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Sc ott Hughes 24 ebruary 2005 Massa chusetts Institute of Technology Dep ar tment of Physics 8.022 Spring 2005 Lecture 7: Current, continuity equa tion, resist ance, Ohm's la w. 7.1 Electric curren t: basic notions The term \electric curren t" is used to describ the harge er unit time that ﬂo ws through region. In cgs units, curren is measured in esu/sec, naturally enough. In SI units, curren is measured in Coulom bs/sec, whic is giv en the name mp er (or amp). It is ery imp ortan to kno ho to con ert et een these units! Amps are used to describ curren ts AR more often than esu/sec: amp ere 998 10 esu/sec. Supp ose ha sw arm of harges, all with the same harge The numb er density of these harges is some alue (i.e., there are harges er unit olume). Supp ose further that all of these harges are mo ving with elo cit Ho uc curren is ﬂo wing through an area gure this out, need to calculate ho man of the harges pass through the area in time This um er is giv en the um er of harges that t in to the oblique prism sk etc hed elo w: The um er of harges in this prism is its olume, jj cos times the um er densit The um er of harges that pass through the area in time is )( cos The total char ge that passes through is times this um er: so that the curren ust 63

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This form of the curren motiv ates the denition of the curr ent density or this (admittedly articial) example of all harges streaming in the same direction with the same elo cit the curren densit is just the harge densit nq times that elo cit The curren is then giv en Caution ear in mind that the curren densit is curren er unit ar harge/(time area). This is hop efully ob vious dimensional analysis: harge densit is harge/(length cub ed); elo cit is length/time; hence curren densit is harge/(time length squared). 7.2 Electric curren t: more details Some of the details assumed ab are ob viously fairly articial. mak them more realistic one one. First, rather than ha ving single kind of harge that is free to mo e, material migh ha bunc of dieren harges that can carry curren t. or example, cean ater conducts electricit largely ecause of the freely mo ving so dium and hlorine ions. Other materials dissolv ed in the cean con tribute bit as ell. Eac ariet of harge ma ha its wn harge, um er densit and elo cit Let the subscript lab el dieren harge \sp ecies" e.g., could refer to electrons, to hlorine ions, etc. The total curren densit comes from com bining them all together: Note, via this form ula, the equiv alence et een ositiv and negativ harges get ﬂipping the elo cit direction: ositiv harge mo ving with con tributes the same amoun to the erall curren densit as negativ harge mo ving with Next, the assumption that al harges of sp ecies are mo ving with the exact same elo cit is prett ob viously ludicrous. curren of amp corresp onds to (for example) 24 10 18 electrons er second streaming past. There's no ev ery single one of those electrons has the exact same elo cit y! So, in our form ula for replace with its alue eraged er all the harges: =1 where is the total um er of harge carriers of sp ecies and is the elo cit of the th harge in sp ecies (admittedly not the sanest lo oking equation in the univ erse). The curren densit is th us giv en (Note: Purcell giv es somewhat dieren form ula for built from the eraged elo cit nd his form ula to rather confusing; it to ok me half an hour to connect what he do es with wn understanding of curren densit and erages. De mortuis nil nisi on um.) Finally do not exp ect in general to encoun ter nice, simple, uniform surfaces through whic our curren ﬂo ws. Not surprisingly generalize to an in tegral: the curren harge/time passing through some general surface is giv en can th us think of total curren as the ﬂux of curren densit through surface. 64

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7.3 Charge conserv ation and con tin uit Supp ose the curren ﬂo ws through close surface: In this case, there must harge piling up inside (or departing from) the olume enclosed the surface (con tained The min us sign in this equation comes from our con en tion on assigning direction to the area elemen since oin ts out, is negativ if the curren densit oin ts in ard, and is ositiv if the curren densit oin ts out ard. Since in ard corresp onds to gain of harge (and vice ersa) clearly need min us sign for the ph ysics to mak sense. Using our skill at ector calculus, can manipulate this equation as follo ws: dV where is the olume enclosed This is just Gauss's theorem. also ha (enclosed dV Putting this together ha dV dV dV dV 65

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This equation is guaran teed to hold only if This result is kno wn as the ontinuity quation it guaran tees conserv ation of electric harge in the presence of curren ts. It also tells us that when curren ts are ste ady no time ariation, so that =@ ust ha 0. 7.4 Ohm's la Electric elds cause harges to mo e. It stands to reason that an electric eld applied to some material will cause curren ts to ﬂo in that material. or an amazingly wide range of materials, an empirical rule called Ohm's la giv es the follo wing relation et een curren densit and applied electric eld: In other ords, the curren densit is directly prop ortional to the electric eld. The constan of prop ortionalit is called the material's onductivity CA UTION: the sym ol stands for BOTH conductivit AND surface harge densit y!!!! This can anno ying, but is con en tion that are stuc with, sadly ou should alw ys kno from the con text of giv en equation the ph ysical meaning of particular Ev en more sadly there are times when oth meanings of need to used sim ultaneously In suc case, suggest using subscripts to distinguish the sym ols e.g., giv es the conductivit is surface harge densit 66

Page 5

Ohm's la comes in ﬂa ors. The ersion already giv en, can though of as the \lo cal" or \microscopic" ersion. It tells me ab out the relation et een curren densit and eld at any given oint in some material. understand the second ersion, consider curren ﬂo wing in some unk of material. The material is erfectly homogeneous, meaning that the conductivit is throughout the unk. It is of length and has cross-section area put this material in uniform electric eld Let's orien this eld parallel to the material's long axis. Then, the oten tial dierence or \v oltage" et een its ends is ev ery oin in the unk's in terior, Ohm's la holds: (dropping the ector signs since ev erything oin ts along the long axis). Since is constan ev erywhere, and since the unk is homogeneous, this alue of holds ev erywhere. The total curren ﬂo wing past an oin ust no lo ok for relationship et een the curren and the oltage The form ula is the \global" or \macroscopic" form of Ohm's la w: it is en tirely equiv alen to but in olv es quan tities whic are in tegrated er the macroscopic exten of our conducting material. The quan tit in paren theses is dened as the esistanc of the unk of material: L In this con text, the sym ol = is the esistivity of the material. Sadly this is et another example of erusing sym ols careful not to confuse as resistivit with as harge densit The ey thing to note ab out the resistance is that it go es prop ortional to the length of material, and in ersely prop ortional to the cross-sectional area. 67

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Units: In SI units, resistance is measured in Ohms, whic is just sp ecial name for olts/Amp eres. In cgs units, the oltage is measured in esu/cm, and the curren in esu/sec, so the resistance is measured in sec/cm, rather dd unit. rom this, can infer the units of resistivit and conductivit y: resistivit is measured in Ohm-meters in SI units and in seconds in cgs. 7.4.1 Example: Resistance of spherical shell deriv ed the form ula for the resistance of piece of material for ery simple geometry Similar calculations apply ev en if things are arranged in go oer Supp ose ha pair of nested spheres; the space et een them is lled with material whose resistivit is The inner sphere is held at constan oten tial the outer sphere is gr ounde so that its oten tial is zero. The radius of the inner sphere is that of the outer sphere is The curren ﬂo ws steadily there is no harge piling up an ywhere. What is the resistance of this shell? an to use Ohm's la in microscopic form, express the eld in terms of the curren densit in terms of curren and then read o the resistance do that, need to kno the oten tial at an oin et een the plates. By spherical symmetry the oten tial ust clearly ha the form =r for some constan ts and ha the oundary conditions that at at 0: With bit of eort, solv these equations to nd ab In practice, this unit is rarely used. The unit Ohm-cm is used uc more often. 68

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No w, use =@ to nd ab So, no kno the electric eld. Note this oin ts radially Ho uc curren passes through shell of radius somewhere et een and ust ha Note that this curren ust constan t! There is no harge piling up, so the same total harge er unit time passs through an giv en shell of radius No w, imp ose the microscopic form of Ohm's la w: (4 )( (4 ab ab Rearranging to put this in the form putting = read o the resistance of the shell: ab 7.4.2 Example: ariation of What happ ens if is not constan t? The answ er is set requiring that the curren ﬂo steady Consider conductors of conductivit and fused together: Area A The curren ust the same in eac half: This can only true if the electric eld is dieren in eac material. The electric eld th us suddenly jumps as cross from one side of the junction to the other, whic means that there ust la er of surface harge at the oundary Whenev er the conductivit aries, there is ossibilit that, in the steady state, some harge accum ulates somewhere. This is necessary to main tain steady curren ﬂo w. Here's where the eruse of the sym ol really gets anno ying. Sorry 69

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7.5 closer lo ok at Ohm's la Ohm's la w, on the face of it, mak es no sense. It tells us that under an applied electric eld, get steady curren densit Since (considering only one curren carrier), this means that the electric eld is causing curren ts to mo at constan elo cit This should bug ou! Electric elds cause for es forces cause ac eler ations Wh don't the harges just accelerate lik mad? The answ er is that the harges are in medium the conductor that prev en ts them from mo ving freely The motion is essen tially lik that in viscous medium. The situation is uc closer to sky div er in free fall after short erio of acceleration, the harges reac \terminal elo cit y", and essen tially ride along at that elo cit from then on. This \terminal elo cit y" is what app ears in the curren densit form ula. The cause of this eectiv ely viscous motion is (roughly) the man collisions that the harge carriers exp erience as they ﬂo through the conductor. The harge carriers accelerate lik hell for short time, smash in to something, get directed in some random direction, harge lik hell again, and rep eat The net aver age eha vior is uniform drift. More concretely let's lo ok at ho the momen tum of harge carriers accum ulate in conductor. Consider single harge rst. Supp ose at collision just ccurred. Its momen tum follo wing this collision where is the random elo cit it has follo wing the collision. The electric eld acting on this harge hanges its momen tum er time in terv al imparting an impulse The total momen tum of this particle, th us gro ws linearly with time un til the next collision ccurs, at whic oin the clo is reset and rep eat. No consider whole sw arm of of these harges. The aver age momen tum of mem er of this ensem ble is =1 The elo cit is the random elo cit imparted to harge follo wing its last collision; is the time in terv al for that harge since that collision. If is large um er, the rst term must extr emely close to zer Wh y? e're adding up enormous um er of randomly orien ted ectors! The result of suc addition is alw ys um er that is ery ery close to zero, pro vided is large What remains is =1 where =1 is the erage time et een collisions. This turns out to prop ert of the material; this, then, is what causes an imp osed electric eld to lead to uniform elo cit If ou'v ev er driv en through Los Angeles during rush hour, ou'v exp erienced the life of harge carrier in conductor. careful analysis of this kind of addition sho ws that there is some error incurred assuming that the ectors sum to precisely zero. Ho ev er, that error is of order As long as is ery large whic is the case for an go conductor, where is of order Av ogadro's um er these small errors are not imp ortan t. Those of ou who tak 8.044 will learn more ab out this. 70

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Notice that from this deriv ation, can just read o the conductivit y: nq nq nq If ha ultiple harge carriers, nd 71

022 Spring 2005 Lecture 7 Current continuity equa tion resist ance Ohms la w 71 Electric curren t basic notions The term electric curren t is used to describ the harge er unit time that 64258o ws through region In cgs units curren is measured in esus ID: 23419

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Page 1

Sc ott Hughes 24 ebruary 2005 Massa chusetts Institute of Technology Dep ar tment of Physics 8.022 Spring 2005 Lecture 7: Current, continuity equa tion, resist ance, Ohm's la w. 7.1 Electric curren t: basic notions The term \electric curren t" is used to describ the harge er unit time that ﬂo ws through region. In cgs units, curren is measured in esu/sec, naturally enough. In SI units, curren is measured in Coulom bs/sec, whic is giv en the name mp er (or amp). It is ery imp ortan to kno ho to con ert et een these units! Amps are used to describ curren ts AR more often than esu/sec: amp ere 998 10 esu/sec. Supp ose ha sw arm of harges, all with the same harge The numb er density of these harges is some alue (i.e., there are harges er unit olume). Supp ose further that all of these harges are mo ving with elo cit Ho uc curren is ﬂo wing through an area gure this out, need to calculate ho man of the harges pass through the area in time This um er is giv en the um er of harges that t in to the oblique prism sk etc hed elo w: The um er of harges in this prism is its olume, jj cos times the um er densit The um er of harges that pass through the area in time is )( cos The total char ge that passes through is times this um er: so that the curren ust 63

Page 2

This form of the curren motiv ates the denition of the curr ent density or this (admittedly articial) example of all harges streaming in the same direction with the same elo cit the curren densit is just the harge densit nq times that elo cit The curren is then giv en Caution ear in mind that the curren densit is curren er unit ar harge/(time area). This is hop efully ob vious dimensional analysis: harge densit is harge/(length cub ed); elo cit is length/time; hence curren densit is harge/(time length squared). 7.2 Electric curren t: more details Some of the details assumed ab are ob viously fairly articial. mak them more realistic one one. First, rather than ha ving single kind of harge that is free to mo e, material migh ha bunc of dieren harges that can carry curren t. or example, cean ater conducts electricit largely ecause of the freely mo ving so dium and hlorine ions. Other materials dissolv ed in the cean con tribute bit as ell. Eac ariet of harge ma ha its wn harge, um er densit and elo cit Let the subscript lab el dieren harge \sp ecies" e.g., could refer to electrons, to hlorine ions, etc. The total curren densit comes from com bining them all together: Note, via this form ula, the equiv alence et een ositiv and negativ harges get ﬂipping the elo cit direction: ositiv harge mo ving with con tributes the same amoun to the erall curren densit as negativ harge mo ving with Next, the assumption that al harges of sp ecies are mo ving with the exact same elo cit is prett ob viously ludicrous. curren of amp corresp onds to (for example) 24 10 18 electrons er second streaming past. There's no ev ery single one of those electrons has the exact same elo cit y! So, in our form ula for replace with its alue eraged er all the harges: =1 where is the total um er of harge carriers of sp ecies and is the elo cit of the th harge in sp ecies (admittedly not the sanest lo oking equation in the univ erse). The curren densit is th us giv en (Note: Purcell giv es somewhat dieren form ula for built from the eraged elo cit nd his form ula to rather confusing; it to ok me half an hour to connect what he do es with wn understanding of curren densit and erages. De mortuis nil nisi on um.) Finally do not exp ect in general to encoun ter nice, simple, uniform surfaces through whic our curren ﬂo ws. Not surprisingly generalize to an in tegral: the curren harge/time passing through some general surface is giv en can th us think of total curren as the ﬂux of curren densit through surface. 64

Page 3

7.3 Charge conserv ation and con tin uit Supp ose the curren ﬂo ws through close surface: In this case, there must harge piling up inside (or departing from) the olume enclosed the surface (con tained The min us sign in this equation comes from our con en tion on assigning direction to the area elemen since oin ts out, is negativ if the curren densit oin ts in ard, and is ositiv if the curren densit oin ts out ard. Since in ard corresp onds to gain of harge (and vice ersa) clearly need min us sign for the ph ysics to mak sense. Using our skill at ector calculus, can manipulate this equation as follo ws: dV where is the olume enclosed This is just Gauss's theorem. also ha (enclosed dV Putting this together ha dV dV dV dV 65

Page 4

This equation is guaran teed to hold only if This result is kno wn as the ontinuity quation it guaran tees conserv ation of electric harge in the presence of curren ts. It also tells us that when curren ts are ste ady no time ariation, so that =@ ust ha 0. 7.4 Ohm's la Electric elds cause harges to mo e. It stands to reason that an electric eld applied to some material will cause curren ts to ﬂo in that material. or an amazingly wide range of materials, an empirical rule called Ohm's la giv es the follo wing relation et een curren densit and applied electric eld: In other ords, the curren densit is directly prop ortional to the electric eld. The constan of prop ortionalit is called the material's onductivity CA UTION: the sym ol stands for BOTH conductivit AND surface harge densit y!!!! This can anno ying, but is con en tion that are stuc with, sadly ou should alw ys kno from the con text of giv en equation the ph ysical meaning of particular Ev en more sadly there are times when oth meanings of need to used sim ultaneously In suc case, suggest using subscripts to distinguish the sym ols e.g., giv es the conductivit is surface harge densit 66

Page 5

Ohm's la comes in ﬂa ors. The ersion already giv en, can though of as the \lo cal" or \microscopic" ersion. It tells me ab out the relation et een curren densit and eld at any given oint in some material. understand the second ersion, consider curren ﬂo wing in some unk of material. The material is erfectly homogeneous, meaning that the conductivit is throughout the unk. It is of length and has cross-section area put this material in uniform electric eld Let's orien this eld parallel to the material's long axis. Then, the oten tial dierence or \v oltage" et een its ends is ev ery oin in the unk's in terior, Ohm's la holds: (dropping the ector signs since ev erything oin ts along the long axis). Since is constan ev erywhere, and since the unk is homogeneous, this alue of holds ev erywhere. The total curren ﬂo wing past an oin ust no lo ok for relationship et een the curren and the oltage The form ula is the \global" or \macroscopic" form of Ohm's la w: it is en tirely equiv alen to but in olv es quan tities whic are in tegrated er the macroscopic exten of our conducting material. The quan tit in paren theses is dened as the esistanc of the unk of material: L In this con text, the sym ol = is the esistivity of the material. Sadly this is et another example of erusing sym ols careful not to confuse as resistivit with as harge densit The ey thing to note ab out the resistance is that it go es prop ortional to the length of material, and in ersely prop ortional to the cross-sectional area. 67

Page 6

Units: In SI units, resistance is measured in Ohms, whic is just sp ecial name for olts/Amp eres. In cgs units, the oltage is measured in esu/cm, and the curren in esu/sec, so the resistance is measured in sec/cm, rather dd unit. rom this, can infer the units of resistivit and conductivit y: resistivit is measured in Ohm-meters in SI units and in seconds in cgs. 7.4.1 Example: Resistance of spherical shell deriv ed the form ula for the resistance of piece of material for ery simple geometry Similar calculations apply ev en if things are arranged in go oer Supp ose ha pair of nested spheres; the space et een them is lled with material whose resistivit is The inner sphere is held at constan oten tial the outer sphere is gr ounde so that its oten tial is zero. The radius of the inner sphere is that of the outer sphere is The curren ﬂo ws steadily there is no harge piling up an ywhere. What is the resistance of this shell? an to use Ohm's la in microscopic form, express the eld in terms of the curren densit in terms of curren and then read o the resistance do that, need to kno the oten tial at an oin et een the plates. By spherical symmetry the oten tial ust clearly ha the form =r for some constan ts and ha the oundary conditions that at at 0: With bit of eort, solv these equations to nd ab In practice, this unit is rarely used. The unit Ohm-cm is used uc more often. 68

Page 7

No w, use =@ to nd ab So, no kno the electric eld. Note this oin ts radially Ho uc curren passes through shell of radius somewhere et een and ust ha Note that this curren ust constan t! There is no harge piling up, so the same total harge er unit time passs through an giv en shell of radius No w, imp ose the microscopic form of Ohm's la w: (4 )( (4 ab ab Rearranging to put this in the form putting = read o the resistance of the shell: ab 7.4.2 Example: ariation of What happ ens if is not constan t? The answ er is set requiring that the curren ﬂo steady Consider conductors of conductivit and fused together: Area A The curren ust the same in eac half: This can only true if the electric eld is dieren in eac material. The electric eld th us suddenly jumps as cross from one side of the junction to the other, whic means that there ust la er of surface harge at the oundary Whenev er the conductivit aries, there is ossibilit that, in the steady state, some harge accum ulates somewhere. This is necessary to main tain steady curren ﬂo w. Here's where the eruse of the sym ol really gets anno ying. Sorry 69

Page 8

7.5 closer lo ok at Ohm's la Ohm's la w, on the face of it, mak es no sense. It tells us that under an applied electric eld, get steady curren densit Since (considering only one curren carrier), this means that the electric eld is causing curren ts to mo at constan elo cit This should bug ou! Electric elds cause for es forces cause ac eler ations Wh don't the harges just accelerate lik mad? The answ er is that the harges are in medium the conductor that prev en ts them from mo ving freely The motion is essen tially lik that in viscous medium. The situation is uc closer to sky div er in free fall after short erio of acceleration, the harges reac \terminal elo cit y", and essen tially ride along at that elo cit from then on. This \terminal elo cit y" is what app ears in the curren densit form ula. The cause of this eectiv ely viscous motion is (roughly) the man collisions that the harge carriers exp erience as they ﬂo through the conductor. The harge carriers accelerate lik hell for short time, smash in to something, get directed in some random direction, harge lik hell again, and rep eat The net aver age eha vior is uniform drift. More concretely let's lo ok at ho the momen tum of harge carriers accum ulate in conductor. Consider single harge rst. Supp ose at collision just ccurred. Its momen tum follo wing this collision where is the random elo cit it has follo wing the collision. The electric eld acting on this harge hanges its momen tum er time in terv al imparting an impulse The total momen tum of this particle, th us gro ws linearly with time un til the next collision ccurs, at whic oin the clo is reset and rep eat. No consider whole sw arm of of these harges. The aver age momen tum of mem er of this ensem ble is =1 The elo cit is the random elo cit imparted to harge follo wing its last collision; is the time in terv al for that harge since that collision. If is large um er, the rst term must extr emely close to zer Wh y? e're adding up enormous um er of randomly orien ted ectors! The result of suc addition is alw ys um er that is ery ery close to zero, pro vided is large What remains is =1 where =1 is the erage time et een collisions. This turns out to prop ert of the material; this, then, is what causes an imp osed electric eld to lead to uniform elo cit If ou'v ev er driv en through Los Angeles during rush hour, ou'v exp erienced the life of harge carrier in conductor. careful analysis of this kind of addition sho ws that there is some error incurred assuming that the ectors sum to precisely zero. Ho ev er, that error is of order As long as is ery large whic is the case for an go conductor, where is of order Av ogadro's um er these small errors are not imp ortan t. Those of ou who tak 8.044 will learn more ab out this. 70

Page 9

Notice that from this deriv ation, can just read o the conductivit y: nq nq nq If ha ultiple harge carriers, nd 71

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