Chapter 3 — Current Electricity

Fig 3.1 — Educational Diagram: Electric current

Definition: Electric current is the rate of flow of electric charge through a cross-section of a conductor. Conventionally the direction of current is the direction of positive charge flow (from higher potential to lower potential), which is opposite to the electron flow in metals.

Macroscopic (circuit) view: In circuits, current I is given by I = dq/dt (charge per unit time). A steady current means I is constant in time. Ohm's law (for ohmic materials) gives a linear relation between potential difference V across a conductor and the current through it: V = IR, where R is the resistance.

Microscopic view: In a conductor of cross-sectional area A, with number density n of charge carriers each of charge q (for electrons q = -e), the drift velocity v_d of carriers produces current

• I = nqAv_d.
• Current density J = I/A = nqv_d (vector in direction of conventional current).

The electric field E inside a conductor causes the carriers to drift; for linear materials J = σE, where σ is conductivity (σ = 1/ρ). Mobility μ relates drift velocity and field: v_d = μE, so σ = nqμ.

Resistance and resistivity: For a uniform conductor of length L and cross-sectional area A, resistance R = ρL/A, where ρ (rho) is resistivity. Resistivity depends on material and temperature. For many metals near room temperature, R(T) = R_0[1 + α(T - T_0)], where α is the temperature coefficient of resistance.

Cells and internal resistance: A real cell of emf ε has an internal resistance r. When delivering current I to an external load R, the terminal (measured) voltage V = ε - Ir. The power delivered to the load is P = I^2R = VI.

Energy and heating (Joule effect): A current I through resistance R for time t dissipates energy W = I^2Rt and heat Q = I^2Rt (or Q = VIt = V^2t/R). This effect is used in heaters and lamps and is the reason wires heat under large current.

Kirchhoff's laws (steady currents): 1) Junction rule (conservation of charge): sum of currents entering a junction equals sum leaving. 2) Loop rule (conservation of energy): sum of potential changes around any closed loop is zero. These are used to solve currents in networks of resistors.

Key points to remember:

• Direction of conventional current is opposite to electron flow in metals.
• Ohm's law is empirical and holds for ohmic materials (linear V–I relation); many devices (diodes, transistors) are non-ohmic.
• Current density and conductivity link microscopic carrier motion with macroscopic electrical behavior.
• Power and Joule heating are central for energy considerations and safety (fuses, circuit breakers).

Fig 3.2 — Educational Diagram: Drift velocity and number density

Drift velocity (v_d)
When an electric field is applied to a conductor, free charge carriers (electrons in metals, electrons and holes in semiconductors) acquire a small average velocity in the direction of the force. This average velocity superimposed on the random thermal motion is called the drift velocity, v_d. For electrons the drift direction is opposite to the conventional current.

Number density (n)
Number density n is the number of charge carriers per unit volume (SI unit: m^-3). For a metal with Z free electrons per atom, mass density ρ (kg m^-3) and molar mass M (kg mol^-1),

n = (Z · ρ · N_A) / M

Relation between current, drift velocity and number density
Consider a conductor of cross-sectional area A carrying current I. If each carrier has charge q and number density n, the current is the charge crossing a section per second:

I = n · q · A · v_d

So the drift velocity is

v_d = I / (n · q · A)

In terms of current density J = I/A,

J = n · q · v_d → v_d = J / (n · q)

Drift velocity and electric field
In a material, the drift velocity is proportional to the applied electric field E. The proportionality constant is the mobility μ:

v_d = μ · E

Current density then becomes J = n · q · μ · E. The electrical conductivity σ is σ = n · q · μ, so Ohm's law in microscopic form is J = σ E.

Microscopic picture and relaxation time
Under a constant field electrons accelerate between collisions. If the mean time between collisions is τ and electron mass is m, a simple Drude-model result is

v_d = (q · E · τ) / m

and μ = (q · τ) / m.

Orders of magnitude and physical insight
Typical drift velocities in everyday wires are extremely small (10^-5 to 10^-3 m s^-1) while thermal speeds of electrons are ~10⁵ m s^-1. The signal (electric field) propagates nearly at the speed of light through the conductor; individual carriers move slowly.

Fig 3.3 — Educational Diagram: Current density and mobility

Definition and physical meaning
Current density (a vector) J at a point in a conductor is the electric current per unit cross-sectional area normal to the flow: J = I/A (for uniform flow). More generally, J = dq/(A dt) and has SI unit ampere per square metre (A m⁻²). It indicates how much charge flows through a unit area every second and in which direction.

Microscopic picture and relation to drift velocity
In a conductor with free charge carriers of number density n (carriers per m³) each carrying charge q and having average drift velocity v_d, the current density is
J = n q v_d
This is a vector equation: J points in the direction of v_d multiplied by the sign of q. For electrons (q = −e) the electron drift velocity is opposite to the conventional current density.

Mobility
Mobility μ of charge carriers is a measure of how quickly carriers drift in response to an electric field E. It is defined by the proportionality between drift speed and applied electric field (for the low-field/Ohmic regime):
v_d = μ E
Hence mobility has SI unit metre² per volt-second (m² V⁻¹ s⁻¹).

Connecting J, μ and E
Combine J = n q v_d and v_d = μ E to get
J = n q μ E
For materials where current density is proportional to electric field (Ohmic materials) we can write J = σ E, where σ (electrical conductivity) = n q μ. The resistivity ρ = 1/σ.

Microscopic origin of mobility
In the simple Drude model, carriers accelerate under E and are scattered after an average relaxation time τ, giving v_d ≈ (q E τ)/m. Thus mobility μ = |q| τ / m (take magnitude of q). Mobility depends on scattering mechanisms (impurities, phonons) and therefore on temperature and material.

Sign and direction
Because electrons have negative charge, their drift velocity is opposite to the electric field; the conventional current density J is in the direction of E for positive carriers. The sign of q must be kept in vector forms.

Steady-state and continuity
Charge conservation connects current density and local charge density ρ via the continuity equation: ∇·J + ∂ρ/∂t = 0. In steady DC conditions ∂ρ/∂t = 0 so ∇·J = 0 (no net accumulation of charge inside the conductor).

Typical orders of magnitude and examples
Electron charge e = 1.602×10⁻¹⁹ C. Example: copper has free electron density n ≈ 8.5×10²8 m⁻3 and conductivity σ ≈ 5.8×10⁷ S m⁻¹, giving electron mobility μ ≈ σ/(n e) ≈ 4.3×10⁻3 m² V⁻¹ s⁻¹ (≈ 43 cm² V⁻¹ s⁻¹). In semiconductors mobilities are much larger (e.g., in silicon at 300 K: μ_e ≈ 0.135 m² V⁻¹ s⁻¹ or 1350 cm² V⁻¹ s⁻¹ for electrons, μ_h ≈ 0.045 m² V⁻¹ s⁻¹ for holes).

When linear relations break down
At very high fields or in non-ohmic materials, v_d may saturate or become nonlinear with E and the simple relations J = n q μ E and v_d = μ E are no longer valid (important in high-field semiconductor devices).

Practical importance
Current density and carrier mobility determine heating (J²ρ losses), electromigration in metal interconnects, switching speed of semiconductor devices (higher μ → faster response), plating rates in electrochemistry (current density controls deposition), and safe current-carrying capacity of wires and fuses.

Fig 3.4 — Educational Diagram: Ohm's law

Ohm's law — statement: For many metallic conductors kept at a constant temperature, the potential difference (V) across the conductor is directly proportional to the current (I) passing through it. Mathematically, V = IR, where R is the constant of proportionality called resistance.

Meaning of terms:

• Current (I): Rate of flow of charge through the conductor (SI unit: ampere, A).
• Potential difference (V): Work done per unit charge across the ends of the conductor (volt, V).
• Resistance (R): Opposition offered by the conductor to the flow of current (ohm, Ω). R = V/I for an ohmic device.

Microscopic form: At the microscopic level, current density J is proportional to electric field E: J = σE, where σ is electrical conductivity. Since J = I/A and E ≈ V/L, this leads to V = (ρL/A)I where ρ = 1/σ is resistivity.

Resistivity and geometry: Resistance depends on material and dimensions: R = ρL/A, where L is length and A is cross-sectional area. For a longer wire resistance increases; for a thicker wire resistance decreases.

Temperature dependence: For most metals, resistance increases approximately linearly with temperature over a limited range: R(T) = R0[1 + α(T - T0)], where α is the temperature coefficient of resistance. Some materials (semiconductors) show decreasing resistance with rising temperature.

Ohmic vs non-Ohmic conductors: Ohmic conductors obey V ∝ I (straight line through origin in V–I graph) for a range of V and constant temperature. Non-ohmic devices (e.g., diodes, filament bulbs) show nonlinear V–I characteristics.

Limitations: Ohm's law is empirical and holds for many conductors under fixed physical conditions (especially constant temperature). It is not a universal law for all materials or all conditions.

Experimental verification (brief): A simple experiment uses a variable source, an ammeter in series and a voltmeter across a test wire. Plot V versus I; if the plot is a straight line through origin, the wire obeys Ohm's law and the slope gives R.

Fig 3.5 — Educational Diagram: Resistance and resistivity

Overview
Resistance and resistivity describe how strongly a material opposes electric current. Resistance (R) is a property of a particular object (wire, resistor) and depends on its shape, size and material. Resistivity (ρ) is an intrinsic property of the material and does not depend on the object's dimensions.

Macroscopic definition
Ohm's law (for an ohmic conductor) relates voltage V across a conductor to the current I through it: V = I R. Resistance R = V / I (SI unit: ohm, Ω).

Dependence on geometry and material
For a uniform conductor of length L and cross-sectional area A, resistance is given by:
R = ρ L / A
where ρ is the resistivity of the material (SI unit: ohm·metre, Ω·m). Thus R increases with length and decreases with larger cross-sectional area.

Microscopic picture
Electric current is due to charge carriers (electrons in metals) drifting under the electric field with a small average drift velocity v_d. Current density J = I / A. Microscopically, J = n e v_d (n = carrier density, e = magnitude of electron charge). The electric field E and current density are related by J = σ E, where σ is conductivity; σ = 1 / ρ.

Derivation of R = ρ L / A (brief)

1. J = I / A and J = σ E.
2. For a uniform conductor, E = V / L, so I / A = σ (V / L).
3. Rearrange: V / I = L / (σ A) = ρ L / A. Hence R = ρ L / A.

Temperature dependence
For metals, resistivity increases approximately linearly with temperature over moderate ranges: ρ(T) = ρ_0 [1 + α (T - T_0)], where α is the temperature coefficient of resistivity at T_0. For semiconductors, resistivity typically decreases with increasing temperature (negative temperature coefficient) because carrier concentration rises.

Other relations
Power dissipated in a resistor: P = V I = I^2 R = V^2 / R. Series resistances add: R_eq = Σ R_i. Parallel resistances: 1 / R_eq = Σ (1 / R_i).

Non-ohmic behavior
Not all devices follow V ∝ I. Examples: diodes, thermistors (temperature-dependent R), filament lamps (R increases with temperature as current rises). For an ohmic conductor the V–I graph is a straight line through the origin.

Practical notes
- Resistivity values help choose materials: copper (low ρ) for wires, nichrome (higher ρ, stable with temperature) for heating elements.
- Thin wires have higher resistance; long transmission lines suffer I^2R losses (use higher voltages to reduce current and losses).

Fig 3.6 — Educational Diagram: Temperature dependence of resistivity

Definition
Resistivity (ρ) is an intrinsic property of a material that measures how strongly it opposes the flow of electric current. It depends on temperature: ρ = ρ(T).

Physical origin of temperature dependence

• Metals: Conduction in metals is by free electrons. As temperature rises, lattice vibrations (phonons) increase, causing more frequent electron scattering. This increases resistivity. Near room temperature the change is approximately linear.
• Semiconductors and insulators: Conduction depends on thermally excited charge carriers. Increasing temperature produces many more carriers (electrons and holes), so resistivity decreases—often very rapidly (approximately exponentially) with temperature.
• Superconductors: Some materials below a critical temperature Tc exhibit zero resistivity abruptly.

Linear approximation (small temperature ranges)
For many metals and resistors over a limited temperature range (for example around room temperature) the resistivity (and resistance) changes approximately linearly with temperature:

ρ(T) = ρ0[1 + α (T − T0)]

R(T) = R0[1 + α (T − T0)]

Here ρ0 (or R0) is the value at reference temperature T0, and α is the temperature coefficient of resistivity (or resistance) at T0. For metals α > 0; for intrinsic semiconductors an effective α is negative (large magnitude).

Nonlinear / wide-range behavior
For intrinsic semiconductors the conductivity σ (and hence resistivity ρ = 1/σ) varies approximately as:

σ(T) ∝ exp(−Eg / (2kT)) ⇒ ρ(T) ∝ exp(+Eg / (2kT))

where Eg is the band gap and k is Boltzmann's constant. This gives a very steep decrease of ρ with increasing T.

Key remarks

• The linear formula is an approximation; α itself can vary with T.
• Materials are selected for particular α: e.g., manganin has a very small α and is used for precision resistors; platinum (Pt100) RTDs have a well-characterized positive α used for temperature sensing.
• Practical consequences: heating of transmission lines raises resistance and power losses; thermistors exploit strong T-dependence for sensors and temperature compensation.

Fig 3.7 — Educational Diagram: V–I characteristics of different elements

What is a V–I characteristic?
The V–I characteristic (voltage-current characteristic) of an element is the graph or relation between the potential difference across the element (V) and the current (I) through it. It tells how I responds to applied V and reveals whether the element obeys Ohm's law (ohmic) or is non-ohmic.

General notes

• Axes convention: usually V on the horizontal axis and I on the vertical axis.
• Slope of the I–V curve (dI/dV) = conductance. Reciprocal of slope = resistance R (if slope is constant).
• If V and I are proportional (straight line through origin) the element is ohmic and R = V/I is constant.

Common elements and their V–I behavior

• Ideal resistor (metal wire at near-constant temperature): Linear V–I relation, straight line through origin. Obeys Ohm's law: V = IR.
• Filament lamp (incandescent bulb): Non-ohmic. As current increases the filament heats up, its resistance increases (positive temperature coefficient). The V–I curve is nonlinear: for increasing V the I increases but at a decreasing rate (curve bending so slope decreases).
• Semiconductor diode (pn junction): Strongly non-linear and asymmetric. In forward bias small current until knee (threshold ~0.7 V for Si, ~0.3 V for Ge); beyond knee current rises steeply (approximately exponential). In reverse bias only a tiny leakage current flows until breakdown (large reverse current) at the breakdown voltage (zener behavior if controlled).
• LED: Similar to diode with a forward threshold determined by the band gap; emits light when forward-biased above threshold.
• Thermistor: NTC (negative temperature coefficient): resistance decreases with temperature, so the V–I curve can be superlinear (I grows faster than linear with V as device heats). PTC: opposite behavior (resistance rises with temperature), producing sublinear I–V curve.
• Photoresistor (LDR): Resistance depends on illumination; under higher light the slope (conductance) increases, so V–I line is still roughly linear for a given illumination but changes with light level.
• Electrolyte / gas discharge / arc lamps: Often nonlinear; gas discharge may show negative differential resistance regions (current increases while voltage drops across device) and threshold voltages for conduction.
• Superconductor: Ideal: V = 0 for I < Ic (critical current). Once I exceeds Ic the device becomes resistive and V appears.

Why nonlinearity arises
Nonlinearity usually comes from temperature dependence of resistance (filament, PTC/NTC), charge-carrier injection and recombination (diodes), energy barriers (electrolytes, pn junctions), phase changes (superconductors), or field-dependent mobility in gases and plasmas.

How to use the V–I curve

• Find resistance: for ohmic elements R = V/I (constant). For non-ohmic elements instantaneous (dynamic) resistance = dV/dI or dynamic conductance = dI/dV.
• Identify thresholds (diode knee, breakdown), signs of heating (filament), or switching (PTC jumping resistance, superconducting transition).

Fig 3.8 — Educational Diagram: Combination of resistors

What it means
A combination of resistors is any arrangement of two or more resistors connected together in a circuit. The main objectives are to (a) find the equivalent (single) resistance seen by the source and (b) know how currents and voltages divide in the network.

Basic types

• Series connection: resistors joined end-to-end so the same current flows through each. The voltages across them add.
• Parallel connection: resistors connected to the same two nodes; the same voltage appears across each branch and the currents through branches add.
• Series–parallel (mixed): networks that can be reduced stepwise by replacing simple series or parallel groups with their equivalents.
• Bridge and non-reducible networks: e.g., Wheatstone bridge or arbitrary meshes that may require Kirchhoff's laws or transformations (Δ–Y / Y–Δ) to analyze.

Series resistors — key facts and derivation
If R1 and R2 are in series, the same current I flows through both. Total voltage V = V1 + V2 = I R1 + I R2. So the equivalent resistance is

Req = R1 + R2 + ... + Rn

Parallel resistors — key facts and derivation
For resistors R1 and R2 in parallel the voltage across each is the same (V). Total current I = I1 + I2 = V/R1 + V/R2. So

1/Req = 1/R1 + 1/R2 + ... + 1/Rn

For two resistors specifically: Req = (R1 R2) / (R1 + R2).

How currents and voltages divide

• In series: I is same; voltage divides in proportion to resistances: V1 = I R1, V2 = I R2, so V1/V2 = R1/R2.
• In parallel: V is same; currents divide inversely proportional to resistances: I1/I2 = R2/R1.

Power in resistor combinations

• Power dissipated by a resistor: P = V I = I^2 R = V^2 / R.
• Total power from source = sum of powers dissipated in each resistor.

Reduction strategy for complex networks

• Identify simple series or parallel groups and replace them by their equivalents stepwise.
• If a bridge element is present (e.g., Wheatstone bridge), check for balance condition. If balanced, the bridge branch carries no current and can be removed.
• If not reducible by simple series/parallel, apply Kirchhoff's laws (KCL, KVL) or use Δ–Y (delta–star) or Y–Δ conversions to simplify.

Wheatstone bridge (balanced)
A typical bridge with resistances R1,R2 in one pair and R3,R4 in the other is balanced (no current through the bridge resistor) when

R1 / R2 = R3 / R4

Δ–Y (delta–star) transformations
Useful to convert between a triangle (Δ) and star (Y) so that series/parallel reductions become possible.

Delta (R_ab, R_bc, R_ca) → Star (R_a, R_b, R_c):

R_a = (R_ab * R_ca) / (R_ab + R_bc + R_ca)

R_b = (R_ab * R_bc) / (R_ab + R_bc + R_ca)

R_c = (R_bc * R_ca) / (R_ab + R_bc + R_ca)

Star (R_a, R_b, R_c) → Delta (R_ab, R_bc, R_ca):

R_ab = (R_a R_b + R_b R_c + R_c R_a) / R_c

R_bc = (R_a R_b + R_b R_c + R_c R_a) / R_a

R_ca = (R_a R_b + R_b R_c + R_c R_a) / R_b

Solving using Kirchhoff's laws (when reduction not possible)

• Assign currents to branches and node voltages. Use KCL at nodes and KVL around independent loops to form linear equations.
• Solve equations (algebraically or using matrix methods) for currents/voltages, then compute equivalent resistance Req = V_source / I_source.

Practical tips

• Two identical resistors R in series give 2R; two identical in parallel give R/2.
• When dealing with many identical resistors, look for symmetry to simplify analysis.
• Always check units and signs when applying KVL/KCL.

Fig 3.9 — Educational Diagram: Kirchhoff's rules

What are Kirchhoff's rules? Kirchhoff's rules (or Kirchhoff's circuit laws) are two fundamental laws for analyzing electric circuits that cannot be simplified by series/parallel reduction alone. They are used to write equations for currents and voltages in circuits with multiple loops and junctions.

1. Kirchhoff's Current Law (KCL) — Junction Rule
At any junction (node) in an electric circuit, the algebraic sum of currents entering the junction equals the sum leaving it. Equivalently, the sum of currents at a node is zero:

• ΣI_in = ΣI_out
• or ΣI = 0 (taking entering as positive and leaving as negative, or vice versa)

2. Kirchhoff's Voltage Law (KVL) — Loop Rule
For any closed loop in a circuit, the algebraic sum of potential differences (voltage gains and drops) around the loop is zero:

• ΣΔV = 0
• When traversing a loop, add +E when you go from negative to positive terminal of an emf, and subtract IR when you cross a resistor in direction of current (voltage drop).

Sign conventions and procedure

• Choose a direction for each current (if sign comes out negative, actual current is opposite).
• For KCL: write one equation per independent node (N nodes → at most N−1 independent equations).
• For KVL: write one independent loop equation per mesh (for M meshes write M independent loop equations).
• Apply Ohm's law V = IR for resistors. Include internal resistance r of batteries: terminal V = E − Ir (when current leaves positive terminal).
• Solve the simultaneous linear equations (substitution, elimination, or matrix methods like Cramer's rule).

Practical notes

• For circuits like a balanced Wheatstone bridge, the current through the bridge branch can be zero — simplifying equations.
• Double-check loop direction and sign for battery crossings: going from − to + is a rise (+E), from + to − is a drop (−E).
• KCL and KVL are exact and apply to DC circuits (steady currents); for AC and time-varying fields, KVL must include changing magnetic flux (Faraday's law) unless negligible.

Step-by-step example outline (two-loop circuit)

• Label currents I1 and I2 in the two loops and a shared branch current as I3 = I1 − I2 (according to chosen directions).
• Write KCL at a node if needed (or express shared current algebraically).
• Write KVL for loop 1: +E1 − I1R1 − (I1 − I2)R3 = 0.
• Write KVL for loop 2: +E2 − I2R2 − (I2 − I1)R3 = 0.
• Solve the two linear equations for I1 and I2. Interpret signs.

Fig 3.10 — Educational Diagram: Electromotive force (EMF) and internal resistance

Definition: Electromotive force (EMF, denoted E or  ε) of a source is the work done per unit charge in moving charge once around the complete circuit inside the source (chemical or other processes). It is the maximum potential difference between the terminals of the source when no current is drawn (open-circuit).

Real (non-ideal) source: A practical source (battery, cell, generator) has some internal resistance r. When current flows, some voltage is lost inside the source across r, so the terminal voltage V is less than the EMF.

Simple circuit model: Represent the source as an ideal EMF E in series with an internal resistor r, connected to an external load R. By Kirchhoff's loop law:

• E = V + I r, where V = terminal voltage across the external resistor R.
• Also V = I R, so E = I(R + r).

From these relations:

• Terminal voltage: V = E - I r. This shows V falls linearly with current; when I increases, internal drop I r increases, lowering V.
• Circuit current: I = E / (R + r).
• Short-circuit current (R = 0): I_sc = E / r.

Power considerations:

• Power delivered to load R: P_R = V I = I^2 R = E^2 R / (R + r)^2.
• Power dissipated inside the source (lost as heat): P_r = I^2 r.
• Total power supplied by the source: P_total = E I = P_R + P_r = E^2 / (R + r).

Maximum power transfer theorem: The load receives maximum power when R = r. Under this condition, P_R,max = E^2 / (4 r). Note: maximum power delivered to the load does not mean maximum efficiency (efficiency is 50% at this point).

Measurement of EMF and internal resistance:

• Open-circuit (no load) voltmeter reading approximates EMF if the voltmeter's resistance is very large (ideally infinite).
• More accurate EMF measurement: potentiometer (draws negligible current) gives true EMF.
• Internal resistance can be found experimentally by measuring terminal voltage V for different load currents I and plotting V vs I. The straight line has intercept E (at I=0) and slope -r (V = E - I r).

Physical origin of EMF: In a battery, chemical reactions do work on charges to separate positive and negative carriers, producing an electric potential difference. In a generator, mechanical work (via magnetic forces) does the work on charges.

Important remarks:

• EMF is not a force but an energy per charge (units volts, V).
• Internal resistance depends on construction, temperature, state of charge (for batteries), and increases as battery ages or at low temperatures.
• Terminal voltage equals EMF only when no current flows (open circuit).

Fig 3.11 — Educational Diagram: Cells in series and parallel

Overview
Cells (batteries) are combined in series or parallel to obtain required voltage, current capacity or power. Each cell is modelled as an ideal emf E in series with an internal resistance r. The terminal (output) voltage across a cell when current I is drawn is V = E - I r.

Cells in series
When n cells are connected in series (positive of one to negative of next), emfs add algebraically and internal resistances add. Equivalent circuit: E_eq = sum of emfs, r_eq = sum of internal resistances. If all cells are identical (E, r): E_eq = nE and r_eq = nr. The current through an external load R is

I = E_eq / (R + r_eq) = (sum E_i) / (R + sum r_i).

Terminal voltage across the series combination under load: V_terminal = E_eq - I r_eq. Short-circuit current (R = 0): I_sc = E_eq / r_eq.

Why use series?
Series connection increases voltage while capacity (in ampere-hours) remains that of a single cell (for identical cells). Common uses: torch/flashlight (two or more AA in series), 6 V lawnmower battery made from 4 x 1.5V cells in series, laptop packs often connect cells in series to reach required battery voltage.

Cells in parallel
Parallel connection means all positive terminals are tied together and all negatives together. If n identical cells (same E, r) are connected in parallel, equivalent emf E_eq = E and equivalent internal resistance r_eq = r / n (since conductances add). The total current delivered to a load R is I_total = E / (R + r_eq). The capacity (mAh) increases approximately by factor n while voltage stays the same.

For non-identical cells in parallel (different E or r) direct connection is not recommended: a difference in emf forces current to flow between cells even with no external load, possibly causing heating, damage or reverse charging. For two cells connected directly in parallel but with different emfs E1 and E2, the internal circulating current (short between cells) is I_between = (E1 - E2) / (r1 + r2).

Power and efficiency
Power delivered to external load R: P_load = I^2 R = (E_eq^2 R) / (R + r_eq)^2. Power dissipated inside cells: P_internal = I^2 r_eq. Maximum power transfer to the load occurs when R = r_eq (maximum power theorem), giving P_max = (E_eq^2) / (4 r_eq).

Practical notes
- Cells in series must be of the same type, age and state of charge to avoid cell reversal and uneven stress.
- Parallel cells should be matched too; otherwise large circulating currents can damage cells.
- For increasing voltage use series; for increasing capacity/current capability use parallel; many packs use series-parallel combinations to get both required voltage and capacity.

Fig 3.12 — Educational Diagram: Potentiometer

Definition & Principle: A potentiometer (in Class 12 physics context) is a null-type instrument used to measure the emf of a cell or to compare emfs without drawing current from the test cell. It works on the principle of a uniform potential gradient along a long, uniform resistance wire. At a particular point (the null or balance point) the potential difference between a terminal of the test cell and a point on the wire equals the cell's emf, and because no current flows through the galvanometer at that point, the measured emf is the true emf (unloaded).

Basic construction: A long uniform resistance wire AB (usually 1–2 m) is stretched on a wooden/plastic board. A constant driver cell (with rheostat to adjust current) is connected across the wire ends to produce a steady potential drop along the wire. One end of the test cell is connected to one end of the wire; the other end is connected to a galvanometer and a movable contact (jockey) which can touch any point on the wire. When the jockey is at the balance point, the galvanometer shows zero deflection.

How it is used:

• First establish a steady current through the wire so the potential drop per unit length (potential gradient) is stable.
• Place the test cell in the left gap and move the jockey until the galvanometer reads zero — that position gives the balance length L where potential difference along the wire equals the cell emf.
• If a standard cell of known emf Es is used to calibrate the wire, the potential gradient k = Es / Ls (Ls being its balance length). Then any unknown emf E = k Lu = Es (Lu / Ls).

Advantages of the null method: No current flows from the test cell at balance, so there is no loading and the measured emf is the true emf. High precision is obtainable by choosing long wire and fine jockey positioning.

Applications: measuring emf of cells, comparing two emfs, determining internal resistance of a cell (by measuring terminal voltage under load and using Vt = E - Ir), calibrating voltmeters, and in precise voltage reference experiments.

Assumptions and practical points: The wire must be uniform; driver current should be constant; temperature effects and contact resistances should be minimized; the galvanometer must be sensitive and the jockey contact clean.

Fig 3.13 — Educational Diagram: Wheatstone bridge and meter bridge

Overview
The Wheatstone bridge and the meter bridge are circuits used to measure unknown resistances accurately by comparing them with known resistances. Both operate on the null-deflection principle: the galvanometer shows zero when the bridge is balanced, eliminating errors due to meter internal resistance.

Wheatstone bridge — Description and working
The Wheatstone bridge is a four-resistor network arranged as a diamond: two known resistances R1 and R2 in one leg, a known standard resistance R3 in the second leg, and the unknown resistance Rx in the fourth leg. A galvanometer connects the two midpoints and a source (battery) is applied across the opposite corners.

When the bridge is balanced (no current through the galvanometer), the potential at the two midpoints is equal and the ratio of resistances satisfies the balance condition:

R1 / R2 = R3 / Rx

Hence the unknown resistance is Rx = (R3 × R2) / R1 (or Rx = R3 × (R2/R1) depending on labeling).

Derivation idea
Balance means the voltage drop across R1 equals the voltage drop across R3 (from the same supply), so the ratios of resistances in the two divider legs are equal. That gives the balance equation above.

Key points and sensitivity
- The bridge gives very accurate results because the measurement is done at null point where galvanometer current is zero.
- Sensitivity (ability to detect small changes in Rx) increases with larger supply voltage and lower galvanometer resistance; it is highest when the arms are of comparable resistance.
- Typical application: precise measurement of resistances, sensor readouts (strain gauges) where small resistance changes are detected.

Meter bridge (slide-wire bridge) — Description and working
The meter bridge is a practical form of Wheatstone bridge using a uniform resistance wire (usually 1 m long) stretched on a scale. The wire serves as two variable resistances proportional to lengths l and (100 − l) cm. A known standard resistance S and unknown resistance X are connected to the two ends of the bridge, and a jockey taps the wire to find the null point (no galvanometer deflection).

At balance: S / X = length corresponding to S / length corresponding to X = l / (100 − l) if the wire length is 100 cm. Thus

X = S × (100 − l) / l or commonly written X = S × l / (100 − l) depending on which side is labelled; be consistent with your connections. If l is the balance length measured from the S side, the usual formula used in CBSE labs is

X = S × l / (100 − l)

Procedure (meter bridge)
1. Connect standard resistor S and unknown X to the two left and right gaps. Place jockey on wire and connect galvanometer between jockey and bridge midpoint.
2. Move jockey to find point l where galvanometer shows null. Repeat several times and take average l.
3. Use X = S × l / (100 − l) to compute unknown resistance.
4. Repeat with different S values to reduce systematic error and compute mean.

Precautions
- Ensure good contacts and clean wire; avoid parallax while reading l.
- Keep galvanometer sensitivity adequate; use null method to avoid heating errors.
- Use moderate source voltage — too high causes wire heating and resistance change.

Applications (brief)
- Wheatstone bridge: precision resistance measurement, strain gauges and load cells (sensor bridges), temperature sensors (RTDs) in bridge circuits.
- Meter bridge: laboratory determination of resistivity of wires, comparison of resistances, educational demonstrations of bridge balancing.

Comparison (short)
- Wheatstone bridge is a circuit concept (four discrete resistors) used for precision measurement and sensor interfaces.
- Meter bridge is a practical implementation using a uniform-length wire to get continuously variable resistances and is widely used in school labs.

Common errors and how to reduce them
- Contact resistance: ensure tight connections.
- Non-uniform wire: choose good quality uniform wires.
- Temperature effects: perform measurements quickly and avoid heating the wire or resistors.

Fig 3.14 — Educational Diagram: Measuring instruments: ammeter and voltmeter

Overview: Ammeter and voltmeter are instruments used to measure electric current and potential difference (voltage) respectively. A sensitive moving-coil galvanometer (a device that shows deflection proportional to current) is the basic sensing element used to make both instruments by appropriate external resistances.

Construction & basic principle:

• Moving-coil galvanometer: a coil suspended in a magnetic field; coil deflection is proportional to the current through it (for small angles).
• Ammeter: made by connecting a low-resistance (shunt) resistor R_s in parallel with the galvanometer so most current bypasses the coil. This lets the instrument measure larger currents without damaging the coil.
• Voltmeter: made by connecting a high-resistance (multiplier) resistor R in series with the galvanometer so only a small current flows through the coil when a voltage is applied, allowing measurement of larger voltages.

Ideal instruments:

• Ideal ammeter: zero internal resistance so it does not change the circuit current; must be connected in series.
• Ideal voltmeter: infinite internal resistance so it draws no current from the circuit; must be connected in parallel.

Internal resistance and loading effect:

• A real ammeter has a small internal resistance r_a; inserting it in series changes the total circuit resistance slightly and thus the current (loading effect).
• A real voltmeter has a finite (large) internal resistance R_v; connecting it across circuit elements draws a small current and can change the voltage being measured (loading effect). Higher R_v reduces this error.

Range extension:

• To increase the current range of a meter with full-scale deflection current I_m and coil resistance r_a, add a shunt R_s in parallel so the meter reads up to I (desired maximum). Choose R_s so only I_m flows through the coil at full-scale.
• To increase the voltage range of a meter with coil resistance r_v and full-scale voltage V_m, add a series resistor R so the meter reads up to V (desired maximum).

Accuracy & sensitivity:

• Voltmeter sensitivity is commonly given as ohm-per-volt: S = R_v / V_fs (higher S means less loading).
• An ammeter should have as low an internal resistance as practical to minimize perturbation of the circuit.
• Digital multimeters typically have very high input impedance for voltage (e.g., 10 MΩ) and use shunts plus electronic circuitry for current measurement. Clamp meters allow non-intrusive current measurement by sensing magnetic field around a conductor.

Connection rules:

• Ammeter: always in series with the load; never place an ammeter directly across a voltage source without a suitable shunt/resistance (risk of short circuit).
• Voltmeter: always in parallel with the component across which the potential difference is measured.

Common sources of error:

• Finite internal resistances (loading) causing reading errors.
• Temperature changes altering resistance of shunt/multiplier and coil.
• Zero error or damping/overshoot for moving-coil instruments.

Practical notes: In labs and everyday use, digital multimeters (DMMs) replace separate analog ammeters/voltmeters. For high currents, shunt resistors or clamp meters are used. For sensitive voltage measurements (e.g., electronics), use meters with very high input impedance to prevent circuit disturbance.

Fig 3.15 — Educational Diagram: Electrical energy and power

Electrical energy is the work done in moving electric charge through an electric potential difference. If a charge q is moved through a potential difference V, the work done (electrical energy) is W = qV. For a steady current I, q = It, so

W = V I t.

Electrical power is the rate at which electrical energy is delivered or converted into other forms (heat, light, mechanical work). Instantaneous power p is defined as:

p = dW/dt = V I.

Using Ohm's law V = IR for a resistor R, we get alternative forms useful in calculations:

p = I^2 R = V^2 / R.

Joule's law of heating: The heat produced (or energy dissipated as heat) in a resistor R carrying current I for time t is

H = I^2 R t (which follows from W = P t = I^2 R t).

Energy units and billing: The SI unit of energy is the joule (J). Practical electrical energy uses the kilowatt-hour (kWh):

1 kWh = 1000 W × 3600 s = 3.6 × 10^6 J. Electricity bills are usually given in kWh.

Source with internal resistance: For a battery (emf ε) with internal resistance r delivering current I to an external resistor R, the current is

I = ε / (R + r).

Power delivered to the external resistor R is

P_R = I^2 R = ε^2 R / (R + r)^2.

This expression leads to the maximum power transfer theorem: P_R is maximum when R = r, and the maximum power is

P_max = ε^2 / (4 r).

Sign conventions and interpretation: If P > 0 for an element, the element absorbs power (converts electrical to other forms). If P < 0, the element delivers power to the circuit (like an ideal emf supplying energy).

Summary of key relationships:

• W = qV = V I t
• P = dW/dt = V I = I^2 R = V^2 / R
• H (heat) = I^2 R t
• For source with internal resistance: I = ε/(R + r), P_R = ε^2 R/(R + r)^2
• 1 kWh = 3.6 × 10^6 J

These formulas let you calculate energy consumption of appliances, design safe fuses and wires (considering I^2R losses), and understand why high-voltage transmission is used to reduce I and therefore reduce transmission losses (P_loss ∝ I^2).

Fig 3.16 — Educational Diagram: Joule heating and applications

Joule heating (Heating by electric current)

Joule heating (also called resistive heating or Ohmic heating) is the phenomenon in which electrical energy is converted into thermal energy when an electric current passes through a conductor. The microscopic origin is the work done by the electric field on charge carriers (electrons) which, through frequent collisions with the lattice ions, transfers kinetic energy to the lattice and raises its temperature.

Joule's law (qualitative and quantitative)

Experimentally, the heat produced in a conductor in a given time depends on three factors: the magnitude of current, the resistance of the conductor and the time for which the current flows. This is expressed by Joule's law:

Heat produced in time t: Q = I^2 R t

Derivation (using macroscopic laws): work done per unit charge across potential difference V is V. In time t the total charge passed is q = I t, so electrical energy supplied is W = V q = V I t. Using Ohm's law V = I R gives W = I^2 R t. Thus the heat produced Q equals the electrical energy converted into thermal energy (neglecting other losses).

Power (rate of heat production)

Instantaneous power dissipated as heat: P = dQ/dt = I^2 R = V I = V^2 / R. These forms are useful depending on which variables are known.

Microscopic form (density)

In terms of field and current density: power density (heat per unit volume) is p = E · J = J^2 / sigma = sigma E^2, where J is current density and sigma is electrical conductivity.

Dependence on temperature and material

Resistance R depends on geometry and material: R = rho L / A, where rho is resistivity, L length and A cross-sectional area. For metals, resistivity typically increases with temperature (rho(T) ≈ rho0 [1 + alpha (T - T0)]), so Joule heating can change R and produce nonlinear behavior (positive feedback).

Practical consequences

Joule heating is exploited deliberately in many devices (heaters, toasters) and is an unwanted loss in others (power transmission lines). Losses in transmission are minimized by reducing current (use higher voltage), using conductors of larger cross-section or low-resistivity materials, or using superconductors where possible.

Fig 3.17 — Educational Diagram: Drift and microscopic conduction mechanisms

Overview
When an electric field is applied to a conductor or semiconductor, charge carriers (electrons in metals, electrons and holes in semiconductors, ions in electrolytes) undergo a slight systematic motion superposed on their large random thermal motion. This systematic component is called the drift velocity and is the microscopic origin of macroscopic electric current.

Drift velocity and current
In a conductor with carrier density n (number of charge carriers per unit volume), charge q (for electrons q = -e), cross-sectional area A and average drift speed v_d, the electric current I is

• I = n q A v_d.

For electrons we often write (using magnitude of charge e): I = n e A v_d and the current density J = I/A = n e v_d.

Microscopic (Drude) description
The classical Drude model pictures free carriers undergoing frequent randomizing collisions with average time between collisions (relaxation time) τ. Under a constant electric field E a carrier accelerates between collisions, so the steady average drift velocity is

• v_d = (e E τ) / m,

where m is the effective mass of the carrier. Define mobility μ = e τ / m, then

• v_d = μ E.

Current density and microscopic form of Ohm's law:
J = n e v_d = n e μ E. Define conductivity σ = n e μ = n e^2 τ / m, so

• J = σ E (microscopic Ohm's law),
• ρ = 1/σ is resistivity.

Collision mechanisms and temperature dependence
Carriers are scattered by:

• lattice vibrations (phonons) — dominant in metals at higher temperatures; scattering increases with T so τ falls and resistivity rises approximately linearly with T in many metals;
• impurities and defects — important at low temperatures; produces a T-independent contribution to resistivity;
• carrier–carrier interactions and surface scattering (in thin films/wires).

In semiconductors the number density n and mobility μ both depend strongly on temperature and doping; thus conductivity can vary greatly with T and applied field.

Random thermal motion vs drift
Thermal velocities of electrons (root-mean-square) are of order v_th ~ sqrt(3kT/m) (~10^5 m/s), whereas typical drift velocities are tiny (~10^-4 to 10^-2 m/s in normal conductors). The net current arises because the tiny drift is superposed on otherwise isotropic random motion.

Transient response
When an electric field is switched on, velocity approaches steady drift value exponentially with time constant τ:

• v(t) = v_d (1 - e^{-t/τ}).

Limits of linear behaviour
At low fields J ∝ E (Ohmic). At high fields, carriers can heat, scatter differently, or saturate (nonlinear conduction) — important in semiconductors and vacuum/gas discharges.

Example numerical estimate (typical copper wire)
For copper: n ≈ 8.5 × 10^28 m^-3. For a wire radius 1 mm (A ≈ 3.14 × 10^-6 m^2) carrying I = 10 A,

• v_d = I / (n e A) ≈ 10 / (8.5×10^28 × 1.6×10^-19 × 3.14×10^-6) ≈ 2.3 × 10^-4 m/s (≈ 0.23 mm/s).

This shows how tiny drift velocity is compared to thermal speeds.

Fig 3.18 — Educational Diagram: Non-ideal effects and limitations

What this means: In real circuits components and sources are not ideal. Non-ideal effects are deviations from ideal behaviour (perfect conductors, fixed resistance, zero internal resistance, infinite input impedance of meters). These effects cause measurement errors, power loss and non-linear behaviour.

Internal resistance of a source: A practical cell or battery has an internal resistance r. If its emf is ε and current I is drawn, the terminal (measured) voltage V across the terminals is V = ε − I r. Graph of V versus I is a straight line with intercept ε and slope −r. Internal resistance dissipates power I²r inside the source, reducing power available to the load.

Power and maximum power transfer: If a load R is connected, current I = ε/(R + r). Power delivered to the load P_R = I²R = ε² R/(R + r)². Maximum power is delivered to R when R = r (maximum power transfer theorem). At that point half the power is lost in internal resistance and half delivered to the load.

Limitations of Ohm's law: Ohm's law (V = IR) holds only for ohmic conductors where resistance is constant (typically at constant temperature). Many devices are non-ohmic: their V–I relation is non-linear (examples: incandescent filament lamp, semiconductor diode, thermistor). For a filament lamp resistance increases with temperature so the V–I curve is curved (current grows less than linearly with voltage).

Non-ideal measuring instruments and loading effect: Ideal ammeter has zero resistance and ideal voltmeter infinite resistance. Real ammeters have small but finite resistance and voltmeters finite large resistance. Connecting them alters the circuit and can produce measurement errors (loading). For example, a voltmeter with finite R_v in parallel with a high-value resistor draws extra current and reduces measured voltage.

Temperature and material dependence: Resistance depends on temperature and material: for metals R ≈ R_0[1 + α(T − T_0)] (α is temperature coefficient). Semiconductors and thermistors show strong non-linear dependence (often exponential). Thus resistance is not constant when current causes heating.

Other practical non-ideal effects: contact resistance at junctions, internal leakage in batteries, finite switching/response times, and at high frequencies skin effect and inductive/capacitive reactances change current distribution so simple DC laws must be extended. Also Kirchhoff's loop law assumes negligible time-varying magnetic flux through the loop; at very high frequencies or rapidly changing fields, corrections (Maxwell’s equations) are needed.

Practical implications & mitigation: Expect voltage drop under heavy load (battery voltage falls), heating losses (I²R) in wires and batteries, measurement errors (use meter with appropriate range and high input impedance), and choose wiring, fusing and cooling to reduce losses. For precision use low internal resistance sources, low-resistance ammeters, high-resistance voltmeters, Kelvin (4-wire) measurements for small resistances, and frequency-aware circuit design for AC.

Fig 3.19 — Educational Diagram: Superconductivity (brief overview)

What is superconductivity?
Superconductivity is a physical state in which a material shows exactly zero electrical resistivity and expels magnetic fields when cooled below a characteristic temperature called the critical temperature (T_c).

Historical note: Discovered by H. Kamerlingh Onnes in 1911 for mercury at about 4.2 K.

Key characteristics

• Zero resistivity: Electrical resistance drops abruptly to zero for T < T_c, allowing persistent currents without applied voltage.
• Meissner effect: A superconductor expels magnetic flux from its interior when it transitions into the superconducting state — it behaves as a perfect diamagnet (magnetic susceptibility χ ≈ −1).
• Critical parameters: Every superconductor has limits: a critical temperature T_c, a critical magnetic field H_c(T), and a critical current density J_c. Exceeding any of these destroys superconductivity.

Types of superconductors

• Type I: Show complete Meissner effect and a single critical field H_c. Mostly elemental metals (e.g., Hg, Pb).
• Type II: Have two critical fields H_c1 and H_c2. Between these fields magnetic flux penetrates as quantized vortices; many alloys and high-T_c ceramics (e.g., YBa₂Cu₃O₇) are Type II.

Microscopic idea (brief): In conventional superconductors, electrons form bound pairs (Cooper pairs) due to an effective attractive interaction mediated by lattice vibrations (phonons). These pairs condense into a single quantum state that flows without scattering. BCS theory provides the quantitative microscopic explanation.

High-Tc superconductors: Some ceramic superconductors have much higher T_c (e.g., YBCO, T_c ≈ 90–93 K), making cooling with liquid nitrogen (77 K) feasible for some applications.

Practical consequences: Zero resistance enables persistent currents and lossless power transport (in principle). The Meissner effect and flux pinning in Type II materials enable magnetic levitation and very strong superconducting magnets.