Chapter 10 — Electricity

Definition: Electric current is the rate of flow of electric charge through a conductor. It is the amount of charge that passes through a cross-section of a conductor per unit time.

Symbol and unit: Current is denoted by I and its SI unit is ampere (A). 1 A = 1 coulomb per second.

Microscopic picture: In metallic conductors, free electrons move randomly; when an electric field is applied they acquire a small average drift velocity superimposed on random motion. This net movement of charge carriers (electrons) constitutes electric current. By convention, current direction is taken as the direction positive charges would move (conventional current), opposite to electron flow in metals.

Mathematical definition: If a charge Q passes a cross-section in time t, the average current I = Q/t. For time-varying current, instantaneous current i(t) = dQ/dt.

Ohm's law (for ohmic conductors): For many conductors at constant temperature, current is proportional to the potential difference (voltage) across it: V = IR, where R is the resistance. Resistances add in series and combine via reciprocal sum in parallel.

Resistance and factors affecting it: Resistance R depends on the material and geometry: R = ρ L / A, where ρ is resistivity (material property), L is length and A is cross-sectional area. For metals resistivity and hence resistance generally increase with temperature; in semiconductors resistivity typically decreases with temperature.

Power dissipation: A current I through an element with voltage V across it dissipates electric power P = VI. Using Ohm's law, P can be written as P = I²R or P = V²/R.

Steady (DC) and time-varying (AC) currents: Direct current (DC) has a constant direction and magnitude (e.g., cells, batteries), while alternating current (AC) periodically reverses direction (e.g., mains supply). The instantaneous current in AC varies with time; RMS values are used to describe effective heating power.

Measurement: Current is measured with an ammeter connected in series with the circuit element. Practical safety devices like fuses and circuit breakers protect circuits from excessive current.

Common misconceptions & notes: (1) Current is not the same as voltage: voltage is the cause (potential difference), current is the effect (flow of charge). (2) The same current flows through all elements in a series circuit; in parallel circuits the total current splits among branches.

Electric potential (V) at a point is defined as the work done by an external agent in bringing a unit positive charge from infinity (where potential is taken as zero) to that point, without acceleration. It is a scalar quantity and its SI unit is the volt (1 V = 1 J/C).

Mathematically: V = W/q, where W is the work done (in joules) and q is the test charge (in coulombs).

Potential difference (ΔV) between two points A and B is the work done by an external agent in taking a unit positive charge from A to B. It is equal to the difference of potentials at the two points:

ΔV = VB − VA = Wext/q, where Wext is the work done by the external force in moving charge q from A to B. If a charge q moves through a potential difference ΔV, its change in electric potential energy is ΔU = q·ΔV.

Relation with electric field: Electric field and potential are related. The potential difference between two nearby points separated by displacement ds in the direction of the field is dV = −E·ds (in one dimension, E = −dV/ds). For a uniform field between parallel plates, the magnitude of potential difference is |ΔV| = E·d, where d is the plate separation.

Potential of a point charge: For an isolated point charge Q (with potential zero at infinity): V(r) = kQ/r, where k = 1/(4πε0) and r is distance from the charge. This potential decreases as r increases (for positive Q).

Equipotential surfaces: Surfaces on which potential is the same everywhere are called equipotential surfaces. No work is required to move a charge along an equipotential. Equipotentials are always perpendicular to electric field lines.

In circuits: The potential difference between two points in a circuit gives the driving force that causes current. The electromotive force (EMF) of a source is the work done per unit charge in moving charge inside the source; terminal potential difference may differ from EMF when internal resistance exists.

What is an electric circuit?
An electric circuit is a closed path that allows electric charges (current) to flow from a source of electrical energy (like a cell or battery) through conductors and electrical components (resistors, lamps, switches, etc.) and back to the source. If the path is broken, the circuit is open and current stops.

Basic components

• Source: cell or battery (provides electromotive force, EMF).
• Conductors: wires (usually copper) that connect components.
• Load: device that consumes electrical energy (bulb, resistor, motor).
• Switch: opens or closes the circuit to control current.
• Protective devices: fuse or circuit breaker to prevent excessive current.
• Measuring instruments: ammeter (measures current) and voltmeter (measures potential difference).

Conventional current direction
Conventional current is taken as the flow of positive charge from the positive terminal of the battery, through the circuit, to the negative terminal.

Pictorial vs schematic (circuit) diagrams
A pictorial diagram shows the actual appearance and relative positions of components; a schematic diagram uses standard symbols to show how components are electrically connected. Schematic diagrams are preferred for analysis because they are clear and unambiguous.

Important schematic symbols (commonly used)

• Cell / Battery (short and long parallel lines)
• Resistor (zig-zag or rectangle)
• Wire (solid line)
• Switch (open/closed break in wire)
• Bulb / Lamp (circle with a cross)
• Ammeter (circle with 'A', connected in series)
• Voltmeter (circle with 'V', connected in parallel)
• Fuse (a strip or small rectangle in series) and earth/ground symbol

Series and parallel connections
Series: components connected end-to-end so the same current flows through each. Equivalent resistance adds: R_eq = R1 + R2 + ...
Parallel: components connected across the same two points so the potential difference across each is the same. Reciprocal rule: 1/R_eq = 1/R1 + 1/R2 + ... . In parallel, total current splits among branches.

Practical consequences

• In series, if one component (like a bulb) fails (opens), the entire circuit stops working.
• In parallel (household wiring), each appliance works independently — one appliance failing does not cut power to others.
• Ammeter must be placed in series (low internal resistance); voltmeter in parallel (high internal resistance).

How to draw a neat circuit diagram

• Use standard symbols and straight lines.
• Show clear connections and junctions (dots) where wires meet.
• Label instruments and polarity of sources (+ and –).
• Indicate where ammeter and voltmeter are connected in experimental circuits.

Simple analysis with Ohm's law
Ohm's law relates voltage (V), current (I) and resistance (R): V = I R. Using this and series/parallel rules you can compute currents, voltages across elements and equivalent resistance of simple circuits.

Ohm's law: For a metallic conductor kept at a constant temperature, the electric current (I) through it is directly proportional to the potential difference (V) across its ends. Mathematically, I ∝ V (at constant temperature), which leads to the fundamental relation V = IR.

Meaning of terms: V is the potential difference in volts (V), I is the current in amperes (A), and R is the electrical resistance in ohms (Ω). Resistance is a measure of how much a conductor opposes the flow of electric charge. For a given conductor R = V / I (constant for that conductor at fixed temperature).

Experimental verification (simple circuit):

• Apparatus: battery or variable DC source, ammeter (in series), voltmeter (in parallel across the conductor), connecting wires, and a resistor or nichrome wire whose V–I behaviour is to be measured.
• Procedure: Vary the source voltage in steps, record the corresponding current each time (keep the conductor’s temperature approximately constant).
• Observation: Plot V (y-axis) versus I (x-axis). For an ohmic conductor the plot is a straight line through the origin. The slope of the V–I line is the resistance R (slope = V / I). Equivalently, the slope of I versus V is the conductance (1/R).

Microscopic idea (brief): In metals, free electrons drift under an applied electric field. Collisions of electrons with the lattice (ions, impurities) dissipate energy and produce resistance. If scattering conditions remain the same (constant temperature), the drift velocity — and thus current — is proportional to the applied field (and hence V), giving Ohm’s law.

Limitations: Ohm's law holds for ohmic materials (most metallic conductors) at constant temperature. Many devices (e.g., semiconductor diodes, filament lamps, thermistors) are non-ohmic — their V–I relationship is not linear because their resistance changes with voltage, current or temperature.

Applications: Ohm's law is used to calculate currents, voltages and resistances in circuits, to design resistor values, to size wiring and fuses, and to analyze simple circuit behavior.

What is resistance?
Resistance is the property of a conductor that opposes the flow of electric current. It determines how much current flows for a given potential difference across the conductor. The SI unit of resistance is the ohm (symbol: Ω).

Ohm's law (for ohmic conductors)
For many conductors (called ohmic), the current through them is directly proportional to the potential difference across them provided temperature and other physical conditions remain constant. Mathematically, V ∝ I or V = IR, where V is potential difference, I is current, and R is resistance. This relation defines the resistance R = V/I.

Factors affecting resistance

• Length (L): Resistance is directly proportional to the length of the conductor. Longer conductor → larger resistance.
• Cross-sectional area (A): Resistance is inversely proportional to the cross-sectional area. Thicker conductor → smaller resistance.
• Material: Different materials offer different intrinsic opposition to current. This property is expressed by resistivity.
• Temperature: For most metals resistance increases with temperature; for some materials (e.g., thermistors) it decreases with temperature.

Resistivity (ρ)
Resistivity is a material property that quantifies how strongly a material resists current. It is defined for a uniform wire by the relation R = ρL/A. The SI unit of resistivity is ohm-meter (Ω·m). Low resistivity means good conductor (e.g., copper, silver); high resistivity means poor conductor or insulator.

Microscopic view
On the microscopic level, electrons moving under an electric field collide with atoms, impurities and lattice vibrations. Those collisions dissipate electrical energy as heat and manifest macroscopically as resistance.

Non-ohmic behavior
Not all devices obey Ohm's law. For example, a filament bulb has resistance that changes with temperature; its V–I graph is nonlinear. Semiconductor devices (diodes, transistors) are also non-ohmic.

Typical equations and units

• Resistance: R = V/I, unit: ohm (Ω).
• Resistivity: R = ρL/A → ρ = RA/L, unit: ohm-meter (Ω·m).
• Temperature dependence (approx. for metals): R = R0(1 + αΔT), where α is temperature coefficient of resistance.

Practical importance
Understanding resistance and resistivity helps in designing electrical wiring, choosing materials for heating elements, making resistors for circuits, preventing excessive heating in appliances, and using sensors (thermistors, RTDs) that rely on resistance change with temperature.

What is resistance? Resistance (R) is a measure of how strongly an object opposes the flow of electric current. Its SI unit is ohm (Ω). Ohm's law relates voltage (V), current (I) and resistance: V = I·R for ohmic conductors.

Main factors that affect resistance

• Length (L): Resistance is directly proportional to the length of the conductor. A longer wire has more collisions between electrons and the atomic lattice, so R increases with L. Mathematically: R ∝ L.
• Cross‑sectional area (A): Resistance is inversely proportional to the cross‑sectional area. A thicker wire provides more paths for electrons, so R decreases as A increases. Mathematically: R ∝ 1/A. For a circular wire A = πd²/4, so R ∝ 1/d² where d is diameter.
• Material (resistivity, ρ): Different materials offer different intrinsic opposition to current, described by resistivity ρ (unit: Ω·m). Metals (copper, aluminum) have low ρ; insulators have very high ρ. R ∝ ρ.
• Temperature (T): Resistance changes with temperature. For most metals resistance increases with temperature (positive temperature coefficient). For some semiconductors and thermistors, resistance decreases as temperature rises (negative coefficient). For small temperature changes: R(T) = R0[1 + α(T − T0)], where α is the temperature coefficient of resistance.
• Impurities, crystal structure and mechanical condition: Alloying, impurities, strain, or defects increase scattering and usually increase resistance. Very thin films or rough wires behave differently from ideal uniform conductors.

Combined formula: For a uniform conductor of length L and cross‑sectional area A made of a material of resistivity ρ, the resistance is given by:

R = ρ · L / A

Practical consequences: Power loss in a conductor carrying current I is P = I²R. Thus using low‑resistance, short, and thick conductors (and low‑ρ materials) reduces energy loss — important in power transmission. Conversely, resistive heating elements use high‑ρ materials and suitable geometry to produce heat.

Basic idea
Resistors can be connected in two simple ways: in series (end-to-end) or in parallel (side-by-side). These combinations change the net (equivalent) resistance seen by a source and how current and voltage distribute in the circuit.

Series combination
When resistors R1, R2, ... are connected one after another, the same current I flows through each resistor. The equivalent resistance R_eq (series) is the sum of resistances:
R_eq = R1 + R2 + ... + Rn

Voltage divides across series resistors: V_total = V1 + V2 + ... where Vi = I * Ri. Larger resistances get proportionally larger voltage drops (voltage divider).

Parallel combination
When resistors are connected with both ends joined (parallel), the voltage across each resistor is the same (V). The total current I_total is the sum of branch currents: I_total = I1 + I2 + ... where Ii = V / Ri. The reciprocal of equivalent resistance is the sum of reciprocals:
1 / R_eq = 1 / R1 + 1 / R2 + ... + 1 / Rn

For two resistors in parallel: R_eq = (R1 * R2) / (R1 + R2).

Key properties & intuition

• Series: R_eq increases as you add resistors (current decreases for fixed voltage).
• Parallel: R_eq decreases as you add resistors (total current increases for fixed voltage).
• In series current is common; in parallel voltage is common.
• Power dissipated: P = V * I = I^2 * R = V^2 / R. In series, P_i = I^2 * Ri. In parallel, P_i = V^2 / Ri.
• Special cases: a branch with resistance = 0 behaves like a short circuit; an open branch (infinite R) carries no current.

Worked numeric examples

Example 1 (series): R1 = 4 Ω, R2 = 6 Ω, battery V = 12 V.
R_eq = 4 + 6 = 10 Ω. Current I = V / R_eq = 12 / 10 = 1.2 A.
Voltage drops: V1 = I * R1 = 1.2 * 4 = 4.8 V, V2 = 1.2 * 6 = 7.2 V (4.8 + 7.2 = 12 V).

Example 2 (parallel): R1 = 4 Ω, R2 = 6 Ω, battery V = 12 V.
1 / R_eq = 1/4 + 1/6 = 5/12 → R_eq = 12/5 = 2.4 Ω.
Total current I_total = V / R_eq = 12 / 2.4 = 5 A. Branch currents: I1 = 12/4 = 3 A, I2 = 12/6 = 2 A (3 + 2 = 5 A).

When circuits are mixed
Many practical circuits combine series and parallel parts. Reduce stepwise: replace a simple series or parallel block by its R_eq, redraw, and repeat until a single R_eq remains.

Safety and applications
Parallel wiring is used in homes so each appliance gets full mains voltage and can operate independently. Series wiring is used where same current is required through components (e.g., some string circuits), but complete series strings are avoided for mains lighting because one failure may turn off the whole string.

Emf (E): Emf (electromotive force) of a cell is the chemical energy converted to electrical energy per unit charge when no current is drawn (open-circuit). It is measured in volts (V). Emf is the maximum potential difference the cell can provide.

Internal resistance (r): A real cell is not an ideal voltage source — it has some internal resistance due to the electrolyte, electrodes and connections. This resistance is called the internal resistance (r) and is measured in ohms (Ω). When current flows, some energy is lost inside the cell across r, so the terminal voltage across the cell drops.

Terminal voltage and relations: If a cell of emf E and internal resistance r is connected to an external resistor R and current I flows, the terminal voltage V (voltmeter reading across the cell terminals) is:

V = E − I r

Ohm's law for the external circuit gives:

I = E / (R + r)

These two relations come from the loop equation: E = I(R + r) = I R + I r. The part I R appears across the external resistor and I r is dropped inside the cell.

Power: Power delivered to the external resistor is P_ext = I² R. Power dissipated inside the cell is P_int = I² r. The total power supplied by the cell is P_total = E I.

Maximum power transfer: The external device receives maximum power when R = r. The maximum power delivered to R then is:

P_max = E² / (4 r) (when R = r)

Determination of E and r (experimental method):

• Measure the open-circuit terminal voltage (no load) — this equals emf E (approx.).
• Connect different known resistances R, measure current I and terminal voltage V for each.
• Plot V (y-axis) versus I (x-axis). The graph is a straight line: V = E − r I. The y-intercept gives E and the slope is −r.

Why internal resistance matters (qualitative): When a device draws heavy current (small R), the I r drop is large, so the terminal voltage falls significantly and the device may underperform. Internal resistance also causes heating and reduces battery life.

What is being measured? Resistance (R) of a conductor tells how much it opposes the flow of electric current. For many conductors (ohmic conductors) at constant temperature, voltage across the conductor is proportional to current through it (Ohm's law): V ∝ I.

Basic principle: R = V / I where V is potential difference across the resistor and I is the current through it. In experiment we vary V, measure corresponding I, plot V versus I and obtain R from the graph.

Common experimental methods:

• Ammeter–Voltmeter method: Connect the unknown resistor, an ammeter in series and a voltmeter in parallel with the resistor. Use a variable source (rheostat or variable supply). For several settings record V and I, plot V (y-axis) vs I (x-axis). For an ohmic conductor the plot is a straight line through origin; slope = R.
• Accuracy notes for ammeter–voltmeter method: Real ammeters and voltmeters have internal resistances (Ra and Rv). Two practical errors arise: (a) some current flows through the voltmeter (if Rv not very large) so ammeter reads total current; (b) the voltmeter may include the ammeter resistance in its reading if connected differently. To correct or reduce errors use a high-resistance voltmeter and a low-resistance ammeter, keep voltmeter always across only the resistor, and prefer the arrangement that prevents voltmeter current from passing through the ammeter.
• Wheatstone bridge (for high accuracy): A balanced bridge compares the unknown resistance Rx with known resistances. At balance (no current through galvanometer) the ratio of two known arms equals the ratio of the other two arms, giving Rx from the known values. This method is preferred for small resistances or high-precision measurement.

Practical procedure (Ammeter–Voltmeter method):

1. Set up circuit with battery, rheostat, ammeter in series, unknown resistor R_x, and voltmeter connected across R_x.
2. Adjust rheostat to get several different currents. For each record ammeter reading I and voltmeter reading V.
3. Plot V (y-axis) vs I (x-axis). For linear graph slope = R_x. Alternatively calculate R = V/I for stable reading.
4. Keep temperature as constant as possible (resistance changes with temperature).

Sources of error and how to reduce them:

• Internal resistances of meters — use meters with suitable ranges (high-range voltmeter, low-resistance ammeter) and apply corrections when required (formulas given below).
• Heating of the resistor as current flows — take readings quickly or use small currents.
• Contact and connection resistances — use clean tight connections, four-terminal (Kelvin) method for very low resistances.

When Ohm's law does not hold: For non-ohmic devices (e.g., filament bulb, diode), V–I graph is non-linear; resistance depends on V or I (or temperature).

Definition: Electric power is the rate at which electrical energy is converted into some other form (heat, light, mechanical work) per unit time. It tells how fast work is done by an electric current.

Basic formula and units: If W is the work (or energy) done in time t, electric power P is P = W / t. The SI unit of power is watt (W), where 1 W = 1 J s^-1. In electrical circuits power can be expressed in terms of voltage (V), current (I) and resistance (R):

• P = V I (power = potential difference × current)
• Using Ohm's law (V = IR): P = I² R
• Also P = V² / R

Derivation (short): The electrical work done in moving charge Q through a potential difference V is W = QV. If Q flows in time t, current I = Q / t, so W / t = V (Q / t) = V I. Hence P = V I. Substituting V = IR gives the alternate forms above.

Electrical energy and units used in billing: Energy consumed by an appliance of power P operating for time t is E = P t. For household billing power is often given in kilowatts (kW) and time in hours (h), so energy is measured in kilowatt-hour (kWh). 1 kWh = 1000 W × 3600 s = 3.6 × 10⁶ J.

Practical meaning & appliances: The power rating on an appliance (for example 60 W for a bulb) is the rate at which it consumes electrical energy. Higher power means more energy use per second and usually a brighter lamp or more heat produced. Choosing appliances and calculating electricity cost requires using P and time: Cost = (P in kW × hours) × rate per kWh.

Safety and efficiency notes: Devices with large power draw require thicker wires, proper fuses, and safe sockets because large current may cause heating. Two appliances with the same resistance behave differently depending on voltage and current: at fixed voltage P decreases with larger resistance (P = V²/R), while at fixed current P increases with R (P = I²R).

Definition: When an electric current passes through a conductor, electrical energy is partly converted into heat energy. This phenomenon is called the heating effect of electric current.

Joule's law (statement): The heat produced in a conductor in a given time is directly proportional to the square of the current passing through the conductor, to the resistance of the conductor, and to the time for which the current flows. In symbolic form: H ∝ I² R t.

Derivation (using Ohm's law and electric power):

• Instantaneous electrical power dissipated as heat: P = V I.
• Using Ohm's law V = I R, P = I (I R) = I² R.
• Energy (heat) produced in time t: H = P t = I² R t. This is Joule's law in equation form.
• Alternate forms: H = V I t and H = V² t / R (using V = I R).

Units: Heat (H) is measured in joules (J). Current I in amperes (A), resistance R in ohms (Ω), time t in seconds (s). Power P in watts (W), where 1 W = 1 J/s.

Physical reason: Charges (electrons) moving through a resistive material collide with ions and lattice atoms, losing kinetic energy that appears as internal energy (heat) of the material.

Relation to temperature rise: The heat produced can raise the temperature of the conductor. If the conductor (or body) of mass m and specific heat capacity c has temperature rise ΔT, then H = m c ΔT. Combining with Joule's law: I² R t = m c ΔT (useful to estimate temperature rise).

Simple experimental demonstration: Connect a length of nichrome wire to a battery and ammeter, measure current I and time t, and observe temperature rise (or melting of wax/fuse). Vary I (by changing voltage or wire length) and note how heat production changes (H ∝ I²).

Important notes:

• For a fixed current, heat produced increases with resistance. For a fixed voltage, heat produced decreases with increasing resistance (H = V² t / R).
• Joule heating is the principle behind many heating devices but is a loss in power-distribution lines. Wires carrying large currents may heat up and must be sized to avoid overheating.

Overview
A household electric circuit supplies electric energy from the mains (typically 230 V AC, 50 Hz) to appliances through three conductors: live (L), neutral (N) and earth (E). Live carries current to the appliance, neutral returns it to the supply, and earth provides a low-resistance path to ground for fault currents to protect people and equipment.

Basic wiring and connections

• Distribution: mains meter → main switch → distribution board (fuse/MCB/ELCB/RCD) → branch circuits for lights, sockets and heavy appliances.
• Parallel connection: Most household loads (lamps, fans, sockets) are connected in parallel so each device receives the full supply voltage and can operate independently.
• Switch placement: Switches are placed in the live wire so that when switched off the appliance is isolated from the live supply.

Common fault types

• Open circuit: wire broken or switch off — device does not work.
• Short circuit: live to neutral/earth contact with very low resistance causing a sudden large current — risk of fire and damage.
• Overload: too many devices/draw exceeds rated current of circuit causing heating and possible fuse/MCB trip.
• Earth fault (leakage): current flows from live to earth — shock hazard unless protective device acts.

Protective devices and how they work

• Fuse: a thin wire (or strip) sized to melt when current exceeds its rating for a short time. It provides simple, inexpensive overcurrent protection. Replace only with the correct rating.
• MCB (Miniature Circuit Breaker): an automatic switch that trips on overcurrent. It can be reset after tripping and protects against overload and short circuit.
• RCD/ELCB (Residual Current Device/Earth Leakage Circuit Breaker): detects imbalance between live and neutral currents (indicating leakage to earth) and trips quickly to prevent electric shock. Typical sensitivity for personal protection is ~30 mA.
• Earthing: connecting the metal body of an appliance to earth ensures that in case of internal insulation failure the fault current flows to earth, causing fuse/MCB to operate and reducing shock risk.

Safety practices

• Use proper fuses/MCBs with correct ratings; do not bypass or replace with higher ratings to avoid fire risk.
• Install RCDs for bathrooms, kitchens and outdoor circuits where leakage/shock risk is higher.
• Ensure all metal-bodied appliances are earthed or are double-insulated if earthing is not provided.
• Do not touch electrical appliances with wet hands, avoid overloading sockets, and use certified plugs and wiring.
• Switch off the main supply before doing wiring work; call a qualified electrician for faults.

Why parallel wiring for household loads?
In a parallel circuit each appliance gets full supply voltage; turning one off does not affect others. If bulbs were in series, voltage and brightness would change and one failure would break the entire string.

Energy and cost
Household energy consumption is calculated from electrical power and time. Managing high-power devices (heaters, air conditioners) and using energy-efficient appliances reduces consumption and risk of overloading circuits.

Overview
Materials are classified by how well they allow electrical charge to move. Conductors allow easy flow of charge (current), insulators resist charge flow, and semiconductors lie between the two and can be engineered (by doping, temperature, or fields) to change their conductivity.

Microscopic reason (simple band picture)

• Atoms have electrons in energy bands. The valence band is the highest filled band and the conduction band is the next higher empty band.
• Conductor (metal): valence and conduction bands overlap or there are free electrons — electrons move easily under an electric field.
• Insulator: a wide band gap (large energy difference) exists between valence and conduction bands — electrons cannot jump the gap under ordinary conditions, so current is negligible.
• Semiconductor: a small band gap (Eg) — only a small energy (thermal or via doping) is needed to promote electrons to the conduction band so conductivity is moderate and strongly temperature-dependent.

Charge carriers and conduction mechanisms

• Conductors: conduction mainly by free electrons (in metals) or by ions in electrolytes.
• Insulators: extremely few free carriers; charge does not flow under normal electric fields.
• Semiconductors: intrinsic (pure) semiconductors have few carriers produced by thermal excitation. Doping with donor (n-type) or acceptor (p-type) atoms increases carrier concentration and hence conductivity.

Temperature dependence

• Metals: as temperature increases, lattice vibrations increase, scattering increases and resistivity (R) increases approximately linearly: R(T) ≈ R0[1 + α(T − T0)], where α is the temperature coefficient of resistivity.
• Semiconductors: increasing temperature generates more electron–hole pairs, so conductivity increases strongly (roughly exponentially) with temperature: σ ∝ e^{−Eg/(2kT)} (Eg = band gap, k = Boltzmann constant).
• Insulators: behave like semiconductors with a very large Eg, so carrier generation is negligible at ordinary temperatures.

Practical implications

• Conductors are used for wires and contacts (e.g., copper, aluminum, silver) because of low resistance.
• Insulators are used to coat wires and prevent leakage (e.g., rubber, plastic, glass).
• Semiconductors are the basis of electronic devices (diodes, transistors, solar cells, LEDs) because their conductivity can be precisely controlled by doping, temperature, and applied voltages.

Key measurable quantities

• Resistance (R) — how much a component resists current (ohms, Ω).
• Resistivity (ρ) — material property: R = ρ(L/A) for a uniform rod of length L and cross-section A.
• Conductivity (σ) — σ = 1/ρ; higher σ means easier current flow.

Note for Class 10: The above band ideas are a simple qualitative model to explain why materials behave differently. Important practical differences are seen in V–I characteristics (Ohmic vs non-Ohmic behavior) and in how conductivity changes with temperature.

Overview

Resistors and heating elements convert electrical energy into thermal energy, but their material requirements differ because their roles differ. Resistors (used to control current or provide precise resistance) require materials with stable, predictable resistance and often a low temperature coefficient of resistance (TCR). Heating elements require materials with relatively high resistivity, high melting/softening temperature, good mechanical strength at high temperature and acceptable oxidation resistance.

Key material properties

• Resistivity (ρ): Resistance R = ρL/A. Higher ρ gives larger resistance for a given geometry.
• Temperature coefficient of resistance (α): Describes how R changes with temperature. Precision resistors need very small α (nearly constant R over temperature).
• Melting point and mechanical strength: Heating elements must withstand high temperature without melting or breaking.
• Oxidation/corrosion resistance: Prolongs life of heating elements at elevated temperatures.
• Workability and cost: Ease of manufacturing (winding, film deposition) and economic factors matter.

Materials commonly used for resistors

• Manganin (Cu–Mn–Ni alloy) and Constantan (Cu–Ni alloy): Very low TCR and stable; used for precision/standard resistors and ammeter/shunt resistors.
• Metal-film and metal-oxide films: Deposited on ceramic cores to make precise, stable fixed resistors used in electronics.
• Carbon composition and carbon film: Used in general-purpose resistors; carbon composition has larger noise and worse stability but can tolerate pulse energy.
• Wire-wound resistors (often using nickel–chromium alloys for the wire): Used when high power dissipation is needed and precise resistance is required.

Materials commonly used for heating elements

• Nichrome (Ni–Cr alloy): Widely used in toasters, electric heaters, hair dryers and ovens. Advantages: high resistivity, high melting point, good mechanical strength and acceptable oxidation resistance.
• Kanthal (Fe–Cr–Al alloy): Used in industrial furnaces and some domestic heaters; excellent high-temperature strength and oxidation resistance.
• Tungsten: Very high melting point; used for incandescent lamp filaments where the element must glow white-hot (also used in vacuum/hard environments).
• Ceramic/metal-sheathed elements: Resistive wire (nichrome/Kanthal) embedded in ceramic or enclosed in metal tubes for immersion heaters, kettles, and industrial heaters.

How choice matches application

Precision electronic circuits use low-TCR alloys (manganin/constantan) or film resistors for stable resistance. Devices meant to heat (kettles, irons) use high-resistivity, high-melting alloys (nichrome, Kanthal) formed to maximize heat generation and durability. Lamp filaments use tungsten because it can operate at very high temperature and emit visible light.

Overview: Practical skills in the Electricity chapter focus on measuring current and potential difference, drawing and interpreting V–I graphs, determining resistance from measurements, assembling series/parallel circuits, and solving numerical problems on resistance, current, power and energy. The numerical problems apply Ohm's law and combination rules to calculate currents, voltages, equivalent resistances and power consumption.

Key practical skills:

• Correctly connecting an ammeter (in series) and a voltmeter (in parallel) to measure current through and potential difference across a component.
• Varying applied voltage, recording corresponding current, and plotting a V–I graph to verify Ohm's law for an ohmic conductor.
• Determining resistance from the slope of the V–I graph (plot V on vertical axis, I on horizontal: slope = R).
• Assembling resistors in series and parallel to find equivalent resistance and measuring currents and voltage drops experimentally.
• Calculating electric power and energy consumption for devices using P = VI, P = I^2R or P = V^2/R, and E = Pt.
• Estimating uncertainties: repeat measurements, use suitable ranges on meters, and account for systematic errors (meter internal resistance, loose connections, contact resistance).

Typical experimental procedure (V–I graph / verifying Ohm's law):

1. Set up circuit: variable DC source → rheostat (or variable resistor) → test wire/resistor → ammeter in series. Connect voltmeter across the test resistor.
2. Start with lowest voltage. Record V (voltmeter) and I (ammeter) for several values as you increase voltage.
3. Tabulate readings and plot V (y-axis) vs I (x-axis). For an ohmic conductor, points lie approximately on a straight line through the origin.
4. Find slope of the best-fit straight line; slope = ΔV/ΔI = R (resistance). Use two distant, well-measured points to reduce error.
5. Compare measured R with nominal/theoretical value; discuss deviations (heating, non-ohmic behavior, meter loading).

Precautions & common sources of error:

• Ensure proper polarity and correct meter ranges; do not exceed meter limits.
• Tighten connections to reduce contact resistance; avoid heating the resistor during quick repeated runs (it changes R).
• Use sufficiently spaced data points for accurate slope; repeat readings for reliability and take average where appropriate.
• Remember ammeter must be in series and have low resistance; voltmeter in parallel and have high resistance to minimise circuit disturbance.

Approach to numerical problems:

• Identify what is given (V, I, individual resistances, time, power rating) and what is asked (I, R, P, energy).
• Choose appropriate formula (Ohm's law, series/parallel rules, power formulas, energy = Pt).
• Show intermediate steps with units, and check answers for physical plausibility (e.g., currents positive and not exceeding battery capability).
• For combination networks, simplify stepwise (series/parallel) and use Kirchhoff ideas only if needed for more complex configurations.