus I for (a) ohmic and (b) nonohmic materials. The resistance R=V/I is independent of I for ohmic materials, as is indicated by the constant slope of the line in (a). Ohmic Nonohmic 5 Resistivity Expected: RL/A The resistance of an ohmic conductor is proportional to its length, L, and inversely proportional to its cross-sectional area, A L Rρ A ρ (“rho”) in Wm is the constant of proportionality and is called the resistivity of the material Example Determine the required length of nichrome (=10-6 Wm) with a radius of 0.65 mm in order to obtain R=2.0 W. R=L/AL=RA/ (2.0W)p (0.00065m) L 2 . 65 m 6 10 Ωm 2 The resistivity depends on the material and the temperature 6 Temperature Variation of Resistivity For most metals, resistivity increases with increasing temperature With a higher temperature, the metal’s constituent atoms vibrate with increasing amplitude The electrons find it more difficult to pass the atoms (more scattering!) Temperature Variation of Resistivity, cont For most metals, resistivity increases approximately linearly with temperature over a limited temperature range ρ ρo [1 α(T To )] ρo is the resistivity at some reference temperature To To is usually taken to be 20° C is the temperature coefficient of resistivity [unit: 1/(C)] Temperature Variation of Resistance Since the resistance of a conductor with uniform cross sectional area is proportional to the resistivity, the temperature variation of resistance can be written R Ro [1 α(T To )] Example The material of the wire has a resistivity of 0=6.810-5 Wm at T0=320C, a temperature coefficient of =2.010-3 (1/C) and L=1.1 m. Determine the resistance of the heater wire at an operating temperature of 420C. Solution =0[1+(TT0)] =[6.810-5 Wm][1+(2.010-3 (C)-1) (420C-320C)]=8.210-5 Wm R=L/A R=(8.210-5 Wm)(1.1 m)/(3.110-6 m2) R=29 W 7 Superconductors A class of materials and compounds whose resistances fall to virtually zero below a certain temperature, TC TC is called the critical temperature (in the graph 4.1 K) “normal” Superconductors, cont The value of TC is sensitive to Chemical composition Pressure Crystalline structure Once a current is set up in a superconductor, it persists without any applied voltage Since R = 0 Superconductor Timeline 1911 1986 High-temperature superconductivity discovered by Bednorz and Müller Superconductivity near 30 K 1987 Superconductivity discovered by H. Kamerlingh Onnes Superconductivity at 92 K and 105 K Current More materials and more applications Tc values for different materials; note the high Tc values for the oxides. It’s magic! 8 Electrical Energy and Power In a circuit, as a charge moves through the battery, the electrical potential energy of the system is increased by ΔQΔV [AsV=Ws=J] The chemical potential energy of the battery decreases by the same amount As the charge moves through a resistor, it loses this potential energy during collisions with atoms in the resistor The temperature of the resistor will increase Electrical Energy and Power, cont The rate of the energy transfer is power (P): DW ΔQ P V IV Dt Δt Units: (C/s)(J/C) =J/s=W 1J=1Ws=1Nm W=AV V Electrical Energy and Power, cont From Ohm’s Law, alternate forms of power are (use V=IR and I=V/R) 2 V P IV I R R Joule heat (I R losses) 2 2 Electrical Energy and Power, final The SI unit of power is Watt (W) I must be in Amperes, R in Ohms and V in Volts The unit of energy used by electric companies is the kilowatt-hour This is defined in terms of the unit of power and the amount of time it is supplied 1 kWh =(103 W)(3600 s)= 3.60 x 106 J 9 Electrical Activity in the Heart Heart beat Initiation Every action involving the body’s muscles is initiated by electrical activity Voltage pulses cause the heart to beat These voltage pulses (1 mV) are large enough to be detected by equipment attached to the skin Electrocardiogram (EKG) A normal EKG P occurs just before the atria begin to contract The QRS pulse occurs in the ventricles just before they contract The T pulse occurs when the cells in the ventricles begin to recover Abnormal EKG, 1 The QRS portion is wider than normal This indicates the possibility of an enlarged heart Abnormal EKG, 2 There is no constant relationship between P and QRS pulse This suggests a blockage in the electrical conduction path between the SA and the AV nodes This leads to inefficient heart pumping Abnormal EKG, 3 No P pulse and an irregular spacing between the QRS pulses Symptomatic of irregular atrial contraction, called The atrial and ventricular contraction are irregular fibrillation Implanted Cardioverter Defibrillator (ICD) Devices that can monitor, record and logically process heart signals Then supply different corrective signals to hearts that are not beating correctly Dual chamber ICD Monitor lead