Current & Resistance

Current & Resistance • - Current and current density • Ohm’s Law • Resistivity • Resistance

Electrical Current CURRENTI 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 + no field E end end start start net displacement

+ + + + + + E + + + + + + L = vdDt Determining the current 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 vdDt (since L=vt) Current: I = DQ/Dt = nqAvdDt /Dt I = nqAvd So, 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

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). Example Take Area = 1 mm2, assume I = 1 A 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. And since I=nqAvd; Where  is a constant called the conductivity of the material. The current density J and the electric field E are bothestablished in a conductor as a result of a potential difference across the conductor.

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

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

Ohm’s Law • Current density  field: J = sE • Current  potential difference: V = IR r = “resistivity”, has units of W m s = “conductivity” units,

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 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 the resistance change?A) increase by 4B) increase by 2C) stay the sameD) decrease by 2E) decrease by 4

Question 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? • resistance • potential difference between the ends • electric field • electron number density • electron drift speed • current density

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

Current & Resistance