7.3 Current Electricity, Ohm's Law, Kirchhoff's Rules & DC Circuits

7.3 Current Electricity, Ohm's Law, Kirchhoff's Rules & DC Circuits

Current electricity deals with continuous motion of electric charges through conductors under electric potential gradients. A firm grasp of circuit laws, internal resistance, and DC bridge measurements is vital for solving complex circuit problems in the AMC Physics test.

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1. Electric Current & Microscopic Drift Velocity

Electric current $I$ is the rate of net charge flow through a conductor cross-section:

$I = \frac{\Delta Q}{\Delta t} \quad [\text{SI Unit: Ampere (A) = C/s}]$

By convention, conventional current flows in the direction of positive charge movement (from higher potential to lower potential). Electronic current consists of negative electrons flowing in the opposite direction.

Microscopic Model of Current

Inside a conductor subjected to an electric field $E$, conduction electrons experience an electric force and acquire an average directional velocity called drift velocity $v_d$ superimposed on their random thermal motion:

$I = n e A v_d$

where:

• $n$ is free electron density (number of conduction electrons per unit volume).
• $e = 1.602 \times 10^{-19} \text{ C}$ is elementary electronic charge.
• $A$ is conductor cross-sectional area.
• $v_d$ is drift velocity (typically small, $\sim 10^{-4} \text{ m/s}$).

Current density $J = \frac{I}{A} = n e v_d$ represents current per unit cross-sectional area.

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2. Ohm's Law, Resistance & Resistivity

Ohm's Law Statement

Ohm's Law states that the current $I$ flowing through a metallic conductor is directly proportional to the potential difference $V$ applied across its ends, provided physical conditions (such as temperature) remain constant:

$V = I R$

Materials obeying this linear relationship are Ohmic conductors (e.g., copper, silver). Non-Ohmic devices exhibit non-linear $I\text{-}V$ curves (e.g., semiconductor diodes, filament lamps, thermistors).

Resistance & Resistivity

Electrical resistance $R$ measures opposition to electric charge flow:

$R = \rho \frac{L}{A} \quad [\text{SI Unit: Ohm (}\Omega\text{)}]$

where $L$ is length, $A$ is cross-sectional area, and $\rho$ is material resistivity (in $\Omega\cdot\text{m}$). Electrical conductivity $\sigma = \frac{1}{\rho}$ (unit: $\Omega^{-1}\cdot\text{m}^{-1}$ or $\text{S/m}$).

Temperature Dependence of Resistance

For metallic conductors, resistance increases with temperature due to increased lattice vibrations impeding electron drift:

$R_T = R_0 (1 + \alpha \Delta T)$

where $\alpha$ is the temperature coefficient of resistance ($K^{-1}$ or $^\circ\text{C}^{-1}$):

• Metals: Positive $\alpha$ (resistance increases as temperature rises).
• Semiconductors & Thermistors: Negative $\alpha$ (resistance decreases rapidly with temperature increase due to generation of additional charge carriers).

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3. Electromotive Force (EMF) & Internal Resistance

Electromotive Force ((\mathcal{E}))

Electromotive force $\mathcal{E}$ is the total energy supplied by a source (e.g., battery or generator) per unit charge passing through it:

$\mathcal{E} = \frac{W}{Q} \quad [\text{SI Unit: Volt (V)}]$

Internal Resistance ($r$) & Terminal Potential Difference ($V$)

Real voltage sources possess inherent internal resistance $r$. When a cell delivers current $I$ to an external load resistor $R$, an internal potential drop $I r$ occurs within the cell:

$V = \mathcal{E} - I r$

where $V = I R$ is the terminal potential difference across the cell terminals.

AMC Special Cases:

• Discharging Battery (normal use): $V = \mathcal{E} - I r \implies V < \mathcal{E}$.
• Charging Battery: $V = \mathcal{E} + I r \implies V > \mathcal{E}$.
• Open Circuit ($I = 0$): $V = \mathcal{E}$.

Electrical Power & Joule Heating

Electrical power $P$ dissipated in a resistor of resistance $R$ is:

$P = V I = I^2 R = \frac{V^2}{R} \quad [\text{SI Unit: Watt (W)}]$

Total energy dissipated over time $t$ is given by Joule's Law: $E_e = I^2 R t$.

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4. Kirchhoff's Circuit Rules

Complex networks unsuitable for simple series-parallel reduction are solved using Kirchhoff's two rules:

1. Kirchhoff's First Rule (Junction / Current Rule)

The algebraic sum of all electric currents entering any circuit node or junction must equal the algebraic sum of currents leaving that node:

$\sum I_{in} = \sum I_{out} \quad \text{or} \quad \sum I = 0$

Physical Foundation: Law of Conservation of Electric Charge.

2. Kirchhoff's Second Rule (Loop / Voltage Rule)

The algebraic sum of all potential differences (EMFs and $I R$ drops) around any completely closed loop in a circuit must be zero:

$\sum \Delta V = 0$

Physical Foundation: Law of Conservation of Energy.

Sign Rules for Traversing Loops:

• Traversing a resistor in direction of current: Potential drop ($-I R$).
• Traversing a resistor opposite to current: Potential rise ($+I R$).
• Traversing a battery from negative to positive terminal: Potential rise ($+\mathcal{E}$).
• Traversing a battery from positive to negative terminal: Potential drop ($-\mathcal{E}$).

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5. DC Measuring Instruments: Wheatstone Bridge & Potentiometer

Wheatstone Bridge

A Wheatstone bridge consists of four resistors $R_1, R_2, R_3, R_4$ arranged in a quadrilateral loop with a galvanometer connected across opposite nodes. At null balance (zero galvanometer current):

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

If $R_4 = X$ is unknown, it can be precisely evaluated as $X = R_3 \left(\frac{R_2}{R_1}\right)$.

Potentiometer

A potentiometer is an instrument designed to measure potential differences without drawing current from the circuit under test (acting as an ideal voltmeter with infinite resistance at null balance).

Key Potentiometer Formulas for AMC:

1. Comparing EMFs of Two Cells: $\frac{\mathcal{E}_1}{\mathcal{E}_2} = \frac{l_1}{l_2}$ where $l_1$ and $l_2$ are balancing lengths on the potentiometer wire.
2. Determining Internal Resistance of a Cell: $r = R \left( \frac{l_1}{l_2} - 1 \right)$ where $l_1$ is open-circuit balancing length and $l_2$ is closed-circuit balancing length across external resistor $R$.

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6. Worked Numerical Examples for AMC Candidates

Example 1: Terminal Potential Difference & Internal Resistance

Problem: A battery of EMF $\mathcal{E} = 12 \text{ V}$ and internal resistance $r = 1 ,\Omega$ is connected to an external load resistor $R = 5 ,\Omega$. Calculate the circuit current and terminal voltage.

Solution:

1. Total circuit resistance: $R_{total} = R + r = 5 + 1 = 6 ,\Omega$.
2. Calculate circuit current: $I = \frac{\mathcal{E}}{R + r} = \frac{12 \text{ V}}{6 \,\Omega} = 2 \text{ A}$
3. Terminal voltage: $V = \mathcal{E} - I r = 12 \text{ V} - (2 \text{ A} \times 1 \,\Omega) = 12 - 2 = 10 \text{ V}$

Example 2: Wheatstone Bridge Null Balance

Problem: In a balanced Wheatstone bridge, $R_1 = 10 ,\Omega$, $R_2 = 30 ,\Omega$, and $R_3 = 15 ,\Omega$. Find the value of unknown resistance $R_4$.

Solution: Apply the balance condition $\frac{R_1}{R_2} = \frac{R_3}{R_4}$: $R_4 = R_3 \left( \frac{R_2}{R_1} \right) = 15 \times \left( \frac{30}{10} \right) = 15 \times 3 = 45 \,\Omega$

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Basic Electronics for FSc Screening

Expect a few definition-level electronics items mixed into current-electricity sets:

• Intrinsic vs extrinsic semiconductors: pure Si/Ge vs donor (n-type) / acceptor (p-type) doping.
• p–n junction diode: forward bias conducts; reverse bias largely blocks; used in rectification.
• Half-wave vs full-wave rectifier: which diodes conduct in which half-cycle; output ripple is qualitative only.
• Transistor (npn/pnp) as amplifier/switch: emitter, base, collector roles at FSc depth—no hybrid-parameter algebra.
• Logic gates: AND, OR, NOT truth tables are occasional non-verbal-adjacent academic extras.

Keep answers conceptual: doping type, bias direction, and gate truth values.