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 : 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 start end start end net displacement + E no field

Charge ΔQ in length L of wire passes through the shaded disk of area A in time Δ t : ΔQ = (number of charge carriers/volume) x (charge on each one) x volume L = v d  t E Determining the current

Current: I =  Q/  t = nqAv d  t /  t So,  Charge:  Q = n q V = n q (AL) = n q A v d  t (since L=vt) I = nqAv d v d = 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 Take Area = 1 mm 2, assume I = 1 A The mobile charges in most metals are electrons, with about one or two electrons per atom being free to move. So there are about charges per cm 3 (or m -3 ). 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 m 2 carries a current of 10A. Find the drift velocity of the electrons in this wire. Copper has a density of 8.95g/cm 3, and atomic weight of 63.5g/mole. Avagadro’s number is 6.02x10 23 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 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. 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/m 2. And since I=nqAv d ;

L Uniform E “Resistance”, R Because J is proportional to the field, current in a wire is proportional to the potential difference between the ends of the wire. A E

RESISTIVITY: the inverse of conductivity. where (Uniform wire, Length L, cross-section area A) (this depends on the type of material) Unit of resistance R is: Ohm’s Law

Current density  field: J =  E Current  potential difference: V = IR units,  = “conductivity”  = “resistivity”, has units of  m

ρ (20°C) (Ω·m) Cu1.7 x Al2.8 x Graphite3500 x Si640 Quartz~ Resistivities of a few materials

Example A copper wire, 2 mm in diameter and 30 m in length, has a current of 5A. Find: a) resistance 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)resistance B)potential difference between the ends C)electric field D)electron number density E)electron drift speed F)current density

Summary Current Density: Conductivity: Resistivity: Resistance: Ohm’s Law: