Electric Current. Electric current is the rate of flow of charge through some region of space The SI unit of current is the ampere (A)


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1 Electric Current Electric current is the rate of flow of charge through some region of space The SI unit of current is the ampere (A) 1 A = 1 C / s The symbol for electric current is I
2 Average Electric Current Assume charges are moving perpendicular to a surface of area A If ΔQ is the amount of charge that passes through A in time Δt, then the average current is I av ΔQ = Δ t
3 Instantaneous Electric Current If the rate at which the charge flows varies with time, the instantaneous current, I, can be found I = dq dt
4 Direction of Current The charges passing through the area could be positive or negative or both It is conventional to assign to the current the same direction as the flow of positive charges The direction of current flow is opposite the direction of the flow of electrons It is common to refer to any moving charge as a charge carrier
5 Current and Drift Speed Charged particles move through a conductor of crosssectional area A n is the number of charge carriers per unit volume na Δx is the total number of charge carriers
6 Current and Drift Speed, cont The total charge is the number of carriers times the charge per carrier, q ΔQ = (na Δx)q The drift speed, v d, is the speed at which the carriers move v d = Δx / Δt Rewritten: ΔQ = (nav d Δt)q Finally, current, I av = ΔQ/Δt = nqv d A
7 Charge Carrier Motion in a Conductor The zigzag black line represents the motion of a charge carrier in a conductor The net drift speed is small The sharp changes in direction are due to collisions The net motion of electrons is opposite the direction of the electric field
8 Motion of Charge Carriers, cont. In spite of all the collisions, the charge carriers slowly move along the conductor with a drift velocity, v d Changes in the electric field that drives the free electrons travel through the conductor with a speed near that of light This is why the effect of flipping a switch is effectively instantaneous
9 Electric current dq I = ( C = A) ( Amp) dt s Q dq = Idt, dq = Idt Q = It 0 0 t I Direction of currentgmotion direction of positive charge Current density A J I A m = ( 2) I = J da A r r I E + r da = danˆ L I J = I L ( A ) m
10 Consider the conductor shown in the figure. It is connected to a battery (not shown) and thus charges move through the conductor. Consider one of the cross sections through the conductor ( aa or bb or cc ). The electric current i is defined as: i = Current = rate at which charge Current SI Unit: C/s known as the "Ampere" flows dq dt i = dq dt (263)
11 Current direction : conductor + q i q i v r conductor v r An electric current is represented by an arrow which has the same direction as the charge velocity. The sense of the current arrow is defined as follows: 1. If the current is due to the motion of positive charges the current arrow is parallel to the r charge velocity v 2. If the current is due to the motion of negative charges the current arrow is antiparallel to the charge velocity v r. (263)
12 conductor v r A + q q J i conductor v r A = i i A J r J r Current density Current density is a vector that is defined as i follows: Its magnitude J = A SIunitforJ:A/2r The direction of J is the same as that of the current. The current through a conductor of cros sectional area A ms is given by the equation: i= JA, if the current density is constant. r r r If J is not constant then: i= J da. (264)
13 conductor v r A + q q J i conductor v r A = i i A J r J r We note that even though the current density is a vector the electric current is not. This is illustrated in the figure to the left. An incoming current branches at i o point a into two currents, Current i = i + i o 1 2 i1 i2, and. This equation expresses the conservation of charge at point a. Please note that we have not used vector addition. (264)
14 Drift speed When a current flows through a conductor the electric field causes the charges to move with a constant drift speed is superimposed on the random motion of the charges. v d. This drift speed J = nv e d r r J = nev d (265)
15 Drift speed L Carrier  : electron + : electric hole q v A random motion Q number of carriers Steady current : I =, Q=η( AL) q, η:, t volume L L I 1 J I = η A q vd = vav = = =. t t Aηq ηq J v = ηqv v = v d ρ. d charge ρ: charge density ( ) volume
16 EX. I = 1A Cu Cu:atomic mass 63.5 g mole density = 8.9 g cm 3 2 A =1mm 1 conducting carrier η =? v =? d atom N mole g 1 η = Number 3 = = cm mole g cm v d I = = 2 19 Aηg (0.1) η cm s
17 Resistance A V E q L v d can be viewed as the result of the carrier under an acceleration in an average time interval t. (average time between two collisions) av v d = v av slpoe = a F = qe = ma E e ma m v m J = = = d 2 q q tav q ηtav t av
18 m 1 Let ρ = resistivity σ = conductivity 2 q ηtav ρ J r r E = ρj =, J = σe. σ I EL = ρjl V = ρ L A L Let R = ρ resistance ( ohm, Ω ) V = IR. A ρ = ρ (1 ) 0 + αδt T : temperature, α : temperature coefficient of ρ. L R= ρ R= R0 (1 + αt). A
19 Resistivity i r E E r +  V r r = ρj J E r L R = ρ A r = σ E Unlike the electrostatic case, the electric field in the conductor of the figure is not zero. We define as resistivity ρ of the conductor E r r the ratio ρ =. In vector form: E = ρj J V/m V SI unit for ρ : = 2 m =Ω m A/m A 1 The conductivity σ is defined as: σ = ρ Using ρ the previous equation takes the form: r r J = σ E (267)
20 i E r Consider the conductor shown in the figure above. The electric field inside the r E +  V r r = ρ J J r = σ E V conductor E =. L i The current density J = A We substitute E and J into equation R E r L = ρ A E ρ = and get: J V / L V A A L ρ = = = R R= ρ i/ A i L L A (267)
21 Variation of resistivity with temperature In the figure we plot the resistivity ρ of copper as function of temperature T. The dependence of ρ on T is almost linear. Similar dependence is observed in many conductors. ( T T ) ρ ρ = υ α o o o (268)
22 (269) Ohm's Law : A resistor was defined as a conductor whose resistance does not change with the voltage V applied across it. In fig.b we plot the current i through a resistor as function of V. The plot (known as " i V curve" ) is a straight line that passes through the origin. Such a conductor is said to be " Ohmic" and it obeys Ohm's law that states: The current i through a conductor is proportional to the voltage V applied across it.
23 (269) Not all conductors abey Ohm's law (these are known as " non  Ohmic" ) An example is given in fig.c where we plot i versus V for a semiconductor diode. The ratio V / i (and thus the resistance R ) is not constant. As a matter of fact the diode does not conduct for negative voltage values. Note : Ohm's "law" is in reality a definition of Ohmic conductors (defined as the conductors that obey Ohm's law)
24 A Microspopic view of Ohm's law : In order to understand why some materials such as metals obey Ohm's law we must look into the details of the conduction process at the atomic level. A schematic of an Ohmic conductor such as copper is shown in the figure. We assume that there are free electrons that move around in random directions with an effective speed v 6 eff = m/s. The free electrons suffer collisions with the stationary copper atoms. F r v r d (2610)
25 A schematic of a free electron path is shown in the figure in the figure using the dashed gray line. The electron starts at point A and ends at point B. We now assume that an electric field E r is applied. The new electron path is indicated by the dashed green line. Under the action of the electric force the electron acquires a small drift speed v d. The electron drifts to the right and ends at point B. F r v r d (2610)
26 Consider the motion of one of the free electrons. We assume that the average time between collisions with the copper atoms is equal to τ. The electic field exerts a force F = ee on the electron, F ee resulting in an acceleration a = =. m m F r v r d The drift speed is given by the equation: v d eeτ = aτ = ( eqs.1) m (2611)
27 J We can also get vd from the equation: J = nevd vd = ( eqs.2) ne If we compare equations 1 and 2 we get: J eeτ m vd = = E =. If we compare the last equation with: 2 J ne m ne τ m E = ρj we conclude tha t : ρ = This is a statement of Ohm's law 2 ne τ (the resistance of the conductor does not depend on voltage and thus E ) This is because m, n, and e are constants. The time τ can also be considered be v r d independent of E since the drift spees vd is so much smaller than v. F r eff (2611)
28 Resistor R V R du = Vdq = VIdt 2 du 2 V P= = VI = I R= dt R ( power : J = Watt = V A) s Kirchhoffs voltage rule ( loop rule ) : The sum of the charge in potential encounted in a complete traversal of any loop of a circuit must be zero. ( conservation of energy )
29 Resistivity Values
30 Resistors Most circuits use elements called resistors Resistors are used to control the current level in parts of the circuit Resistors can be composite or wirewound
31 Resistor Values Values of resistors are commonly marked by colored bands
32 Resistance of a Cable, Example Assume the silicon between the conductors to be concentric elements of thickness dr The resistance of the hollow cylinder of silicon is ρ dr = dr 2πrL
33 Resistance of a Cable, Example, cont. The total resistance across the entire thickness is R b ρ b = dr = ln a 2πL a This is the radial resistance of the cable This is fairly high, which is desirable since you want the current to flow along the cable and not radially out of it
34 Conduction Model, final Using Ohm s Law, expressions for the conductivity and resistivity of a conductor can be found: 2 nq τ 1 me σ = ρ = = 2 m σ nq τ e Note, the conductivity and the resistivity do not depend on the strength of the field The average time is also related to the free mean path: τ = l/v av
35 Resistance and Temperature Over a limited temperature range, the resistivity of a conductor varies approximately linearly with the temperature ρ = ρ [1 + α( T T )] o ρ o is the resistivity at some reference temperature T o T o is usually taken to be 20 C α is the temperature coefficient of resistivity SI units of α are o C 1 o
36 Temperature Variation of Resistance Since the resistance of a conductor with uniform cross sectional area is proportional to the resistivity, you can find the effect of temperature on resistance R = R o [1 + α(t  T o )]
37 Resistivity and Temperature, Graphical View For metals, the resistivity is nearly proportional to the temperature A nonlinear region always exists at very low temperatures The resistivity usually reaches some finite value as the temperature approaches absolute zero
38 Residual Resistivity The residual resistivity near absolute zero is caused primarily by the collisions of electrons with impurities and imperfections in the metal High temperature resistivity is predominantly characterized by collisions between the electrons and the metal atoms This is the linear range on the graph
39 Semiconductors Semiconductors are materials that exhibit a decrease in resistivity with an increase in temperature α is negative There is an increase in the density of charge carriers at higher temperatures
40 Superconductors A class of materials and compounds whose resistances fall to virtually zero below a certain temperature, T C T C is called the critical temperature The graph is the same as a normal metal above T C, but suddenly drops to zero at T C
41 Superconductors, cont The value of T C is sensitive to: chemical composition pressure molecular structure Once a current is set up in a superconductor, it persists without any applied voltage Since R = 0
42 Superconductor Application An important application of superconductors is a superconducting magnet The magnitude of the magnetic field is about 10 times greater than a normal electromagnet Used in MRI units
43 Electrical Power Assume a circuit as shown As a charge moves from a to b, the electric potential energy of the system increases by QΔV The chemical energy in the battery must decrease by this same amount
44 Electric Power, final The power is given by the equation: = IΔV Applying Ohm s Law, alternative expressions can be found: 2 2 V = IΔ V = I R = R Units: I is in A, R is in Ω, V is in V, and is in W
45 Electric Power Transmission Real power lines have resistance Power companies transmit electricity at high voltages and low currents to minimize power losses
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