13.2 Electric Current, Resistance, Ohm's Law, Kirchhoff's Rules & Potentiometer

Electric Current, Resistance, Ohm's Law, Kirchhoff's Rules & Potentiometer

In the Intermediate Physics Part II syllabus and the UET ECAT entry test, current electricity and multiloop circuit analysis are staple topics. ECAT questions frequently demand instantaneous calculations of microscopic drift velocities, temperature coefficients, internal resistance drops, balanced bridge conditions, and potentiometer comparisons without calculator assistance. Mastering these fundamental circuit theorems and their underlying conservation principles ensures speed and accuracy under intense exam time pressure.

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1. Microscopic Electric Current & Drift Velocity

Macroscopic and Microscopic Definitions of Current

Electric current $I$ is macroscopically defined as the time rate of transfer of net electric charge across a given cross-sectional area of a conductor:

$I = \lim_{\Delta t \to 0} \frac{\Delta Q}{\Delta t} = \frac{dQ}{dt}$

The SI unit of electric current is the ampere (A) ($1\text{ A} = 1\text{ coulomb/second}$). By convention, conventional current is defined along the direction in which positive charges would drift, flowing from higher potential to lower potential, whereas actual conduction electrons in metallic lattices drift in the opposite direction.

Microscopic Transport: Derivation of $I = n A v_d e$

In a metallic conductor (such as copper), free conduction electrons undergo chaotic thermal agitation, colliding frequently with stationary ionic lattice sites at speeds on the order of $10^5\text{ m/s}$. Because this thermal motion is completely isotropic, the net thermal velocity vector is zero ($\langle \mathbf{v}_{\text{th}} \rangle = \mathbf{0}$), producing zero net electric current.

When a potential difference $V$ is applied across a wire of length $l$, an internal electric field $\mathbf{E} = \frac{V}{l}$ is established throughout the conductor. This field accelerates electrons between lattice collisions, superimposing a steady, slow net directional migration known as the drift velocity $\mathbf{v}_d$.

Consider a cylindrical conducting wire of cross-sectional area $A$ containing $n$ free conduction electrons per unit volume (number density of charge carriers):

1. In time interval $\Delta t$, the electrons drift forward by a distance $\Delta l = v_d \Delta t$.
2. The total volume of electrons passing through a cross-section in time $\Delta t$ is $\Delta \mathcal{V} = A \Delta l = A v_d \Delta t$.
3. The total number of mobile electrons in this volume is $N = n \Delta \mathcal{V} = n A v_d \Delta t$.
4. Because each electron carries charge $e = 1.602 \times 10^{-19}\text{ C}$, the total charge passing through is:

$\Delta Q = N e = n A v_d e \Delta t$

1. Dividing by $\Delta t$ yields the fundamental microscopic current equation:

$I = \frac{\Delta Q}{\Delta t} = n A v_d e$

Rearranging for drift velocity $v_d$:

$v_d = \frac{I}{n A e} = \frac{J}{n e}$

where $J = \frac{I}{A}$ is the current density (in $\text{A/m}^2$).

ECAT Conceptual Insight: For typical copper wires carrying normal household currents ($1$–$5\text{ A}$), the drift velocity $v_d$ is remarkably small—approximately $0.1$ to $1\text{ mm/s}$ ($10^{-4}$–$10^{-3}\text{ m/s}$). Yet, when you flick a light switch, the lamp turns on instantaneously. This occurs because the electric field propagates through the wire at nearly the speed of light ($c \sim 3 \times 10^8\text{ m/s}$), initiating drift of free electrons simultaneously throughout the entire circuit.

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2. Ohm's Law, Resistivity & Temperature Dependence

Ohm's Law

Published by Georg Simon Ohm in 1827, Ohm's law states that the current flowing through a metallic conductor is directly proportional to the potential difference applied across its terminals, provided the physical temperature and mechanical state of the conductor remain strictly constant:

$V \propto I \implies V = I R$

where $R$ is the electrical resistance of the conductor, measured in SI units of ohms ($\Omega$) ($1\,\Omega = 1\text{ volt/ampere}$):

• Ohmic Conductors: Materials that obey Ohm's law strictly across wide operating ranges; their $V$-$I$ characteristic graph is a straight line passing through the origin (e.g., copper, silver, aluminum, nichrome at constant temperature).
• Non-Ohmic Devices: Systems where the $V$-$I$ characteristic is non-linear; resistance varies with voltage or current (e.g., semiconductor diodes, transistors, thermistors, filament lamps where temperature rises with current).

Resistivity and Conductivity

At a fixed temperature, the resistance $R$ of a uniform conductor is directly proportional to its length $l$ and inversely proportional to its cross-sectional area $A$:

$R = \rho \frac{l}{A}$

The constant of proportionality $\rho$ is the resistivity (specific resistance) of the material:

$\rho = \frac{R A}{l}$

SI units of resistivity: ohm-meter ($\Omega\cdot\text{m}$). Resistivity is an intrinsic property of the material and temperature, completely independent of the dimensions or geometry of the specimen.

• Electrical Conductance ($G$): The reciprocal of resistance: $G = \frac{1}{R}$, measured in siemens (S) or $\Omega^{-1}$ (mho).
• Electrical Conductivity ($\sigma$): The reciprocal of resistivity: $\sigma = \frac{1}{\rho}$, measured in $\text{S/m}$ or $(\Omega\cdot\text{m})^{-1}$.

Microscopically, Ohm's law is expressed vectorially as $\mathbf{J} = \sigma \mathbf{E}$.

Temperature Dependence of Resistance and Resistivity

As temperature increases, the amplitude of thermal vibrations of positive metal ions in the crystal lattice increases. This escalates the collision cross-section encountered by drifting electrons, increasing resistivity.

Over normal working temperature ranges, resistance $R_t$ at temperature $t\,(^\circ\text{C})$ varies linearly with temperature according to:

$R_t = R_0(1 + \alpha t)$

where $R_0$ is resistance at $0^\circ\text{C}$ and $\alpha$ is the fractional temperature coefficient of resistance:

$\alpha = \frac{R_t - R_0}{R_0 t} = \frac{\Delta R}{R_0 \Delta t}$

Similarly, in terms of resistivity:

$\rho_t = \rho_0(1 + \alpha t) \implies \alpha = \frac{\rho_t - \rho_0}{\rho_0 t}$

The SI unit of $\alpha$ is per Kelvin ($\text{K}^{-1}$) or per degree Celsius ($^\circ\text{C}^{-1}$):

• Positive Temperature Coefficient (PTC, $\alpha > 0$): Conductors and metals (e.g., copper, iron, platinum, tungsten). Resistance increases as temperature rises.
• Negative Temperature Coefficient (NTC, $\alpha < 0$): Semiconductors (silicon, germanium), carbon, and electrolytes. In semiconductors, rising temperature provides thermal energy to rupture covalent bonds, generating new electron-hole pairs and dramatically multiplying carrier density $n$, which outweighs lattice scattering and causes resistance to drop steeply with temperature.

Color Coding of Carbon Resistors

Carbon composition resistors indicate their nominal resistance and tolerance via standardized colored bands using the mnemonic: "B. B. ROY of Great Britain had a Very Good Wife":

Calculation Formula: $R = (\text{Band 1})(\text{Band 2}) \times 10^{\text{Band 3}} \pm \text{Tolerance}\%$. Example: Yellow (4), Violet (7), Orange ($10^3$), Gold ($\pm 5\%$) $= 47 \times 10^3\,\Omega \pm 5\% = 47\,\text{k}\Omega \pm 5\%$.

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3. Electromotive Force, Internal Resistance & Maximum Power Transfer

Electromotive Force ($\mathcal{E}$)

The electromotive force (emf) $\mathcal{E}$ of an electrical source is the total energy supplied by the source per unit charge transferred through it from the lower potential terminal to the higher potential terminal:

$\mathcal{E} = \frac{dW}{dq}$

Emf is not a mechanical force; it is an electric potential difference measured in volts (V) ($1\text{ V} = 1\text{ joule/coulomb}$). It represents the open-circuit terminal voltage of the source when zero current is drawn.

Internal Resistance ($r$) and Terminal Potential Difference ($V_t$)

Every real practical chemical cell or generator exhibits an intrinsic internal resistance $r$ arising from electrolytic resistance and internal electrode geometry.

1. Discharging Cell (Source Delivering Power)

When a cell of emf $\mathcal{E}$ and internal resistance $r$ delivers current $I$ to an external load resistor $R$:

$I = \frac{\mathcal{E}}{R + r}$

The internal potential drop across the internal resistance is $v_{\text{int}} = Ir$. The measurable terminal potential difference $V_t$ available across the battery terminals is:

$V_t = \mathcal{E} - Ir = I R$

Because $Ir > 0$ during discharge, the terminal voltage of a discharging battery is always strictly less than its emf ($V_t < \mathcal{E}$).

2. Charging Cell (Source Absorbing Power)

When an external charger forces current into the positive terminal of a secondary accumulator cell in reverse:

$V_t = \mathcal{E} + Ir$

During charging, terminal voltage exceeds the cell's emf ($V_t > \mathcal{E}$).

3. Open-Circuit vs. Short-Circuit Limits

• Open-Circuit ($R \to \infty$): Current $I = 0 \implies V_t = \mathcal{E}$. An ideal voltmeter measuring an open circuit reads the true emf.
• Short-Circuit ($R = 0$): Maximum current flows: $I_{\text{sc}} = \frac{\mathcal{E}}{r}$. Terminal potential difference collapses to zero: $V_t = 0$.

Maximum Power Transfer Theorem

The electrical power $P$ delivered by a practical source of emf $\mathcal{E}$ and internal resistance $r$ to a variable external load resistor $R$ is:

$P = I^2 R = \left(\frac{\mathcal{E}}{R + r}\right)^2 R = \frac{\mathcal{E}^2 R}{(R + r)^2}$

To find the load resistance $R$ that maximizes power dissipation in the load, differentiate $P$ with respect to $R$ and set the derivative to zero:

$\frac{dP}{dR} = \mathcal{E}^2 \left[\frac{(R+r)^2 - 2R(R+r)}{(R+r)^4}\right] = \mathcal{E}^2 \left[\frac{(R+r) - 2R}{(R+r)^3}\right] = 0$

$(R + r) - 2R = 0 \implies R = r$

Theorem: An electrical source delivers maximum power to an external load when the load resistance equals the internal resistance of the source ($R_{\text{load}} = r_{\text{int}}$).

Substituting $R = r$ into the power formula gives the maximum power delivered:

$P_{\max} = \frac{\mathcal{E}^2 r}{(2r)^2} = \frac{\mathcal{E}^2}{4r}$

At maximum power transfer, internal power loss equals external load power ($P_{\text{int}} = P_{\text{load}}$). Thus, the electrical transfer efficiency is exactly $50\%$ ($\eta = 0.50$).

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4. Kirchhoff's Rules for Complex Circuit Networks

When a circuit network contains multiple intersecting loops and cross-connected voltage sources that cannot be reduced by simple series-parallel combinations, we apply Gustav Kirchhoff's two fundamental circuit laws.

Kirchhoff's First Rule (Kirchhoff's Current Law - KCL)

Statement: The algebraic sum of all electric currents meeting at any circuit junction (node) is identically zero:

$\sum_{k=1}^n I_k = 0 \iff \sum I_{\text{entering}} = \sum I_{\text{leaving}}$

Sign Convention: Currents flowing toward the junction are assigned positive values ($+I$); currents flowing away from the junction are assigned negative values ($-I$).

Physical Basis: Kirchhoff's First Rule is an exact manifestation of the Law of Conservation of Electric Charge. Because electric charge cannot be created, destroyed, or accumulate indefinitely at an infinitesimal point node, the charge entering per second must equal the charge leaving per second.

Kirchhoff's Second Rule (Kirchhoff's Voltage Law - KVL)

Statement: The algebraic sum of all potential changes (emfs and resistive voltage drops) encountered in traversing completely around any closed conducting loop is identically zero:

$\sum \Delta V = 0 \iff \sum \mathcal{E} = \sum I R$

Traversing Sign Conventions:

1. Emf Sources: When traversing a source of emf from the negative terminal to the positive terminal (in direction of potential rise), the emf is taken as positive ($+\mathcal{E}$). When traversing from positive to negative (potential drop), the emf is taken as negative ($-\mathcal{E}$).
2. Resistors: When traversing a resistor in the same direction as the assumed branch current, potential drops across the resistor, so the change is negative ($-IR$). When traversing against the direction of assumed branch current, the change is positive ($+IR$).

Physical Basis: Kirchhoff's Second Rule is an exact manifestation of the Law of Conservation of Energy. Because the electrostatic field is conservative, taking a test charge through an arbitrary closed circuit path and returning to the exact starting point performs zero net mechanical work: $\oint \mathbf{E} \cdot d\mathbf{r} = 0$.

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5. Wheatstone Bridge and Potentiometer

The Wheatstone Bridge

Invented by Samuel Hunter Christie and popularized by Sir Charles Wheatstone, the Wheatstone bridge is an electrical bridge network consisting of four resistors ($R_1, R_2, R_3, R_4$) arranged in a closed loop, with a galvanometer $G$ bridging across two opposing nodes and a DC voltage source bridging the other two nodes.

The Balanced Condition

When the resistances are adjusted such that the potential difference across the galvanometer is zero, no current flows through the galvanometer ($I_g = 0$, null deflection). Under this balanced state:

$V_B = V_D \implies I_1 R_1 = I_2 R_2 \quad \text{and} \quad I_1 R_3 = I_2 R_4$

Dividing the two equations yields the Wheatstone Bridge Balance Criterion:

$\frac{R_1}{R_2} = \frac{R_3}{R_4} \iff \frac{R_1}{R_3} = \frac{R_2}{R_4}$

If three resistances are known with high precision, an unknown fourth resistor $R_x$ can be determined with exceptional accuracy: $R_x = R_2\left(\frac{R_3}{R_1}\right)$. Because balance is detected at null current, the measurement is completely independent of the internal resistance of the galvanometer or battery.

The Potentiometer: Principle and Superiority

A potentiometer consists of a uniform high-resistance slide wire $AB$ of length $L$ (typically $1$ to $4\text{ meters}$) through which a steady driver current $I_0$ is maintained by an accumulator. Because the wire has uniform cross-section and resistivity, the potential drop across any length $l$ of the wire is directly proportional to that length:

$V_l = k l$

where $k = \frac{V_{AB}}{L} = \frac{I_0 R_{\text{wire}}}{L}$ is the constant potential gradient of the wire (in $\text{V/m}$).

Primary Applications of the Potentiometer

1. Comparing Emfs of Two Cells: A two-way key connects cell $\mathcal{E}_1$ to the galvanometer, finding balance length $l_1$ where $I_g = 0$:

$\mathcal{E}_1 = k l_1$

Connecting cell $\mathcal{E}_2$ yields balance length $l_2$:

$\mathcal{E}_2 = k l_2$

Dividing the two relations yields the fundamental comparison formula:

$\frac{\mathcal{E}_1}{\mathcal{E}_2} = \frac{l_1}{l_2}$

1. Measuring Internal Resistance ($r$) of a Cell:
   • Open-circuit balance length $l_1$ (measures emf): $\mathcal{E} = k l_1$.
   • With a known shunt resistance $R$ connected across the cell, terminal voltage balance length $l_2$ is found: $V_t = k l_2$.
   • Because $\frac{\mathcal{E}}{V_t} = \frac{I(R+r)}{IR} = \frac{R+r}{R} = 1 + \frac{r}{R}$:

$\frac{l_1}{l_2} = 1 + \frac{r}{R} \implies r = R\left(\frac{l_1 - l_2}{l_2}\right)$

Why a Potentiometer is Superior to a Voltmeter

A standard moving-coil voltmeter requires an operating current to produce needle deflection. When placed across a high-resistance source, this current causes an internal voltage drop $Ir$, resulting in a measured terminal voltage that is systematically lower than the true emf (the loading effect).

By contrast, the potentiometer operates via the null-balance method: at the balance point, $I_g = 0$, meaning zero current is drawn from the cell under test. Consequently, $Ir = 0$, and the instrument measures the true, undisturbed open-circuit electromotive force $\mathcal{E}$. The potentiometer behaves as an ideal voltmeter with infinite internal resistance ($R_{\text{in}} \to \infty$).

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6. Circuit Laws and Components Comparison Table