Notes on Current Electricity (Chapter 3)

3.1 INTRODUCTION

  • In Chapter 1, charges (free or bound) were considered at rest. Currents arise from moving charges and constitute electric current.

  • Currents occur naturally (e.g., lightning) and in everyday devices where charges flow steadily (e.g., torch, clock with a cell).

  • Aim of this chapter: study basic laws governing steady electric currents.

3.2 ELECTRIC CURRENT

  • Consider a small area normal to the direction of charge flow.

    • Let q+ be the net amount of positive charge crossing the area in the forward direction during time t.

    • Let q− be the net amount of negative charge crossing the area in the forward direction during time t.

    • Net forward charge across the area in time t is q = q+ − q−.

  • For steady current, q ∝ t, so the current across the area in the forward direction is I=qt(3.1)I = \frac{q}{t} \quad (3.1)

    • If this quantity is negative, it implies a current in the backward direction.

  • General definition (time-varying currents):

    • Let ∆Q be the net charge flowing across a cross-section during ∆t (between t and t+∆t).

    • The current at time t is
      I(t) = \lim_{\∆t \to 0} \frac{∆Q}{∆t} \equiv I \,.

  • SI unit of current: ampere (A). An ampere is defined through magnetic effects of currents (to be studied in the next chapter). Typical magnitudes:

    • Domestic currents: tens of amperes

    • Lightning: tens of thousands of amperes

    • Nerves: microamperes

3.3 ELECTRIC CURRENTS IN CONDUCTORS

  • Free charges experience a force in an electric field and contribute to current if free to move.

  • In atoms/molecules, electrons and nuclei are bound; bulk matter consists of many molecules. In metals, some electrons are effectively free to move; in electrolytes both ions can move.

  • For solid conductors (metals): current is carried by electrons, with positive lattice ions fixed.

  • Behavior without field:

    • Thermal motion causes random collisions with ions; post-collision velocity is random, so there is no net current on average.

  • Behavior with an applied field (thinking of a cylindrical conductor):

    • Apply a steady field; electrons move toward the positive side to neutralize charges, creating a brief current.

    • If charges at the ends are replenished (e.g., by a battery), a steady field exists inside the conductor, resulting in a steady current.

  • Concept of steady current sources: cells or batteries maintain a steady electric field inside conductors.

3.4 OHM’S LAW

  • Ohm’s law (discovered by G. S. Ohm, 1828): for a conductor with current I and end-to-end potential difference V,
    V∝IorV=RI (3.3)V \propto I \quad\text{or}\quad V = RI \,(3.3)
    where R is the resistance of the conductor.

  • Resistance depends on material and geometry. Consider a slab of length l and cross-sectional area A.

    • If two identical slabs are placed end-to-end, total length 2l, current through both is the same, and total voltage is 2V. Therefore, the resistance doubles:
      R_{C} = \frac{V}{I} = \frac{2V}{I} = 2R \Rightarrow R \propto l. \tag{3.4}

    • Halving cross-sectional area (A → A/2) doubles the resistance: R \propto \frac{1}{A}. \tag{3.7}

    • Combining, for a conductor of length l and cross-section A, R = \rho \frac{l}{A}, \tag{3.9}
      where ρ is resistivity (material-dependent).

  • Ohm’s law in terms of current density and electric field:

    • With V = El (for length l), and current I through area A, current density j = I/A, the relation becomes
      I = j A, \quad E = \frac{V}{l} \Rightarrow E = j\rho \quad\text{or}\quad E = \rho j, \tag{3.11}

    • In vector form, j = σE with conductivity σ ≡ 1/ρ. Ohm’s law can be written as
      \mathbf{E} = \rho \mathbf{j} \quad\text{or}\quad \mathbf{j} = \sigma \mathbf{E}, \tag{3.12} {{ (equivalently) } }</p></li></ul></li><li><p>OriginofOhm’slaw(Drude−likepicture):driftofelectronsunderanelectricfield</p><ul><li><p>Inthepresenceofanelectricfield,electronsacceleratebetweencollisions.Ifτistheaveragetimebetweencollisions(relaxationtime),theaveragedriftvelocityis<br></p></li></ul></li><li><p>Origin of Ohm’s law (Drude-like picture): drift of electrons under an electric field</p><ul><li><p>In the presence of an electric field, electrons accelerate between collisions. If τ is the average time between collisions (relaxation time), the average drift velocity is<br> \mathbf{v}_d = \frac{e\mathbf{E}\tau}{m}, \tag{3.17} </p></li><li><p>Thecurrentdensityfromdriftisrelatedtothenumberdensitynoffreeelectrons:</p></li><li><p>ChargetransportacrossareaAintime∆t:∆Q=−neA∣vd∣∆t,leadingtocurrentIandcurrentdensityj.</p></li><li><p>Fromthis,onederivesOhm’slawintheformj=σEwith<br></p></li><li><p>The current density from drift is related to the number density n of free electrons:</p></li><li><p>Charge transport across area A in time ∆t: ∆Q = −neA|v_d|∆t, leading to current I and current density j.</p></li><li><p>From this, one derives Ohm’s law in the form j = σE with<br> \sigma = \frac{ne^2\tau}{m}. \tag{3.23} </p></li></ul></li><li><p>Mobility(µ):magnitudeofdriftvelocityperunitelectricfield,μ=∣vd∣/∣E∣.</p><ul><li><p>Fromvd=eEτ/m,mobilityis<br></p></li></ul></li><li><p>Mobility (µ): magnitude of drift velocity per unit electric field, μ = |v_d|/|E|.</p><ul><li><p>From vd = eEτ/m, mobility is<br> \mu = \frac{e\tau}{m}. \tag{3.25} </p></li></ul></li></ul><h3id="0579360f−a78e−4c67−9e3d−c103ee60ef52"data−toc−id="0579360f−a78e−4c67−9e3d−c103ee60ef52"collapsed="false"seolevelmigrated="true">3.5DRIFTOFELECTRONSANDORIGINOFRESISTIVITY</h3><ul><li><p>Inzerofield,averagevelocityofelectronsiszeroduetorandomdirections(Eq.3.14).</p></li><li><p>Withanelectricfield,electronsgainvelocityinthefieldbutrandomcollisionskeepasteadydriftvelocity(Eq.3.17).</p></li><li><p>Driftvelocityistypicallysmall,e.g.,forcopper,vd≈few×10−3ms−1(Example3.1resultsbelow).</p></li><li><p>Derivationshowsdriftcurrentisduetoelectronsmovingoppositetothefieldwhileoverallcurrentisdirectedwiththefieldduetopositivecharges,andcollisionsheatthelattice(dissipation).</p></li></ul><h3id="2b4462d7−a504−42b7−9eb3−a7cf1128b44f"data−toc−id="2b4462d7−a504−42b7−9eb3−a7cf1128b44f"collapsed="false"seolevelmigrated="true">3.5.1Mobility</h3><ul><li><p>Mobilityμdefinedasvd/E;formetalsμ=eτ/m,withunitsm2/(V⋅s).</p></li><li><p>Typicalorderofmagnitude:μ≈10−4to10−2m2/(V⋅s)formetals(varieswithmaterialandτ).</p></li></ul><h3id="c7859b7e−ace9−4cac−8733−47f5135d2963"data−toc−id="c7859b7e−ace9−4cac−8733−47f5135d2963"collapsed="false"seolevelmigrated="true">3.6LIMITATIONSOFOHM’SLAW</h3><ul><li><p>Ohm’slawholdsformanymaterialsbutnotall;deviationsinclude:</p><ul><li><p>(a)VisnotproportionaltoI(nonlinear,asinsomedevices).</p></li><li><p>(b)TheV–IrelationdependsonthesignofV(e.g.,diodes).</p></li><li><p>(c)TheV–Irelationisnotunique(multipleVforsameI,e.g.,GaAs).</p></li></ul></li><li><p>SomedevicesdonotobeyOhm’slaw,butmanychaptersstudyOhmicmaterialsthatdoobeyit.</p></li></ul><h3id="a7daf16d−0e81−43f7−9a32−04f5d79e0df8"data−toc−id="a7daf16d−0e81−43f7−9a32−04f5d79e0df8"collapsed="false"seolevelmigrated="true">3.7RESISTIVITYOFVARIOUSMATERIALS</h3><ul><li><p>Resistivityscaleplacesconductors,semiconductors,andinsulatorsinincreasingresistivityorder.</p></li><li><p>Metalstypicallyhaveresistivitiesρintherange 10−8to10−6Ω⋅m.</p></li><li><p>Insulatorshaveveryhighresistivities(e.g.,glass,rubber,etc.upto 1016Ω⋅mandbeyond).</p></li><li><p>Temperaturecoefficients(α)describehowresistivitychangeswithtemperature(ρ<em>T=ρ</em>0[1+α(T−T0)]).</p></li><li><p>Somerepresentativevalues(ρat0°C,αat0°C):</p><ul><li><p>Conductors:Silverρ≈1.6×10−8Ω⋅m,α≈0.0041°C−1;Copperρ≈1.7×10−8Ω⋅m,α≈0.0068°C−1;Aluminiumρ≈2.7×10−8Ω⋅m,α≈0.0043°C−1;etc.</p></li><li><p>Semiconductors:Graphiteρ≈3.5×10−5Ω⋅m;Germaniumρ≈0.46Ω⋅m;Siliconρ≈2300Ω⋅m;(note:thesevaluesvarywithtemperatureanddoping).</p></li><li><p>Insulators:PureWaterρ≈2.5×105Ω⋅m;Glassρ≈1010–1014Ω⋅m;FusedQuartzρ≈1016Ω⋅m.</p></li></ul></li><li><p>Resistorsfordomesticuse:twomajortypes</p><ul><li><p>Wire−boundresistors:manganin,constantan,nichrome;relativelytemperature−insensitive.</p></li><li><p>Carbonresistors:compactandinexpensive;theirvaluesarecodedbycolourbands(Table3.2).</p></li></ul></li></ul><h3id="48302e06−57a7−4f32−a5e1−c5ae248a3bff"data−toc−id="48302e06−57a7−4f32−a5e1−c5ae248a3bff"collapsed="false"seolevelmigrated="true">3.2RESISTORCOLOURCODES(Table3.2)</h3><ul><li><p>Colourbandsmaptodigitsandmultiplier/tolerance:</p><ul><li><p>Black0,Brown1,Red2,Orange3,Yellow4,Green5,Blue6,Violet7,Gray8,White9</p></li><li><p>Multiplier:Black100,Brown101,Red102,Orange103,Yellow104,Green105,Blue106,Violet107,Gray108,White109;Gold10−1;Silver10−2</p></li><li><p>Tolerance:Gold5</p></li></ul></li></ul><h3 id="0579360f-a78e-4c67-9e3d-c103ee60ef52" data-toc-id="0579360f-a78e-4c67-9e3d-c103ee60ef52" collapsed="false" seolevelmigrated="true">3.5 DRIFT OF ELECTRONS AND ORIGIN OF RESISTIVITY</h3><ul><li><p>In zero field, average velocity of electrons is zero due to random directions (Eq. 3.14).</p></li><li><p>With an electric field, electrons gain velocity in the field but random collisions keep a steady drift velocity (Eq. 3.17).</p></li><li><p>Drift velocity is typically small, e.g., for copper, vd ≈ few × 10^−3 m s^−1 (Example 3.1 results below).</p></li><li><p>Derivation shows drift current is due to electrons moving opposite to the field while overall current is directed with the field due to positive charges, and collisions heat the lattice (dissipation).</p></li></ul><h3 id="2b4462d7-a504-42b7-9eb3-a7cf1128b44f" data-toc-id="2b4462d7-a504-42b7-9eb3-a7cf1128b44f" collapsed="false" seolevelmigrated="true">3.5.1 Mobility</h3><ul><li><p>Mobility μ defined as vd/E; for metals μ = eτ/m, with units m^2/(V·s).</p></li><li><p>Typical order of magnitude: μ ≈ 10^−4 to 10^−2 m^2/(V·s) for metals (varies with material and τ).</p></li></ul><h3 id="c7859b7e-ace9-4cac-8733-47f5135d2963" data-toc-id="c7859b7e-ace9-4cac-8733-47f5135d2963" collapsed="false" seolevelmigrated="true">3.6 LIMITATIONS OF OHM’S LAW</h3><ul><li><p>Ohm’s law holds for many materials but not all; deviations include:</p><ul><li><p>(a) V is not proportional to I (nonlinear, as in some devices).</p></li><li><p>(b) The V–I relation depends on the sign of V (e.g., diodes).</p></li><li><p>(c) The V–I relation is not unique (multiple V for same I, e.g., GaAs).</p></li></ul></li><li><p>Some devices do not obey Ohm’s law, but many chapters study Ohmic materials that do obey it.</p></li></ul><h3 id="a7daf16d-0e81-43f7-9a32-04f5d79e0df8" data-toc-id="a7daf16d-0e81-43f7-9a32-04f5d79e0df8" collapsed="false" seolevelmigrated="true">3.7 RESISTIVITY OF VARIOUS MATERIALS</h3><ul><li><p>Resistivity scale places conductors, semiconductors, and insulators in increasing resistivity order.</p></li><li><p>Metals typically have resistivities ρ in the range ~10^−8 to 10^−6 Ω·m.</p></li><li><p>Insulators have very high resistivities (e.g., glass, rubber, etc. up to ~10^16 Ω·m and beyond).</p></li><li><p>Temperature coefficients (α) describe how resistivity changes with temperature (ρ<em>T = ρ</em>0[1 + α(T − T_0)]).</p></li><li><p>Some representative values (ρ at 0°C, α at 0°C):</p><ul><li><p>Conductors: Silver ρ ≈ 1.6×10^−8 Ω·m, α ≈ 0.0041 °C^−1; Copper ρ ≈ 1.7×10^−8 Ω·m, α ≈ 0.0068 °C^−1; Aluminium ρ ≈ 2.7×10^−8 Ω·m, α ≈ 0.0043 °C^−1; etc.</p></li><li><p>Semiconductors: Graphite ρ ≈ 3.5×10^−5 Ω·m; Germanium ρ ≈ 0.46 Ω·m; Silicon ρ ≈ 2300 Ω·m; (note: these values vary with temperature and doping).</p></li><li><p>Insulators: Pure Water ρ ≈ 2.5×10^5 Ω·m; Glass ρ ≈ 10^10–10^14 Ω·m; Fused Quartz ρ ≈ 10^16 Ω·m.</p></li></ul></li><li><p>Resistors for domestic use: two major types</p><ul><li><p>Wire-bound resistors: manganin, constantan, nichrome; relatively temperature-insensitive.</p></li><li><p>Carbon resistors: compact and inexpensive; their values are coded by colour bands (Table 3.2).</p></li></ul></li></ul><h3 id="48302e06-57a7-4f32-a5e1-c5ae248a3bff" data-toc-id="48302e06-57a7-4f32-a5e1-c5ae248a3bff" collapsed="false" seolevelmigrated="true">3.2 RESISTOR COLOUR CODES (Table 3.2)</h3><ul><li><p>Colour bands map to digits and multiplier/tolerance:</p><ul><li><p>Black 0, Brown 1, Red 2, Orange 3, Yellow 4, Green 5, Blue 6, Violet 7, Gray 8, White 9</p></li><li><p>Multiplier: Black 10^0, Brown 10^1, Red 10^2, Orange 10^3, Yellow 10^4, Green 10^5, Blue 10^6, Violet 10^7, Gray 10^8, White 10^9; Gold 10^−1; Silver 10^−2</p></li><li><p>Tolerance: Gold 5%, Silver 10%, No colour 20%</p></li></ul></li><li><p>Example: four colours orange, blue, yellow, gold → 36 × 10^4 Ω with 5% tolerance.</p></li></ul><h3 id="12b67b29-5c5a-444e-ad07-012ca9339fcd" data-toc-id="12b67b29-5c5a-444e-ad07-012ca9339fcd" collapsed="false" seolevelmigrated="true">3.8 TEMPERATURE DEPENDENCE OF RESISTIVITY</h3><ul><li><p>Metallic conductors: ρ<em>T ≈ ρ</em>0 [1 + α (T − T_0)], α &gt; 0. This is approximately linear over moderate ranges around T0.</p></li><li><p>Nichrome, Manganin, Constantan: very weak temperature dependence (α small).</p></li><li><p>Semiconductors: resistivity decreases with increasing temperature due to increasing carrier concentration n with T; in metals, n is almost temperature independent, so ρ increases because τ decreases with T.</p></li><li><p>Figures (3.9–3.11) illustrate ρ_T vs T for copper, nichrome, and semiconductors.</p></li></ul><h3 id="de5d41a7-430a-4f85-a328-1974da1661e9" data-toc-id="de5d41a7-430a-4f85-a328-1974da1661e9" collapsed="false" seolevelmigrated="true">EXAMPLES AND PROBLEMS (selected summary)</h3><h3 id="c0761918-b995-4182-8c2e-eeac19a3b447" data-toc-id="c0761918-b995-4182-8c2e-eeac19a3b447" collapsed="false" seolevelmigrated="true">EXAMPLE 3.1: AVERAGE DRIFT SPEED IN COPPER</h3><ul><li><p>Given: Copper wire cross-sectional area A = 1.0×10^−7 m^2; current I = 1.5 A; assume one conduction electron per Cu atom.</p></li><li><p>Density of Cu: ρ_mass = 9.0×10^3 kg/m^3; atomic mass M = 63.5 u.</p></li><li><p>Compute electron density n from copper mass density and atomic mass; n ≈ 8.5×10^28 m^−3.</p></li><li><p>Drift speed from Eq. (3.18): I = neA vd → vd = I/(neA).</p></li><li><p>Result: vd ≈ 1.1×10^−3 m/s (≈ 1.1 mm/s).</p></li></ul><h3 id="fff2e59a-ddb4-49d1-bdcb-f5777fc7a6f1" data-toc-id="fff2e59a-ddb4-49d1-bdcb-f5777fc7a6f1" collapsed="false" seolevelmigrated="true">EXAMPLE 3.1 (part b): COMPARISONS</h3><ul><li><p>(i) Thermal speeds of Cu atoms at 300 K are ~2×10^2 m/s (much larger than vd).</p></li><li><p>(ii) Speed of electromagnetic field along a conductor is ~3×10^8 m/s; drift speed is ~10^−11 times this magnitude.</p></li></ul><h3 id="d7a1471c-125d-42d5-acee-c259ce92a562" data-toc-id="d7a1471c-125d-42d5-acee-c259ce92a562" collapsed="false" seolevelmigrated="true">EXAMPLE 3.2: RATIONALE BEHIND INSTANT CURRENT ESTABLISHMENT</h3><ul><li><p>Question: If drift speed is small, how is current established almost instant? Answer: The electric field is established nearly instantaneously throughout the circuit (speed of light). Local drift begins immediately at every point without waiting for charge to travel from one end to the other.</p></li><li><p>Other parts: drift arises from acceleration and collisions; current remains steady because the average drift velocity is constant due to relaxation time τ.</p></li><li><p>(c) Large currents arise because electron density n ≈ 10^29 m^−3.</p></li><li><p>(d) Drift velocity is superposed on random thermal motion; not all free electrons move in the same direction.</p></li><li><p>(e) Paths: straight lines between collisions in the absence of E; curved paths when E present.</p></li></ul><h3 id="d96538a6-39cd-41ac-9eb5-1b69d86ee8eb" data-toc-id="d96538a6-39cd-41ac-9eb5-1b69d86ee8eb" collapsed="false" seolevelmigrated="true">3.5.1 MOBILITY (revisited)</h3><ul><li><p>Mobility μ = vd/E; from vd = (eEτ)/m, μ = eτ/m. (Unit: m^2/(V·s).)</p></li></ul><h3 id="bb85edb0-d67b-4788-aaae-3996b04a8130" data-toc-id="bb85edb0-d67b-4788-aaae-3996b04a8130" collapsed="false" seolevelmigrated="true">3.6 LIMITATIONS OF OHM’S LAW (revisited)</h3><ul><li><p>Non-Ohmic materials/devices exist (diodes, GaAs, etc.).</p></li></ul><h3 id="af6a9a3c-79d6-4551-a5e6-857a50c85e72" data-toc-id="af6a9a3c-79d6-4551-a5e6-857a50c85e72" collapsed="false" seolevelmigrated="true">3.7–3.9 RESISTIVITY, RESISTORS, TEMPERATURE COEFFICIENTS (summary)</h3><ul><li><p>Temperature dependence and practical resistor types (wire-wound vs carbon) with typical values and tolerances.</p></li><li><p>Color codes (Table 3.2) used to denote resistor values.</p></li></ul><h3 id="12a838d1-ce72-49db-a0e0-928cb07cf1b5" data-toc-id="12a838d1-ce72-49db-a0e0-928cb07cf1b5" collapsed="false" seolevelmigrated="true">3.9 ELECTRICAL ENERGY, POWER</h3><ul><li><p>For a circuit with current I, across ends A and B with potentials V(A) and V(B):</p><ul><li><p>The energy change for a charge ∆Q = I∆t is associated with potential change and energy transfer.</p></li></ul></li><li><p>Power dissipated in a conductor due to resistance: <br> P = IV = I^2 R = \frac{V^2}{R}. \tag{3.32-3.33} </p></li><li><p>Thispowerlossisheatintheconductorandmustbesuppliedbyanexternalsource(e.g.,chemicalenergyofacell).</p></li><li><p>Transmissionlines:powerlossPcinthewireswithresistanceRcisminimizedbyincreasingtransmissionvoltageV(Pc=I2Rc,andP=VI).HigherVallowslowercurrentforthesamepower,reducinglosses.Atransformerisusedtostepup/downvoltageasneeded.</p></li></ul><h3id="f63f2290−475e−4d51−a9bf−b7edcba27755"data−toc−id="f63f2290−475e−4d51−a9bf−b7edcba27755"collapsed="false"seolevelmigrated="true">3.10COMBINATIONOFRESISTORS—SERIESANDPARALLEL</h3><ul><li><p>Series:samecurrentthroughallresistors;V=IR1+IR2+…;Req=R1+R2+….</p></li><li><p>Parallel:I=I1+I2+…;Vacrosseachresistoristhesame;fortworesistorsinparallel,1/R<em>eq=1/R1+1/R2;generalizationtonresistors:</p></li><li><p>This power loss is heat in the conductor and must be supplied by an external source (e.g., chemical energy of a cell).</p></li><li><p>Transmission lines: power loss Pc in the wires with resistance Rc is minimized by increasing transmission voltage V (Pc = I^2 Rc, and P = VI). Higher V allows lower current for the same power, reducing losses. A transformer is used to step up/down voltage as needed.</p></li></ul><h3 id="f63f2290-475e-4d51-a9bf-b7edcba27755" data-toc-id="f63f2290-475e-4d51-a9bf-b7edcba27755" collapsed="false" seolevelmigrated="true">3.10 COMBINATION OF RESISTORS — SERIES AND PARALLEL</h3><ul><li><p>Series: same current through all resistors; V = IR1 + IR2 + …; R_eq = R1 + R2 + … .</p></li><li><p>Parallel: I = I1 + I2 + …; V across each resistor is the same; for two resistors in parallel, 1/R<em>eq = 1/R1 + 1/R2; generalization to n resistors: R{eq} = \left(\sum{i=1}^n \frac{1}{Ri}\right)^{-1}. \tag{3.51} </p></li><li><p>Examplesshowcombiningseries/parallelblockstoreducecomplexnetworks(e.g.,Fig.3.17andEq.3.54).</p></li></ul><h3id="2b3d85ac−0cfe−4ea6−902b−7987820a50d3"data−toc−id="2b3d85ac−0cfe−4ea6−902b−7987820a50d3"collapsed="false"seolevelmigrated="true">3.11CELLS,EMF,INTERNALRESISTANCE</h3><ul><li><p>AcellhastwoelectrodesP(positive)andN(negative)inanelectrolyte.Theemfεistheopen−circuitpotentialdifference:ε=V++V−.InpresenceofafiniteloadR(externalcircuit),theterminalpotentialdifferenceis<br></p></li><li><p>Examples show combining series/parallel blocks to reduce complex networks (e.g., Fig. 3.17 and Eq. 3.54).</p></li></ul><h3 id="2b3d85ac-0cfe-4ea6-902b-7987820a50d3" data-toc-id="2b3d85ac-0cfe-4ea6-902b-7987820a50d3" collapsed="false" seolevelmigrated="true">3.11 CELLS, EMF, INTERNAL RESISTANCE</h3><ul><li><p>A cell has two electrodes P (positive) and N (negative) in an electrolyte. The emf ε is the open-circuit potential difference: ε = V+ + V−. In presence of a finite load R (external circuit), the terminal potential difference is<br> V = ε - I r,
      where r is the internal resistance and I is the current. In general, Ohm’s law for the external circuit is IR = ε − I r.

    • Maximum current from a cell (short-circuit) occurs at R = 0: I_max = ε/r.

    • In practical calculations, internal resistance r is often neglected when ε >> I r.

    3.12 CELLS IN SERIES AND IN PARALLEL

    • Series: εeq = ε1 + ε2 + …; req = r1 + r2 + … (assuming current leaves each cell from the positive terminal).

    • If current leaves a cell from the negative terminal in a series chain, the emf term gets a negative sign (as in ε1 − ε2 when appropriate).

    • Parallel: equivalent εeq and req are given by
      r{eq} = \left(\sum{i} \frac{1}{ri}\right)^{-1}, \quad ε{eq} = \frac{\sumi εi / ri}{\sumi 1/r_i}. </p></li><li><p>Forncellsinparallel,moregeneralexpressionsapply(Eq.3.77–3.78).</p></li></ul><h3id="0beeb7b8−706c−4770−b11f−1354895183d5"data−toc−id="0beeb7b8−706c−4770−b11f−1354895183d5"collapsed="false"seolevelmigrated="true">3.13KIRCHHOFF’SRULES</h3><ul><li><p>Junctionrule(conservationofchargeatanode):sumofcurrentsenteringequalssumleaving.</p></li><li><p>Looprule(Kirchhoff’svoltagelaw):thealgebraicsumofpotentialchangesaroundanyclosedloopiszero.</p></li><li><p>Theserulesallowanalysisofcomplexnetworksbeyondsimpleseries/parallelcombinations.</p></li></ul><h3id="143455a4−5f49−44c9−a7de−f97ea4387d68"data−toc−id="143455a4−5f49−44c9−a7de−f97ea4387d68"collapsed="false"seolevelmigrated="true">3.14WHEATSTONEBRIDGE</h3><ul><li><p>AbridgewithfourresistorsR1,R2,R3,R4connectedinadiamond;ACacrossonediagonal,galvanometeracrosstheother.</p></li><li><p>Balance(Ig=0)condition:R1/R2=R3/R4(equivalently2R1?dependingonlabeling;thestandardformisR1/R2=R3/R4).</p></li><li><p>Ifunbalanced,thegalvanometercurrentIgcanbefoundbyKirchhoff’srulesorothermethods.</p></li><li><p>Example3.8yieldsagalvanometercurrentof4.87mAforgivenvalues;thebalanceconditionallowssolvingforanunknownresistor.</p></li></ul><h3id="8e5c66fb−a167−4b79−b552−c563ea763797"data−toc−id="8e5c66fb−a167−4b79−b552−c563ea763797"collapsed="false"seolevelmigrated="true">3.15METERBRIDGE</h3><ul><li><p>Aone−meterlonguniformwireactsasaresistivedivider.AjockeyatdistancelfromendAbalancesthegalvanometeragainstaknownresistanceS.</p></li><li><p>BalanceconditiongivestherelationbetweenR(unknown)andS:<br></p></li><li><p>For n cells in parallel, more general expressions apply (Eq. 3.77–3.78).</p></li></ul><h3 id="0beeb7b8-706c-4770-b11f-1354895183d5" data-toc-id="0beeb7b8-706c-4770-b11f-1354895183d5" collapsed="false" seolevelmigrated="true">3.13 KIRCHHOFF’S RULES</h3><ul><li><p>Junction rule (conservation of charge at a node): sum of currents entering equals sum leaving.</p></li><li><p>Loop rule (Kirchhoff’s voltage law): the algebraic sum of potential changes around any closed loop is zero.</p></li><li><p>These rules allow analysis of complex networks beyond simple series/parallel combinations.</p></li></ul><h3 id="143455a4-5f49-44c9-a7de-f97ea4387d68" data-toc-id="143455a4-5f49-44c9-a7de-f97ea4387d68" collapsed="false" seolevelmigrated="true">3.14 WHEATSTONE BRIDGE</h3><ul><li><p>A bridge with four resistors R1, R2, R3, R4 connected in a diamond; AC across one diagonal, galvanometer across the other.</p></li><li><p>Balance (Ig = 0) condition: R1/R2 = R3/R4 (equivalently 2R1? depending on labeling; the standard form is R1/R2 = R3/R4).</p></li><li><p>If unbalanced, the galvanometer current Ig can be found by Kirchhoff’s rules or other methods.</p></li><li><p>Example 3.8 yields a galvanometer current of 4.87 mA for given values; the balance condition allows solving for an unknown resistor.</p></li></ul><h3 id="8e5c66fb-a167-4b79-b552-c563ea763797" data-toc-id="8e5c66fb-a167-4b79-b552-c563ea763797" collapsed="false" seolevelmigrated="true">3.15 METER BRIDGE</h3><ul><li><p>A one-meter long uniform wire acts as a resistive divider. A jockey at distance l from end A balances the galvanometer against a known resistance S.</p></li><li><p>Balance condition gives the relation between R (unknown) and S:<br> \frac{R}{S} = \frac{l}{100 - l}. \tag{3.85} </p></li><li><p>Therefore,</p></li><li><p>Therefore, R = S \frac{l}{100 - l}. \tag{3.86} </p></li><li><p>Forgoodaccuracy,balancenearthecentre(l≈50cm)tominimizeerrors.</p></li></ul><h3id="ad35d501−886b−4d97−b230−2d58ce57e142"data−toc−id="ad35d501−886b−4d97−b230−2d58ce57e142"collapsed="false"seolevelmigrated="true">3.16POTENTIOMETER</h3><ul><li><p>Apotentiometerusesalonguniformwirewithastandardcellofemfεwiredacrossit.Thepotentialdropfromoneendtoapointatdistancelisεφ(l)whereφisthepotentialdropperunitlength.</p></li><li><p>Emfcomparison:balanceatl1foremfε1andatl2foremfε2gives<br></p></li><li><p>For good accuracy, balance near the centre (l ≈ 50 cm) to minimize errors.</p></li></ul><h3 id="ad35d501-886b-4d97-b230-2d58ce57e142" data-toc-id="ad35d501-886b-4d97-b230-2d58ce57e142" collapsed="false" seolevelmigrated="true">3.16 POTENTIOMETER</h3><ul><li><p>A potentiometer uses a long uniform wire with a standard cell of emf ε wired across it. The potential drop from one end to a point at distance l is εφ(l) where φ is the potential drop per unit length.</p></li><li><p>Emf comparison: balance at l1 for emf ε1 and at l2 for emf ε2 gives<br> \frac{ε1}{ε2} = \frac{l1}{l2}. \tag{3.92} </p></li><li><p>Tomeasureanunknowninternalresistancerofacellusingapotentiometer:whenK2isopen,balanceatl1givesε=φl1;whenK2isclosed,thecellcurrentpassesthrougharesistancebox;thebalanceatl2givesV=φl2.Fromthese,ε/V=l1/l2andε=I(r+R)withV=IR,leadingtoarelationtodeterminer.</p></li><li><p>Advantage:nocurrentdrawnfromthetestsource,sointernalresistancedoesnotaffectthemeasurement.</p></li></ul><h3id="7f915c16−0ea7−4858−936b−0d7c35741d9b"data−toc−id="7f915c16−0ea7−4858−936b−0d7c35741d9b"collapsed="false"seolevelmigrated="true">3.10–3.16SELECTEDEXAMPLEPROBLEMS(CONCEPTUALSUMMARIES)</h3><ul><li><p>Example3.3onwardillustrateenergytransferandpowerdissipation,P=IV=I2R=V2/R,andtherelationofpowertocircuitelements.</p></li><li><p>Example3.4demonstratestemperaturedependenceofresistivity:aheatingelement(nichrome)withR2=230V,I=2.68A;usingρchangewithtemperatureandα,thesteadytemperatureiscalculated(847°C)forgivenparameters.</p></li><li><p>Example3.5analyzesanetworkofresistorswithinternalresistanceandshowshowtocomputeequivalentresistance,branchcurrents,andnodevoltages.</p></li><li><p>Example3.6–3.7illustrateKirchhoff’srulesinmorecomplexnetworks;symmetrycansimplifysolvingcurrentsinacubenetwork(Example3.6).</p></li><li><p>Wheatstonebridgeexamples(3.15–3.16)showthebalanceconditionfornullgalvanometerdeflectionandhowtodetermineunknownresistances.</p></li><li><p>Meterbridgeandpotentiometerexamples(3.15–3.16)demonstratepracticalresistancemeasurementtechniquesusingbalancingmethods.</p></li></ul><h3id="e6887184−5775−4549−9dbe−86ee1cb4e160"data−toc−id="e6887184−5775−4549−9dbe−86ee1cb4e160"collapsed="false"seolevelmigrated="true">IMPORTANTNOTATIONRECAP</h3><ul><li><p>I:current(A)</p></li><li><p>V:potentialdifference(V)</p></li><li><p>R:resistance(Ω)</p></li><li><p>ρ:resistivity(Ω⋅m)</p></li><li><p>A:cross−sectionalarea(m2)</p></li><li><p>l:lengthofconductor(m)</p></li><li><p>j:currentdensity(A/m2)</p></li><li><p>E:electricfield(V/m)</p></li><li><p>σ:conductivity(S/m)=1/ρ</p></li><li><p>τ:averagecollisiontime(s)</p></li><li><p>n:densityoffreechargecarriers(m−3)</p></li><li><p>vd:driftvelocity(m/s)</p></li><li><p>µ:mobility(m2/(V⋅s))</p></li><li><p>ε:emfofasource(V)(open−circuitpotentialdifference)</p></li><li><p>r:internalresistance(Ω)</p></li><li><p>Ig:currentthroughgalvanometer;φ:potentialdropperunitlengthinpotentiometer;l:balancinglengthonpotentiometerwire</p></li></ul><h3id="ceed1902−4af6−47c6−b5ab−cc6f5836051c"data−toc−id="ceed1902−4af6−47c6−b5ab−cc6f5836051c"collapsed="false"seolevelmigrated="true">SUMMARYOFKEYRELATIONSHIPS(ESSENTIALEQUATIONS)</h3><ul><li><p>Current:</p></li><li><p>To measure an unknown internal resistance r of a cell using a potentiometer: when K2 is open, balance at l1 gives ε = φ l1; when K2 is closed, the cell current passes through a resistance box; the balance at l2 gives V = φ l2. From these, ε/V = l1/l2 and ε = I(r + R) with V = IR, leading to a relation to determine r.</p></li><li><p>Advantage: no current drawn from the test source, so internal resistance does not affect the measurement.</p></li></ul><h3 id="7f915c16-0ea7-4858-936b-0d7c35741d9b" data-toc-id="7f915c16-0ea7-4858-936b-0d7c35741d9b" collapsed="false" seolevelmigrated="true">3.10–3.16 SELECTED EXAMPLE PROBLEMS (CONCEPTUAL SUMMARIES)</h3><ul><li><p>Example 3.3 onward illustrate energy transfer and power dissipation, P = IV = I^2R = V^2/R, and the relation of power to circuit elements.</p></li><li><p>Example 3.4 demonstrates temperature dependence of resistivity: a heating element (nichrome) with R2 = 230 V, I = 2.68 A; using ρ change with temperature and α, the steady temperature is calculated (847 °C) for given parameters.</p></li><li><p>Example 3.5 analyzes a network of resistors with internal resistance and shows how to compute equivalent resistance, branch currents, and node voltages.</p></li><li><p>Example 3.6–3.7 illustrate Kirchhoff’s rules in more complex networks; symmetry can simplify solving currents in a cube network (Example 3.6).</p></li><li><p>Wheatstone bridge examples (3.15–3.16) show the balance condition for null galvanometer deflection and how to determine unknown resistances.</p></li><li><p>Meter bridge and potentiometer examples (3.15–3.16) demonstrate practical resistance measurement techniques using balancing methods.</p></li></ul><h3 id="e6887184-5775-4549-9dbe-86ee1cb4e160" data-toc-id="e6887184-5775-4549-9dbe-86ee1cb4e160" collapsed="false" seolevelmigrated="true">IMPORTANT NOTATION RECAP</h3><ul><li><p>I: current (A)</p></li><li><p>V: potential difference (V)</p></li><li><p>R: resistance (Ω)</p></li><li><p>ρ: resistivity (Ω·m)</p></li><li><p>A: cross-sectional area (m^2)</p></li><li><p>l: length of conductor (m)</p></li><li><p>j: current density (A/m^2)</p></li><li><p>E: electric field (V/m)</p></li><li><p>σ: conductivity (S/m) = 1/ρ</p></li><li><p>τ: average collision time (s)</p></li><li><p>n: density of free charge carriers (m^−3)</p></li><li><p>vd: drift velocity (m/s)</p></li><li><p>µ: mobility (m^2/(V·s))</p></li><li><p>ε: emf of a source (V) (open-circuit potential difference)</p></li><li><p>r: internal resistance (Ω)</p></li><li><p>Ig: current through galvanometer; φ: potential drop per unit length in potentiometer; l: balancing length on potentiometer wire</p></li></ul><h3 id="ceed1902-4af6-47c6-b5ab-cc6f5836051c" data-toc-id="ceed1902-4af6-47c6-b5ab-cc6f5836051c" collapsed="false" seolevelmigrated="true">SUMMARY OF KEY RELATIONSHIPS (ESSENTIAL EQUATIONS)</h3><ul><li><p>Current: I = \frac{∆Q}{∆t} \quad \text{with} \quad Q = q+ - q- </p></li><li><p>Steadycurrent:</p></li><li><p>Steady current: I = \lim_{\∆t\to 0} \frac{∆Q}{∆t} </p></li><li><p>Ohm’slaw:</p></li><li><p>Ohm’s law: V = IR </p></li><li><p>Resistanceofauniformconductor:</p></li><li><p>Resistance of a uniform conductor: R = ρ \frac{l}{A} \ (\text{or equivalently } ρ = R \frac{A}{l}) </p></li><li><p>Currentdensity:</p></li><li><p>Current density: j = \frac{I}{A} </p></li><li><p>Electricfieldandresistivityrelation:</p></li><li><p>Electric field and resistivity relation: E = ρ j </p></li><li><p>Conductivity:</p></li><li><p>Conductivity: σ = \frac{1}{ρ} \quad\text{and}\quad j = σE </p></li><li><p>Driftvelocity(Drude−like):</p></li><li><p>Drift velocity (Drude-like): v_d = \frac{eEτ}{m} </p></li><li><p>Mobility:</p></li><li><p>Mobility: μ = \frac{v_d}{E} = \frac{eτ}{m} </p></li><li><p>Energy/powerinaconductor:</p></li><li><p>Energy/power in a conductor: P = IV = I^2R = \frac{V^2}{R} </p></li><li><p>Temperaturedependenceofresistivity:</p></li><li><p>Temperature dependence of resistivity: ρT = ρ0 [1 + α (T - T_0)] </p></li><li><p>Seriesresistors:</p></li><li><p>Series resistors: R{eq} = R1 + R2 + \cdots + Rn </p></li><li><p>Parallelresistors:</p></li><li><p>Parallel resistors: \frac{1}{R{eq}} = \sum{i=1}^n \frac{1}{R_i} </p></li><li><p>Seriescells:</p></li><li><p>Series cells: ε{eq} = ε1 + ε2 + \cdots , \quad r{eq} = r1 + r2 + \cdots </p></li><li><p>Parallelcells:complexbutstandardcombining:</p></li><li><p>Parallel cells: complex but standard combining: r{eq} = \left( \sum \frac{1}{ri} \right)^{-1}, \quad ε{eq} = \frac{\sum εi / ri}{\sum 1/ri} </p></li><li><p>Wheatstonebridgebalance(Ig=0):</p></li><li><p>Wheatstone bridge balance (Ig = 0): \frac{R1}{R2} = \frac{R3}{R4} </p></li><li><p>Potentiometerbalance:</p></li><li><p>Potentiometer balance: \frac{ε1}{ε2} = \frac{l1}{l2} $$

    NOTES

    • The content covers both fundamental definitions (what current is, how it is measured) and practical tools (Ohm’s law, resistivity, temperature dependence, and measurement devices like Wheatstone and potentiometer).

    • While examples provide numeric practice, the essential ideas are the relationships among V, I, R, and the network rules (Kirchhoff) that allow solving complex circuits.

    • Ethical, philosophical, or practical implications revolve around efficiency and safety in power transmission (e.g., high voltage to reduce I^2R losses) and the design choices for materials (temperature stability, dopant effects in semiconductors).