# Chapter 24: Electric Current

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1 Chapter 24: Electric Current Current Definition of current A current is any motion of charge from one region to another. Suppose a group of charges move perpendicular to surface of area A. The current is the rate that charge flows through this area: dq dt ; dq amount of charge that flows during the time interval dt Units: 1 A 1 ampere 1 C/s

2 Current Microscopic view of current

3 Current Microscopic view of current (cont d)

4 Current Microscopic view of current (cont d) n time t the electrons move a distance x υ There are n particles per unit volume that carry charge q The amount of charge that passes the area A in time tis Q q( na υ t) d The current is defined by: d t dq dt lim t 0 Q t nqυ A d The current density J is defined by: J A r J nqυ nq r υ d d Current per unit area Units: A/m 2 Vector current density

5 Ohm s law Resistivity The current density J in a conductor depends on the electric field E and on the properties of the material. This dependence is in general complex but for some material, especially metals, J is proportional to E. E J Ohm s law V/A ohm ρ ρ : resistivity, Units conductivi ty σ Substance ρ (Ωm) 8 silver copper gold steel (V/m)/(A/m ) V m/a Ωm J σe 1/resisitivity Substance graphite silicon glass teflon ρ (Ωm) >

6 Resistivity Conductors, semiconductors and insulators Good electrical conductors such as metals are in general good heat conductors as well. n a metal the free electrons that carry charge in electrical conduction also provide the principal mechanism for heat conduction. Poor electrical conductors such as plastic materials are in general poor thermal conductors as well. Semiconductors have resistivity intermediate between those of metals and those of insulators. A material that obeys Ohm s law reasonably well is called an ohmic conductor or a linear conductor.

7 Resistivity Resistivity and temperature The resistivity of a metallic conductor nearly always increases with increasing temperature. ρ( T ) ρ0[1 + α( T T0 )] reference temp. (often 0 o C) temperature coefficient of resistivity Material Material α ( o C) -1 α ( o C) -1 aluminum iron brass lead graphite manganin copper silver

8 Resistivity vs. temperature Resistivity The resistivity of graphite decreases with the temperature, since at higher temperature more electrons become loose out of the atoms and more mobile. This behavior of graphite above is also true for semiconductors. Some materials, including several metallic alloys and oxides, has a property called superconductivity. Superconductivity is a phenomenon where the resistivity at first decreases smoothly as the temperature decreases, and then at a certain critical temperature T c the resistivity suddenly drops to zero. ρ ρ ρ metal T semiconductor T T T c superconductor

9 Resistance Resistance For a conductor with resistivity r ρ, the r current density J at a point where the electric field is E : E ρj When Ohm s law is obeyed, ρ is constant and independent of the magnitude of the electric field. Consider a wire with uniform cross-sectional area A and length L, and let V be the potential difference between the higher-potential and lower-potential ends of the conductor so that V is positive. A E r J r L JA, V r r E ρj E V L EL ρ V A R ρl A As the current flows through the potential difference, electric potential is lost; this energy is transferred to the ions of the conducting material during collisions. V resistance 1 V/A1Ω

10 Resistance Resistance (cont d) As the resistivity of a material varies with temperature, the resistance of a specific conductor also varies with temperature. For temperature ranges that are not too great, this variation is approximately a linear relation: R( T ) R0[1 + α( T T0 )] A circuit device made to have a specific value of resistance is called a resistor. Ι Ι V V resistor that obeys Ohm s law semiconductor diode

11 Resistance Example: Calculating resistance A b a r Consider a hollow cylinder of length L and inner and outer radii a and b, made of a material with ρ. The potential difference between the inner and outer surface is set up so that current flows radially through cylinder. Now imagine a cylindrical shell of radius r, length L, and thickness dr. A dr 2πrL ρdr 2πrL :area of a cylinder represented by a circle from which the current flows : resistance of this shell dashed R ρ dr b dr ρ L 2πL a r 2π ln b a

12 Electromotive Force (emf) and Circuit Complete circuit and steady current For a conductor to have a steady current, it must be part of a path that forms a closed loop or complete circuit. E r E r + E r J r J r J r E r + E r Electric field E r 1 produced inside conductor causes current Current causes charge to build up at ends, producing opposing field E r 2 and reducing current r After a very short time E r the same magnitude as E r total field 0 and E total current stops completely. 2 1 : has

13 Electromotive Force (emf) and Circuit Maintaining a steady current and electromotive force When a charge q goes around a complete circuit and returns to its starting point, the potential energy must be the same as at the beginning. But the charge loses part of its potential energy due to resistance in a conductor. There needs to be something in the circuit that increases the potential energy. This something to increase the potential energy is called electromotive force (emf). Units: 1 V 1 J/C Emf makes current flow from lower to higher potential. A device that produces emf is called a source of emf. source of emf f a positive charge q is moved from b to a inside the F re b - E r a source, the non-electrostatic force F n does a positive + F r amount of work W n qv ab on the charge. n This replacement is opposite to the electrostatic force E r E r F e, so the potential energy associated with the charge increases by qv ab. For an ideal source of emf F e F n current flow in magnitude but opposite in direction. W n qε qv ab, so V ab ε R for an ideal source.

14 Electromotive Force (emf) and Circuit nternal resistance Real sources in a circuit do not behave ideally; the potential difference across a real source in a circuit is not equal to the emf. V ab ε r (terminal voltage, source with internal resistance r) So it is only true that V ab E only when 0. Furthermore, ε r R or ε / (R + r)

15 Electromotive Force (emf) and Circuit Real battery c R d a b r Battery ε ba + Real battery has internal resistance, r. Terminal voltage, Voutput (Va Vb) ε r. V ε r ε R R R r out +

16 Electromotive Force (emf) and Circuit Potential in an ideal resistor circuit c d a b b a c d b

17 Electromotive Force (emf) and Circuit Potential in a resistor circuit in realistic situation c R d ab r Battery ε b a + - V R r ε ε R + - r 0 d c b a a b

18 Energy and Power in Electric Circuit Electric power Electrical circuit elements convert electrical energy into 1) heat energy( as in a resistor) or 2) light (as in a light emitting diode) or 3) work (as in an electric motor). t is useful to know the electrical power being supplied. Consider the following simple circuit. du dq V e dqv ab due is electrical potential energy lost as dq traverses the resistor and falls in V by V. Electric power rate of supply from Ue. dq 2 Electric power P Vab Vab R dt Units :(1 J/C)(1C/s) 1J/s 1 W (watts) V R 2 ab V ab R V ab

19 Energy and Power in Electric Circuit Power output of a source Consider a source of emf with the internal resistance r, connected by ideal conductors to an external circuit. The rate of the energy delivered to the external circuit is given by: P V ab For a source described by an emf and an internal resistance r V ab ε r + - net electrical power output of the source P V ε ab rate of conversion of nonelectrical energy to electrical energy in the source 2 r rate of electrical energy dissipation in the internal resistance of the source battery (source) a + b - headlight (external circuit)

20 Energy and Power in Electric Circuit Power input of a source Consider a source of emf with the internal resistance r, connected by ideal conductors to an external circuit. total electrical power input + - to the battery battery small emf V ab ε + r P V ε + ab 2 r a + v F n + b - rate of conversion of electrical energy into noneletrical energy in the battery rate of dissipation of energy in the internal resistance in the battery alternator large emf

21 Electromotive Force (emf) and Circuit Examples: r 2 Ω, ε 12 V, R 4 Ω a V cd V ab The rate of The rate of V t is also given by voltmeter dissipation of (2 A) A ammeter The electrical power output is ε The power output is also given by V 2 ε R + r V V. V V ab cd ab cd ε energy conversion in the battery is ε R b energy in the battery is 2 A 2 bc (4 Ω) 16 W. 12 V 4 Ω + 2 Ω 2 A. R (2 A)(4 Ω) 8 V. : measures potential difference r 16 W. r 12 V - (2 A)(2 Ω) 8 V. r (12 V)(2 A) 2 V (2 A) (8 V)(2 A) 16 W. : measures current through it 2 24 W. (2 Ω) 8 W.

22 Drude s model Electric Conduction

23 Drude s model (cont d) Electric Conduction

24 Drude s model (cont d) Electric Conduction

25 Exercise 1 Calculate the resistance of a coil of platinum wire with diameter 0.5 mm and length 20 m at 20 C given ρ Ω m. Also determine the resistance at 1000 C, given that for platinum α / C. R l 20 m ρ (11 10 Ω m) 11 Ω A π[0.5( m)] To find the resistance at 1000 C: ρ ρ α ( T T 0) l But R ρ A so we have : R R 1 + α ( T T ) 0 0 Where we have assumed l and A are independent of temperature - could cause an error of about 1% in the resistance change. 3 1 R(1000 C) (11 )[1 ( C )(1000 C 20 C)] 53 Ω + Ω

26 Exercise 2 A 1000 W hair dryer manufactured in the USA operates on a 120 V source. Determine the resistance of the hair dryer, and the current it draws. P 1000 W 8.33A V 120 V V 120V V R R 14.4 Ω 8.33A The hair dryer is taken to the UK where it is turned on with a 240 V source. What happens? P 2 2 ( V ) (240V) R 14.4Ω 4000 W This is four times the hair dryer s power rating BANG and SMOKE!

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