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Current & Resistance - Current and current density - Ohm’s Law - Resistivity - Resistance Electrical Current CURRENT I is the amount of positive charge flowing past a fixed point in the wire per unit time : dQ I dt if charge dQ flows in time dt Units: 1 ampere (A) = 1 C/s Direction: by convention, current is the direction of movement of positive charge + + + + I + + - I - Electron Velocities • Random velocities of electrons are large (several km/s) • Drift velocity is a slow, average motion parallel to E no field E + F ( e) E end end start start net displacement Determining the current E + + + + + + + + + + + + L = vd Dt Charge ΔQ in length L of wire passes through the shaded disk of area A in time Δt : ΔQ = (number of charge carriers/volume) x (charge on each one) x volume Charge: DQ = n q V = n q (AL) = n q A vd Dt (since L=vt) Current: I = DQ/Dt = nqAvd Dt /Dt So, I = nqAvd vd = average (“drift”) velocity of each charge q = charge on each particle n = number of charge carriers per unit volume A = cross section area L = length AL = volume Example The mobile charges in most metals are electrons, with about one or two electrons per atom being free to move. So there are about 1023 charges per cm3 (or 10 29 m-3). n 1029 electrons/m3 nq ne 1.6 1010 C/m3 Take Area = 1 mm2, assume I = 1 A vdrift 0.06 mm/s Which way are the electrons moving? E = 0 inside ? Note: in electrostatics, we had E=0 inside a conductor, if not, charges would move, the conductor would not be in equilibrium and there would be a current. For a wire to carry a current, we must have an electric field inside the conductor, which is caused by the potential difference between the ends of the wire. This is no longer electrostatics! Example A copper wire of cross sectional area 3x10-6 m2 carries a current of 10A. Find the drift velocity of the electrons in this wire. Copper has a density of 8.95g/cm3, and atomic weight of 63.5g/mole. Avagadro’s number is 6.02x1023 atoms. Assume each atom contributes one free electron. Questions •When you turn on a flashlight, how long does it take for the electrons from the battery to reach the bulb? • What happens to the wire as the electrons go through it? •If you double the electric field in the wire, does the acceleration of the electrons double? •If the electric field remains constant, how do the electron kinetic energies change with time? Current Density J The current density is defined as the current per unit area in a conductor, where A is the cross section of conductor. The current density is a vector quantity, units: Amps/m2. I J A And since I=nqAvd; J nqv d now v d is proportional to the electric field so J E Where is a constant called the conductivity of the material. The current density J and the electric field E are both established in a conductor as a result of a potential difference across the conductor. Because J is proportional to the field, current in a wire is proportional to the potential difference between the ends of the wire. L A E V Uniform E J E V EL I V L V I A A L “Resistance”, R RESISTIVITY: the inverse of conductivity. L V I IR A where 1 (this depends on the type of material) Ohm’s Law L (Uniform wire, Length L, R A cross-section area A) Unit of resistance R is: 1 ohm ( ) 1 volt amp Ohm’s Law •Current density field: J=E •Current potential difference: V = IR = “resistivity”, has units of m A = “conductivity” units, m2 1 V m m Resistivities of a few materials ρ (20°C) (Ω·m) Cu 1.7 x 10-8 Al 2.8 x 10-8 Graphite 3500 x 10-8 Si 640 Quartz ~ 1018 Example A copper wire, 2 mm in diameter and 30 m in length, has a current of 5A. Find: a) resistance 1.7 108 m for Cu b) potential difference between the ends c) electric field d) current density Quiz The wire in the previous example is replaced with a wire of the same length and half the diameter, carrying the same current. By what factor will each of the following change? A) B) C) D) E) F) resistance potential difference between the ends electric field electron number density electron drift speed current density Summary Current Density: Conductivity: : J I A nqvd J E Resistivity: 1 L Resistance: R A Ohm’s Law: V IR (defines )