IGCSE, Edexcel, IB and AP Science Courses | Dr Dabour Hub Academy

Overview

🌳 Concept tree

Lesson 1→The Hot-Wire Ammeter and Single Element AC Circuits

├── Alternating current

├── Advantages of AC

├── The hot-wire ammeter

├── Non-inductive ohmic resistor

├── Pure inductor (coil of zero ohmic resistance)

├── Inductive reactance

├── Capacitor and DC

├── Capacitor and AC

├── Capacitive reactance

└── Combinations of capacitors

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Alternating current

In the previous chapter we studied the dynamo.
The current it produces, an alternating current (AC), changes in intensity and direction periodically: it rises from zero to a maximum, falls to zero during a half cycle, reverses, rises to a maximum in the opposite direction and falls to zero again in the second half cycle.
This repeats every cycle.

Comparison aspect	Main idea	Keywords
•	It is represented graphically by a sine wave, because the voltage and current vary sinusoidally with the angle (0° to 360° per cycle): V = V_max sin ωt, I = I_max sin ωt, ω = 2πf. The quantity ωt is the phase angle.	representedgraphicallysinewave
Frequency (f)	the number of complete cycles per second (hertz) equal to the number of revolutions per second of the dynamo coil producing it. In Egypt f = 50 Hz. The period is T = 1/f = 0.02 s for 50 Hz	numbercompletecyclessecond
Effective values	Ieff = 0.707 Imax Veff = 0.707 Vmax. Meters read effective values. A domestic supply of "220 V" is an effective value its peak is about 311 V	IeffImaxVeffVmax
Example 1a.	A supply has V = 311 sin(100π t). Its frequency is 100π/2π = 50 Hz and its period is 0.02 s. At t = 0.004 s the phase angle is 100π × 0.004 = 1.2566 rad = 72° so V = 311 sin 72° = 296 V. The effective value is 311 × 0.707 = 220 V	supply100πfrequencyperiod

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Advantages of AC

Its voltage can easily be stepped up or down by transformers.
It can be transmitted over long distances at high voltage with small power loss (via transformers).
It can be used in many applications but is not suitable for some, such as electrolysis and electroplating (which need DC).
It can be rectified into DC.
Like DC it has a heating effect, which does not depend on the direction of the current (heat ∝ I²).

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The hot-wire ammeter

Why a moving-coil ammeter cannot be used for AC:

its pointer deflects only if the coil is acted on by a field of constant direction and strength; with AC the torque reverses every half cycle, the coil cannot follow (it just vibrates), and the average reading would be zero.
Instead the heating effect of the current is used to measure the effective value.
Construction:

A thin wire AB of platinum–iridium alloy (expands considerably when hot, resists oxidation) is fixed tightly between two fixed ends A and B.
A silk thread is attached to the middle of the wire, passes round a pulley (P) and is held by a spring (S) fixed to the case. The pulley carries a pointer moving over a scale.
A small resistance (shunt) is connected in parallel with the wire AB.
Action connect the ammeter in series with the circuit. The current heats the wire, which expands, so the tension in the wire falls; the spring pulls the thread, turning the pulley a little, and the pointer deflects. The pointer settles at a steady reading when the temperature becomes constant (rate of heat produced = rate of heat lost by radiation). The reading corresponds to the effective value, and the instrument works for both AC and DC.

Calibration and scale calibrated with DC by connecting it in series with a moving-coil ammeter. The heat produced ∝ I², so the deflection is not proportional to I: the scale is not uniform — divisions are crowded at low values and become wider as the current increases. For example, if 2 A produces 4 scale divisions, 4 A produces 16 divisions.
Disadvantages:

The pointer moves slowly and returns slowly to zero (it takes time to heat and cool).
The wire is affected by changes in room temperature, causing a zero error. The remedy: mount the working wire on a plate of metal that has the same linear expansion coefficient as the wire so the effect on both is the same and compensated.

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Non-inductive ohmic resistor

For a resistor R across an AC source V = V_max sin ωt, the instantaneous current is
I = V/R = (V_max/R) sin ωt = I_max sin ωt

Comparison aspect	Main idea	Keywords
•	Current and voltage have the same phase angle ωt so they are in phase (they reach maximum and zero together). Their phasors (vectors) lie along the same direction	Currentvoltagesamephase
Ohm’s law applies to effective or maximum values	Veff = Ieff R	VeffIeff
Power consumed	P = I²R (average over the cycle using effective values)	averageovercycleusing
Example 1b.	A heater of resistance 24 Ω is connected to a 120 V (effective) supply. I_eff = 120/24 = 5 A, I_max = 5/0.707 = 7.07 A, and P = I²R = 25 × 24 = 600 W.	heaterresistanceconnectedeffective

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Pure inductor (coil of zero ohmic resistance)

When AC flows in a coil, the current changes continuously, so a back emf −L ΔI/Δt is induced by self-induction; it opposes the source emf and has the same frequency.
Applying Kirchhoff’s second law for a coil with negligible resistance, the source voltage balances the induced emf: V = L ΔI/Δt.

The voltage is proportional to the rate of change of current (the slope of the I–t curve): it is maximum when the current passes through zero (steepest slope) and zero when the current is at maximum (zero slope); it is negative when the current is decreasing.
Plotting the slope gives a curve shifted by a quarter of a cycle.
Therefore:
In an inductor the voltage leads the current by 90° (equivalently, the current lags the voltage by 90°).

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Inductive reactance

The opposition offered by a coil to the flow of AC, due to its self-inductance, is the inductive reactance X_L.
It is found to be directly proportional to the frequency of the current and to the self-inductance:
XL = 2π f L (ohm; f in Hz, L in henry)

X_L ∝ f and ∝ L. For DC (f = 0) X_L = 0: the coil offers only its ohmic resistance. A coil passes DC easily and hinders high frequencies (a "choke").
Current in the coil I = V/XL (effective values).
Inductors in series XL = XL1 + XL2 + … (for n equal, n XL1). In parallel: 1/XL = 1/XL1 + 1/XL2 + … (for n equal, XL1/n). (Valid when there is no mutual induction between the coils.)
Example 1. A coil of inductance 0.5 H and negligible resistance is connected to a 100 V, 50 Hz supply.
XL = 2π × 50 × 0.5 = 157 Ω; I = 100/157 = 0.64 A.
On a 100 V DC supply an ideal coil would carry a very large current (limited only by its resistance).

Comparison aspect	Main idea	Keywords
Example 1c.	A coil of L = 0.3 H and negligible resistance is connected to a 220 V supply. At 50 Hz: X_L = 2π × 50 × 0.3 = 94.2 Ω, I = 220/94.2 = 2.34 A. At 100 Hz the reactance doubles to 188.5 Ω and the current halves to 1.17 A.	coilnegligibleresistanceconnected
Example 1d — networks of coils.	Three coils have reactances 12 Ω, 24 Ω and 8 Ω at some frequency. In series X_L = 12 + 24 + 8 = 44 Ω. In parallel 1/X_L = 1/12 + 1/24 + 1/8 = (2 + 1 + 3)/24 = 6/24, so X_L = 4 Ω.	Threereactancessomefrequency
Example 1e — a real coil on DC and on AC.	A coil of L = 0.15 H and resistance 20 Ω is connected to a 40 V supply. On DC: I = 40/20 = 2 A. On 50 Hz AC: X_L = 2π × 50 × 0.15 = 47.1 Ω, Z = √(20² + 47.1²) = 51.2 Ω and I = 40/51.2 = 0.78 A. The same coil passes less than half as much AC as DC.	resistanceconnectedsupplysame

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Capacitor and DC

A capacitor consists of two parallel metal plates separated by an insulator.
Connected to a battery it charges: the battery pushes electrons to the plate joined to its negative pole, which repels electrons away from the other plate and towards the positive pole, so one plate becomes negative and the other positive.
The charge on one plate is Q = CV (C in farad, V in volt).
A momentary current flows until the capacitor voltage equals the battery voltage; then the current is zero.
A capacitor blocks steady DC.
Example 2a. A 50 μF capacitor charged by a 12 V battery holds Q = CV = 50×10⁻⁶ × 12 = 6×10⁻⁴ C.

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Capacitor and AC

With an alternating source, in the first half cycle the capacitor charges until its voltage equals the peak of the source; as the source voltage falls, the capacitor (now at higher voltage than the source) discharges back into it; in the second half cycle the plates charge with the opposite polarity, and so on.
Charge flows in and out of the plates, so an alternating current passes in the circuit (although no charge crosses the insulator).

The current at any instant is proportional to the rate of change of charge, I = ΔQ/Δt = C ΔV/Δt.
It is maximum when the voltage is passing through zero (steepest slope) and zero when the voltage is at its maximum.
In a capacitor the voltage lags the current by 90° (the current leads the voltage by 90°).

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Capacitive reactance

The opposition of a capacitor to AC, due to its capacitance:
(ohm)

Comparison aspect	Main idea	Keywords
•	X_C is inversely proportional to f and to C. For DC (f = 0) X_C is infinite (blocks DC); for high frequencies X_C is small. Current: I = V/X_C.	inverselyproportionalinfiniteblocks
Contrast with the coil	raising the frequency makes a coil oppose the current more but a capacitor less	raisingfrequencymakesoppose
Example 2b.	For C = 20 μF: at 50 Hz X_C = 1/(2π × 50 × 20×10⁻⁶) = 159 Ω; at 500 Hz X_C = 15.9 Ω.

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Combinations of capacitors

Series the same charge Q on each; V = V₁ + V₂ + …; 1/C = 1/C₁ + 1/C₂ + … (n equal: C₁/n). The total capacitance is less than the smallest.
Parallel the same voltage on each; Q = Q₁ + Q₂ + …; C = C₁ + C₂ + … (n equal: nC₁).
(Note the contrast with resistors for capacitors the rules are the other way round.)
Example 2c. 30 μF and 20 μF in series: 1/C = 1/30 + 1/20 = 5/60, C = 12 μF. In parallel: 50 μF.
Example 2. Three capacitors of 10, 30 and 60 μF are connected in parallel across a 220 V, 50 Hz supply.
C = 10 + 30 + 60 = 100 μF = 1.0×10⁻⁴ F.
X_C = 1/(2π × 50 × 1.0×10⁻⁴) = 31.8 Ω.
I = 220/31.8 = 6.9 A.
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Key Scientific Tables & Anatomical Diagrams Extracted from Official Curricula

💡 Core Diagrams & Reference Data→All data, reference comparison tables, and anatomical figures have been accurately digitized from official curriculum textbooks.

🔬 hot_wire_ammeter_and_ac_phasors

Scientific details and precise observational breakdown:

x2

Where (c) is a constant value
then the relation (y-x) is represented graphically as follows

y

x

The slope of the tangent

y

at a point on the curve

Increases by increasing the value of x

) Vectors

) Adding two vectors

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Part 13

📖

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📌 Comprehensive Scientific Terminology Dictionary:

Hot:
Wire Ammeter An electrical instrument designed to measure both DC and the effective (RMS) value of alternating current (Ieff), operating on the thermal effect of current ($Q = I2 R t$).
Inductive Reactance ($X_L$):
The opposition encountered by alternating current when passing through an induction coil due to its self-inductance, measured in ohms (Ω).
Capacitive Reactance ($X_C$):
The opposition encountered by alternating current when flowing through a capacitor due to its capacitance, measured in ohms (Ω).
Effective Value of AC (I_eff):
The steady DC current value that generates the same amount of thermal energy in a given resistor in the same time interval as the alternating current.

Phase Angle (φ):
The angular lead or lag between the alternating voltage vector and alternating current vector in an AC circuit.
⚖️ Governing Scientific Laws, Theorems & Mathematical Models:
Thermal Energy Formula:
[ Formula Q = I2 R t = Ieff2 R t (Thermal power is proportional to the square of effective current, resulting in non-uniform scale spacing on hot-wire ammeters). ]
Inductive Reactance Law:
[ Formula XL = 2π f L = ω L (Where $XL$ is directly proportional to frequency f and self-inductance L). ]
Capacitive Reactance Law:
[ Formula XC = 1 / (2π f C) = 1 / (ω C) (Where $XC$ is inversely proportional to frequency f and capacitance C). ]

Phase Relations:
[ Formula Pure Resistor: V and I are in phase (φ = 0). Pure Inductor: V leads I by 90° (π/2). Pure Capacitor: I leads V by 90° (π/2). ]
📌 Exhaustive In:
Depth Pedagogical Breakdown:
◄ I. Hot:
Wire Ammeter Physics, Thermal Expansion, and Non-Uniform Scaling
Unlike moving:
coil galvanometers which rely on magnetic torque and can only deflect in one direction for DC, the Hot-Wire Ammeter depends entirely on the thermal effect of electric current.
When alternating or direct current flows through an Ir-Pt (Platinum-Iridium) alloy wire, electrical energy is dissipated as heat, raising the wire temperature and causing it to expand.
A silk thread connected to a spring pulls the wire down over a light pulley, rotating an indicator pointer across a calibrated scale until thermal equilibrium is established.

The scale is non:
uniform because thermal power is proportional to the square of current (P ∝ I2).
Thus, doubling current quadruples pointer deflection ($22 = 4$), and tripling current increases deflection by nine times ($32 = 9$).
An essential drawback is slow pointer movement and thermal zero error caused by ambient room temperature fluctuations, resolved by mounting the alloy wire on a base plate with an identical thermal expansion coefficient.
Figure (hot_wire_ammeter_and_ac_phasors.png):
Figure (1) Internal structure of Hot-Wire Ammeter and phasor diagrams for R, L, and C circuits
◄ II. AC Passing Through an Ohmic Resistor (R Circuit)

When an AC source is connected to an induction:
free ohmic resistor R, both voltage and current follow sinusoidal waveforms: v = Vmax sin (ω t) and i = Imax sin (ω t).
According to Ohm's law, the current and voltage reach their positive peaks, zeros, and negative peaks simultaneously.
Therefore, the phase angle φ = 0, meaning voltage and current are in phase.
The resistance R = ρe l / A is independent of frequency f.
Electric power is continuously dissipated in the resistor as Joulean heat at a rate P = Ieff2 R = Veff2 / R.
◄ III. AC Passing Through a Pure Inductor (L Circuit) and Self:

Inductance

When an alternating voltage is applied across an induction coil of negligible resistance, the continuously changing current generates a varying magnetic flux, inducing a back EMF:
emf = -L (Δ i) / (Δ t).
By Lenz's law, this back EMF opposes current growth.
As a consequence, current lags behind voltage by a quarter cycle (90° or π/2 radians).

The opposition is inductive reactance XL = 2π f L.
Significantly, pure inductors do not consume or dissipate electrical energy; instead, energy is temporarily stored during the first quarter cycle in the surrounding magnetic field and returned to the circuit during the subsequent quarter cycle.
At very high frequencies, XL → ∞, causing the inductor to behave as an open switch.
◄ IV. AC Passing Through a Pure Capacitor (C Circuit) and Charge Dynamics
A capacitor consists of two conducting plates separated by an insulator dielectric.
When connected to an AC generator, current flows back and forth to charge and discharge the plates.
The instantaneous charge is q = C v.
Current represents the rate of charge flow:
i = (Δ q) / (Δ t) = C (Δ v) / (Δ t).

When voltage is zero and changing at its steepest rate, current reaches its maximum peak; when voltage reaches its maximum, its rate of change is zero, and current drops to zero.
Consequently, current leads voltage by 90°.
The capacitive reactance XC = 1 / (2π f C) drops dramatically at high frequencies (f → ∞ ⇒ XC → 0), allowing the capacitor to act virtually as a closed switch or short circuit.
🧮 Exemplary Step:
by-Step Worked Problems & Applied Exercises:
Worked Example 1:
Worked Problem 1 – Calculating Reactance and Current for Inductor and Capacitor
Problem Statement:
Problem An AC dynamo provides an RMS voltage of 220 V at frequency 50 Hz. Calculate:
1) The inductive reactance and RMS current if connected to an inductor of L = 0.7 H (negligible resistance).
2) The capacitive reactance and RMS current if connected to a capacitor of C = 14 $\mu$F.
Detailed Solution:
1) X_L = 2π f L = 2 × 22 / 7 × 50 × 0.7 = 220 Ω.
I_eff = V_eff / X_L = 220 / 220 = 1.0 A.
2) X_C = 1 / (2π f C) = 1 / (2 × 22 / 7 × 50 × (14 × 10^-6)) = 1 / (4400 × 10^-6) = 10⁶ / 4400 ≈ 227.3 Ω.
I_eff = V_eff / X_C = 220 / 227.3 ≈ 0.968 A.

💡 Master Teacher Insights & Critical Exam Cautions:
Exam Trap:
In pure inductors and pure capacitors, the average power dissipated over a complete cycle is EXACTLY ZERO (P_av = 0) because energy is stored as magnetic/electric fields and fully returned.
High:
Yield Rule At very high frequencies (f → ∞), an inductor acts as an OPEN switch (XL → ∞), while a capacitor acts as a CLOSED switch (XC → 0).
Scaling Secret:
The Hot-Wire Ammeter scale divisions expand quadratically 1A = 1 unit, 2A = 4 units, 3A = 9 units, 4A = 16 units.
🌍 Practical Real:
World, Industrial & Societal Connections:
✔ Radio Frequency Filters:
LC network filters are universally employed in smartphone transceivers to separate carrier waves from audio signals based on frequency-dependent reactances.

✔ Power Factor Correction in Power Grids:
Industrial facilities with heavy motor loads install capacitor banks to neutralize inductive reactance, raising the power factor to near unity.

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🔍 In-Depth Explanations, Deductions & Multi-Book Insights:

★ Comprehensive Study & Detailed Pedagogical Guide ★
Comprehensive Explanation The Hot-Wire Ammeter and Single Element AC Circuits (Resistor, Inductor, and Capacitor)

| | 2026/2027 |

🎯 Targeted Learning Outcomes & Pedagogical Objectives:
• Explain the scientific principle and calibration of the Hot-Wire Ammeter based on the thermal effect of electric current.
• Analyze the phase relationship between AC voltage and current in an ohmic resistor, pure inductor, and pure capacitor.
• Derive formulas for inductive reactance (X_L = 2π f L) and capacitive reactance (X_C = 1 / (2π f C)).
• Compare AC circuit behaviors at very low and very high frequencies.

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Lesson summary

Comparison aspect	Main idea	Keywords
•	★ High-Yield Summary Mind Maps & Comparison Tables ★	HighYieldSummaryMind
•	🗺️ Core Concept Map & Knowledge Architecture	CoreConceptKnowledgeArchitecture
✔ Hot-Wire Ammeter	Measures RMS AC and DC \| Thermal effect (Q ∝ I2) \| Non-uniform scale	MeasuresThermaleffectuniform
✔ Resistor Circuit	V and I in phase (φ = 0) \| Consumes energy as heat (P = I2 R)	phaseConsumesenergyheat
✔ Inductor Circuit	V leads I by 90° \| XL = 2π f L \| Zero energy dissipated	leadsZeroenergydissipated
✔ Capacitor Circuit	I leads V by 90° \| XC = 1 / (2π f C) \| Zero energy dissipated	leadsZeroenergydissipated
•	📊 Classification & Comparative Summary Tables	ClassificationComparativeSummaryTables
Table	Comparative Analysis of Pure AC Circuit Elements (R and C)	ComparativeAnalysisPureCircuit

Feature	Ohmic Resistor (R)	Pure Inductor (L)	Pure Capacitor (C)
Opposition Type	Ohmic Resistance R	Inductive Reactance X_L = 2π f L	Capacitive Reactance X_C = 1 / (2π f C)
Phase Angle φ	φ = 0 (V and I in phase)	φ = +90° (V leads I)	φ = -90° (I leads V)
Frequency Dependence	Independent of frequency f	Directly proportional (X_L ∝ f)	Inversely proportional (X_C ∝ 1/f)
Energy Dissipation	Dissipated as heat (I_eff² R)	Stored as magnetic field (Zero loss)	Stored as electric field (Zero loss)
Behavior at f → ∞	Resists current normally	Acts as an open circuit (X_L → ∞)	Acts as a short circuit (X_C → 0)

💡 Exam Flashcard Capsules & High-Yield Secrets:

★ RMS current I_eff = I_max / √2 = 0.707 I_max.
★ A DC current of 5A produces the exact same heating rate in a resistor as an AC current of I_eff = 5A.
★ The Hot-Wire Ammeter can be used to measure DC, while moving-coil meters fail with AC above 10 Hz.
★ Reactance is measured in Ohms (Ω) just like resistance, but reactances do not convert energy into Joulean heat.

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Curriculum

Physics

Chapter 1 - Electric Current, Ohms Law and Kirchhoffs Laws

Chapter 2 - Magnetic Effects of Electric Current and Measuring Instruments

Chapter 3 - Electromagnetic Induction and AC Dynamo and Transformer

Chapter 4 - Alternating Current AC Circuits

Chapter 5 - Dual Nature of Wave and Particle

Chapter 6 - Atomic Spectra and X-Rays

Chapter 7 - Lasers and Principles

Chapter 8 - Modern Electronics and Semiconductors

More lessons

Lesson 1: The Hot-Wire Ammeter and Single Element AC Circuits