Chapter 25 Electric Current & DC Circuits

Chapter 25Electric Current &DC Circuits

Topics • Electron Gas • Electric Currents • Resistance & Ohm’s Law

Electron Gas • Metals contain electrons, about 1 per atom, that are free to move about. • The free electrons move in random directions at about 106 m/s • Therefore, on average, there is no net motion of this electron gas

Electron Gas • When an electric field is applied the free electrons accelerate in the direction opposite the field • However, because of the frequent collisions between the electrons and the lattice ions, the electrons very quickly reach a steady drift speed,vd. The steady drift of charge is called an electric current

Electric Currents CurrentI is the rate at which charge flows through a given area A Unit of current 1 ampere (A) = 1C/s

Electric Currents In a short time interval Dt, all particles with drift speed vd in the volume vdDt Across area A Suppose there are n particles per unit volume, each with charge q…

Electric Currents …then the total charge crossing area A is So the current is

Electric Currents If we divide by the area A, we get the current per unit area which is the magnitude of the current density vector

Estimating the Drift Speed Assume Current: 1.0 A Wire radius: 0.08 cm Material: Copper r = 8.93 g/cm3 M= 63.5 g/mol Drift speed:

Estimating the Drift Speed We need an estimate of the electron density: number of electrons/cm3~ no. of atoms/cm3 = 8.93 g/cm3 / Mass per atom = 8.93 g/cm3 / (63.5 g/mol /6x1023/mol) = (8.93/63.5) * 6 x 1023 Avogadro’s No. n= 8.4 x 1022/cm3

Estimating the Drift Speed Current I = -1.0 C/s Electron density n = 8.4 x 1022/cm3 Electron charge q = -1.6 x 10-19 C Area of wire A = p r2 = 2x10-2 cm2 vd = I/ (q nA) = 3.7 x 10-3 cm/s A very slow drift! So why do the lights come on instantly?

Resistance & Ohm’s Law Assume the electric field E is uniform then the potential dropV is given by

Resistance & Ohm’s Law The ratio of the potential drop V in the direction of the current I is called the resistance, measured in ohms (W)

V Slope = R I Resistance & Ohm’s Law For many materials, the resistance R is very nearly independent of both the potential drop, V, and the current, I. For such ohmic materials, V is therefore proportional to I: Ohm’s law

Resistivity From numerous experiments, it has been found that the resistance of a conducting wire can be written as L is the length of the wire A is the cross-sectional area r is called the resistivity (not to be confused with charge density)

Drift speed, revisited • Electric field  electrons feel force F = e E, • acceleration a = F/m = eE/m • Collisions with lattice act like friction  limiting speed vd • Expectation: vd  a  ( = mean time between collision • Calculation shows: vd = a 

Drift speed and resistivity • vd = (e/m) E  = eE (e = •(e/m) = “mobility” ) • I = neA vd= neA e E = neA eV/L =V •nee• A/L = V •A/L •  = nee= “conductivity” •  = 1/ = 1/(nee) = resistivity •  I = V/R, with R =  L /A • “ohmic” materials: e independent of electric field

Resistivity of Common Materials

Summary • Definition of Current • I = dQ/dt • Relation to drift speed vdof charge • I = qnvdA • Definition of Resistance • R = V / I • Ohm’s Law • V = I R

Chapter 25 Electric Current & DC Circuits