24. Electric Current

1. 24. Electric Current

2. Topics • Electric Currents • Ohm’s Law • Conduction • Electric Power

3. Electric Current

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

5. Electric Current 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…

6. Electric Current then the charge crossing area A is So the current is

7. Estimating the Drift Speed Assume Current: I =5.0 A Wire radius: r = 0.08 cm Material: Copper n = 8.93 g/cm3 M= 63.5 g/mol Drift speed:

8. Estimating the Drift Speed We need an estimate of the electron density in copper: 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) x 6 x 1023 Avogadro’s No. n= 8.4 x 1022/cm3

9. Estimating the Drift Speed Current I = -5.0 C/s Electron densityn = 8.4 x 1022/cm3 Electron charge q = -1.6 x 10-19 C Area of wire A = pr2 = 2 mm2 vd = I/ (q nA) = 0.18 mm/s A very slow drift!So why do the lights come on instantly?

10. Conduction

11. Conduction Conduction occurs in • Metals • Ionic solutions • Plasmas • Semiconductors • Superconductors

12. Conduction in Metals The free electrons in metals behave like a gas The electrons move at about 106 m/s. But since the directions are random, on average there is no net motion of the electron gas

13. Conduction in Metals When an electric field is applied the free electrons accelerate in the direction opposite the field. But the frequent collisions between the electrons and the lattice ions, causes the net drift velocity opposite the field to be very small, as we found.

14. Conduction in Metals For metals, it is found that the conductivity is independent of the electric field. How can this be explained? The conductivity depends on the electron-ion collision rate, which depends on the speed of the electrons (~ 106 m/s). But the drift barely alters the collision rate

15. Conduction in Metals What’s the pressure of the electron gas in copper? For copper, r = 8.47 x 1028/m3 E = 7.06 eV = 1.12 x 10-18 J therefore, the electron pressure in copper is P = (2/5)rE = (2/5) x (8.47 x 1028) x (1.12 x 10-18) = 3.8 x 1010 Pa that is, about 400,000 times the pressure in this room!

16. Resistance & Ohm’s Law

17. Current Density If we divide the current by the area A, we get the current per unit area which is the magnitude of the current density vector

18. Current Density If we know the current density J, that is, the current per unit area, we can find the current by integrating the current density: If the area element and the current density vector are always in the same direction, then we can write: See optional problem 24.64 hint: let dA be the area of a thin annulus. Write J = b – c rand first find b and c in terms of J0 and a

19. Ohm’s Law For many materials, it is found that the current density vector is proportional to the electric field s is called the conductivity of the material. r = 1/s is called the resistivity ohm (W) = 1 Volt/Amp Resistivity is measured in W m

20. Resistance & Ohm’s Law The ratio of the potential drop Vin the direction of the current I is called the resistance, measured inohms (W)

21. 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

22. Resistance & Resistivity From numerous experiments, it has been found that the resistance of a conductor of uniform cross-sectional area A and length L is given by See question 24.62 L is the length of the conductor A is the cross-sectional area r is the resistivity

23. A Simple Theory of Resistivity In the presence of and electric field, E, the acceleration of an electron is a = F/m = eE/m If the average time between electron/ion collisions is t, then the electrons speed increases by vd = at, that is, the drift speed. We can write vd = eEt/m = me E where me = t (e/m) is called the mobility

24. A Simple Theory of Resistivity Recall that the current is given by I = nAq vd= nAq e E = nAq e V/L But, R = V/I =rL/A, therefore, r = AV/IL = 1/nqe This shows that the larger the charge per free charge carrier, and the more of them there are, the lower the resistivity.

25. Typical Resistivities MaterialResistivity (W m) Silver 1.59 x 10-8 Copper 1.68 x 10-8 Human blood 0.70 Seawater 0.22 Silicon 23.0 Polystyrene 1015 - 1017

26. Electric Power

27. Energy in Electrical Circuits As they drift, the free electrons collide constantly with the lattice ions. Consequently, much of the energy they gain from the electric field is dissipated as thermal energy

28. Energy in Electrical Circuits The heating of a conductor in this way is called Joule heating Consider a positive charge DQ that drifts a distance L. Its changein potential energy is

29. Energy in Electrical Circuits In the limit DQ→ 0, the rate loss of potential energy is given by

30. Energy in Electrical Circuits Because of energy conservation, the rate of potential energy loss equals the rate at which energy is transferred to the conductor P is the electric power in watts

31. Energy in Electrical Circuits The expression for the electrical power holds true for all devices. But for a resistor, we know that V = IR, so

32. Electric-Power Transmission Power lines are usually at very high voltage. Why? Suppose we want to transmit P = 696 kW of power to a factory along power lines with total resistance R = 2.13 W. What’s the power loss along the transmission lines?

33. Electric-Power Transmission If V is the transmission voltage, then the transmission current is I = P/V So the power lost in transmission is Ploss = I2R = (P/V)2 R For V = 13.4 kV, Ploss = 5.746 kW For V = 55.9 kV, Ploss = 0.330 kW

34. Summary • Definition of Current • I = dQ/dt • Relation to drift speed vdof charge • I = nAvdq • Definition of Resistance • R = V / I • Ohm’s Law • V = I R • Electric Power • P = I V