Current electricity deals with the flow of electric charge through conductors. When a conductor like a metal wire is connected across a battery, free electrons drift steadily through it, constituting a current. This chapter develops the microscopic picture of current in terms of drift velocity and then builds the macroscopic laws that govern circuits: Ohm's law, resistance, resistivity, electrical power, and the rules for combining resistors.
We then study sources of EMF - cells and batteries - with their internal resistance, and learn how the terminal voltage depends on the current drawn. Kirchhoff's rules provide a systematic method for analysing complicated circuits, and the Wheatstone bridge and the potentiometer are important applications of these principles.
The chapter connects electrostatics with practical engineering. The concepts of resistance, current, and power are used in every electrical device, and Kirchhoff's laws are the foundation of all circuit analysis in electronics and electrical engineering.
Electric current is the rate of flow of electric charge through a cross-section of a conductor:
I = Q / t
The SI unit of current is the ampere (A), where 1 A = 1 C/s. By convention, the direction of current is the direction of flow of positive charge, opposite to the direction of motion of electrons.
In a conductor, electrons move randomly with high speeds (about 10^6 m/s) but their net displacement is zero in the absence of an electric field. When a field is applied, the electrons acquire a small average drift velocity vd in the direction opposite to the field. The current is related to the drift velocity by:
I = n A e vd
where n is the number of free electrons per unit volume, A is the cross-sectional area, and e is the electron charge. The current density, defined as current per unit area, is:
J = I / A = n e vd
The drift velocity is directly proportional to the applied electric field: vd = (e E / m) tau, where tau is the average relaxation time between collisions.
Ohm's law states that the current through a conductor is directly proportional to the potential difference across its ends, provided the physical conditions remain constant:
V = I R
The constant R is the resistance of the conductor, measured in ohms (omega), where 1 ohm = 1 V/A. The resistance of a conductor depends on its length l, area A, and the material through its resistivity rho:
R = rho * l / A
The resistivity is a property of the material and is measured in ohm metre. For metals, the resistivity increases with temperature approximately as:
rho_t = rho_0 (1 + alpha Delta T)
where alpha is the temperature coefficient of resistivity. Resistivity is inversely related to conductivity sigma = 1/rho. Materials are classified by resistivity: conductors have very small rho, insulators very large rho, and semiconductors intermediate values.
When current flows through a resistor, electrical energy is converted into heat. The power delivered to a resistor is:
P = V I = I^2 R = V^2 / R
The unit of power is the watt (W). The heat produced in time t is:
H = I^2 R t
The SI unit of electrical energy is the joule. In practice, electrical energy is measured in kilowatt-hour (kWh), where 1 kWh = 3.6 x 10^6 J. The heating effect of current is used in electric heaters, toasters, and bulbs. Ohm's law combined with P = VI gives the three equivalent power formulas, and the appropriate one is chosen depending on known quantities.
Resistors can be combined in series or in parallel. In series, the same current flows through each resistor and the voltage drops add:
R_series = R1 + R2 + R3 + ...
In parallel, the same voltage appears across each resistor and the currents add:
1/R_parallel = 1/R1 + 1/R2 + 1/R3 + ...
For two resistors in parallel, R = R1 R2/(R1 + R2). The series combination always has a resistance larger than the largest individual resistor, while the parallel combination is always smaller than the smallest individual resistor. These results are used to achieve a desired resistance in a circuit.
A cell converts chemical energy into electrical energy. The electromotive force (EMF) E of a cell is the potential difference between its terminals when no current is drawn; it is the maximum energy delivered per unit charge. When a current I flows, the terminal voltage V is less than the EMF because some energy is lost across the internal resistance r:
V = E - I r
The internal resistance arises because the electrolyte and electrodes offer resistance to the flow of ions inside the cell. When cells are combined in series, the total EMF is the sum, and internal resistances also add; when combined in parallel, the EMF is the same as one cell but the internal resistance is reduced.
Kirchhoff's rules apply to any circuit and are the basis of circuit analysis:
Junction rule (conservation of charge): The sum of currents entering a junction equals the sum of currents leaving it.
Loop rule (conservation of energy): The algebraic sum of the changes in potential around any closed loop is zero.
The loop rule is applied by assigning directions to currents, and potential drops (IR) are taken negative when traversed in the direction of current, while EMFs are positive when traversed from negative to positive terminal. These rules allow the determination of all currents in a network.
The Wheatstone bridge is a circuit for comparing resistances precisely. It consists of four resistances arranged in a diamond, with a galvanometer across one diagonal and a battery across the other. The bridge is balanced when the galvanometer shows no deflection, which occurs when:
R1 / R2 = R3 / R4
At balance, no current flows through the galvanometer. The meter bridge is a practical form of the Wheatstone bridge using a uniform wire of 1 m length; the unknown resistance is found from the balance point. The metre bridge is also used to compare and measure resistances and to find the resistivity of a material.
The potentiometer is a device used to compare EMFs of cells and to measure the internal resistance of a cell. It consists of a uniform resistance wire across which a steady current is maintained by a driver cell. The potential difference across any portion of the wire is proportional to its length:
V = K l
where K is the potential gradient. At the balance point, the EMF of a cell equals the potential drop across the balancing length. If l1 and l2 are the balancing lengths for a cell with and without a shunt resistance S, the internal resistance is:
r = (l1 - l2) / l2 * S
The potentiometer measures EMF accurately because it draws no current from the cell at balance.
| Quantity | Formula | Unit |
|---|---|---|
| Current | I = Q/t | ampere (A) |
| Current density | J = n e vd | A/m^2 |
| Ohm's law | V = I R | ohm |
| Resistance | R = rho l/A | ohm |
| Resistivity | rho = R A/l | ohm metre |
| Power | P = VI = I^2 R = V^2/R | watt (W) |
| Heat | H = I^2 R t | joule (J) |
| Terminal voltage | V = E - I r | volt (V) |
| Internal resistance (potentiometer) | r = (l1 - l2)S/l2 | ohm |
| Combination | Resistors | Cells |
|---|---|---|
| Series | R = R1 + R2 + ... | E = E1 + E2 + ..., r adds |
| Parallel | 1/R = 1/R1 + 1/R2 + ... | E same, 1/r = 1/r1 + 1/r2 + ... |
In this chapter we understood current as the flow of charge, with I = Q/t and I = nAevd relating macroscopic current to microscopic drift velocity. Ohm's law V = IR and the resistance formula R = rho l/A connected geometry, material, and circuit behaviour, while P = VI = I^2R = V^2/R described electrical power. We studied cells with internal resistance, terminal voltage V = E - Ir, and the series and parallel combination of cells and resistors. Kirchhoff's junction and loop rules provided a systematic method for circuit analysis, and the Wheatstone bridge and potentiometer gave practical techniques for measuring resistance and EMF accurately. These fundamentals govern all electrical devices and circuits studied in later chapters on magnetism, alternating current, and electronics.