Physics Cycle

19/06/2026

Be Charged. Don't get drained.

19/06/2026

19/06/2026

PHYSICS NOTE: ELECTRIC CHARGES — PRODUCTION, TYPES, DISTRIBUTION & STORAGE
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Syllabus Alignment: WAEC SSCE | UTME/JAMB | AP Physics | Cambridge IGCSE (0625)
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1. DEFINITION OF ELECTRIC CHARGE
Electric charge (Q or q) is a fundamental intrinsic property of matter that causes it to experience a force when placed in an electromagnetic field. It is carried by subatomic particles — protons carry positive charge, electrons carry negative charge. Charge is quantized, meaning it exists only in integer multiples of the elementary charge e.
2. TYPES OF ELECTRIC CHARGES
There are exactly TWO types of electric charges:
Table
Type Carrier How Produced Symbol
Positive (+) Proton Object LOSES electrons +Q or +q
Negative (-) Electron Object GAINS electrons -Q or -q
Key Facts:
Like charges REPEL each other (+ repels +, - repels -)
Unlike charges ATTRACT each other (+ attracts -)
A NEUTRAL object has equal numbers of protons and electrons
Standard Reference Experiments:
Glass rod rubbed with silk → glass becomes POSITIVELY charged
Ebonite rod rubbed with fur → ebonite becomes NEGATIVELY charged
Polythene rubbed with fur → polythene becomes NEGATIVELY charged
Cellulose acetate rubbed with silk → becomes POSITIVELY charged
3. METHODS OF PRODUCING ELECTRIC CHARGES
METHOD 1: CHARGING BY FRICTION (Triboelectric Effect)
When two different insulating materials are rubbed together, electrons are transferred from one material to the other.
Only ELECTRONS are transferred (protons are locked in the nucleus)
The material that GAINS electrons becomes NEGATIVELY charged
The material that LOSES electrons becomes POSITIVELY charged
Example: Rubbing a plastic comb through dry hair:
Comb gains electrons → becomes NEGATIVE
Hair loses electrons → becomes POSITIVE
The charged comb can then attract small pieces of paper
METHOD 2: CHARGING BY CONDUCTION (Contact)
When a charged object touches a neutral conductor, charge flows from the charged object to the neutral one until both reach the same potential.
Direct physical contact is required
The final charge on both objects is of the SAME sign
Charge distributes according to the sizes/shapes of the conductors
METHOD 3: CHARGING BY INDUCTION
Charging an object WITHOUT direct contact, using a charged object brought near it.
Steps for charging a conductor by induction:
Step 1: Bring a charged rod near (but not touching) the conductor
Step 2: Charges in the conductor redistribute (opposite charges attracted, like charges repelled)
Step 3: Earth the conductor while the charged rod is still near it
Step 4: Remove the earth connection
Step 5: Remove the charged rod — the conductor retains an induced charge of OPPOSITE sign to the inducing charge
4. FUNDAMENTAL LAW OF ELECTROSTATICS
"Like charges repel each other; unlike charges attract each other."
Mathematically, the force between two point charges is given by COULOMB'S LAW:
COULOMB'S LAW FORMULA:
plain
F = k * |q1 * q2| / r^2
Where:
F = electrostatic force (N)
q1, q2 = magnitudes of the two charges (C)
r = distance between the charges (m)
k = Coulomb's constant = 8.99 × 10^9 N·m²/C² (or approximately 9.0 × 10^9 N·m²/C²)
Alternative form using permittivity of free space (ε₀):
plain
k = 1 / (4πε₀)
where ε₀ = 8.854 × 10^-12 C²/(N·m²)
Direction of Force:
Same signs (both + or both -): REPULSIVE force (F > 0, pushes apart)
Opposite signs (+ and -): ATTRACTIVE force (F < 0, pulls together)
5. SI UNIT, DIMENSIONS, AND FORMULAS
SI Unit of Charge: COULOMB (C)
Table
Property Value/Symbol
SI Unit Coulomb (C)
Symbol Q or q
Definition 1 C = 1 A × 1 s (the charge transferred by 1 ampere of current in 1 second)
Elementary charge (e) e = 1.602 × 10^-19 C
Charge on electron q_e = -e = -1.602 × 10^-19 C
Charge on proton q_p = +e = +1.602 × 10^-19 C
DIMENSIONAL FORMULA OF CHARGE:
Derived from: Q = I × T (Charge = Current × Time)
plain
[Q] = [I] × [T]
[I] = [M^0 L^0 T^0 I^1] (dimension of current)
[T] = [M^0 L^0 T^1 I^0] (dimension of time)

Therefore:
[Q] = [M^0 L^0 T^1 I^1]
Dimensional formula of charge: [M⁰ L⁰ T¹ I¹]
Important Derived Formulas:
Table
Formula Expression Variables
Charge from current Q = I × t I = current (A), t = time (s)
Number of electrons n = Q / e e = 1.602 × 10^-19 C
Force (Coulomb's Law) F = k·q₁·q₂ / r² k = 9 × 10⁹ N·m²/C²
Electric field E = F / q = k·Q / r² E = N/C or V/m
Electric potential V = k·Q / r V = J/C = Volt
Potential energy U = k·q₁·q₂ / r U = Joule (J)
6. DISTRIBUTION OF CHARGES ON CONDUCTORS
Key Principles:
On an isolated charged conductor, charges reside ONLY on the outer surface.
Inside a hollow conductor, the electric field is ZERO
No net charge exists inside the conductor
Charge density is HIGHEST at sharp points or regions of SMALLER radius of curvature.
Sharp edges and pointed surfaces have the greatest concentration of charge
Flat surfaces have lower charge density
This is called the "power of points"
Surface Charge Density (σ):
plain
σ = Q / A
Where:
σ = surface charge density (C/m²)
Q = total charge (C)
A = surface area (m²)
For a conducting sphere:
Charge distributes uniformly over the surface
Inside the sphere: E = 0, V = constant
Outside the sphere: behaves like a point charge at the center
7. STORAGE OF ELECTRIC CHARGES
Device 1: LEYDEN JAR (Early Capacitor)
Invented in 1745
Consists of a glass jar coated inside and outside with metal foil
Stores charge on the metal plates separated by glass (dielectric)
Can deliver a sudden discharge (electric shock)
Device 2: ELECTROPHORUS
A device for transferring and storing charges by electrostatic induction
Consists of a metal plate with an insulating handle placed on a charged insulating base
Used to produce multiple charges from a single initial charge
Device 3: CAPACITOR (Modern)
A device that stores electric charge and energy in an electric field
Consists of two conducting plates separated by a dielectric (insulator)
Capacitance: C = Q / V
Where C = capacitance (Farad, F), Q = charge (C), V = potential difference (V)
Device 4: GOLD-LEAF ELECTROSCOPE
Used to DETECT and TEST charges
Consists of a metal rod with a brass cap and a thin gold leaf inside a metal case
When charged, the leaf diverges (moves away from the rod)
Greater charge = greater divergence
8. CONDUCTORS AND INSULATORS
Table
Conductors Insulators
Allow electrons to flow freely Do NOT allow electrons to flow easily
Have FREE electrons (delocalized) Electrons are tightly bound to atoms
Examples: metals (Cu, Al, Fe), graphite, salt solution, human body, damp air Examples: plastic, rubber, glass, wood, silk, paper, ebonite, Bakelite, dry air, oils
9. QUANTIZATION AND CONSERVATION OF CHARGE
Quantization of Charge:
Any charge Q can be expressed as:
plain
Q = n × e
Where:
n = integer (..., -2, -1, 0, 1, 2, ...)
e = 1.602 × 10^-19 C (elementary charge)
Law of Conservation of Charge:
"Charge can neither be created nor destroyed, only transferred from one body to another."
Total charge in an isolated system remains constant
When two bodies are rubbed together, the total charge before = total charge after
10. CALCULATION TIPS & PROBLEM-SOLVING STRATEGIES
Tip 1: Always Use SI Units
Convert ALL quantities to SI units before substituting into formulas
cm → m, μC → C (× 10^-6), nC → C (× 10^-9)
Tip 2: Coulomb's Law Calculations
plain
F = k·|q₁·q₂| / r²
k = 9 × 10⁹ N·m²/C²
Remember: force is ALWAYS attractive for opposite charges, repulsive for like charges
The force on q₁ equals the force on q₂ (Newton's 3rd Law)
Tip 3: Finding Number of Electrons
plain
n = Q / e
Example: If Q = -3.2 × 10^-19 C, then n = (-3.2 × 10^-19) / (-1.6 × 10^-19) = 2 electrons
Tip 4: Net Force from Multiple Charges
Use the PRINCIPLE OF SUPERPOSITION:
Calculate each force vector separately
Add them vectorially (consider direction!)
Tip 5: Charge Redistribution
When two identical conducting spheres touch:
plain
Q_final = (Q₁ + Q₂) / 2 (for each sphere)
Tip 6: Dimensional Analysis Check
If a formula gives [Q] = [M⁰ L⁰ T¹ I¹], it is dimensionally correct for charge.
11. WORKED EXAMPLES
EXAMPLE 1: Finding Force Between Charges
Two point charges, q₁ = +3.0 μC and q₂ = -5.0 μC, are placed 0.30 m apart in air. Calculate the force between them.
Solution:
q₁ = 3.0 × 10^-6 C
q₂ = 5.0 × 10^-6 C
r = 0.30 m
k = 9.0 × 10^9 N·m²/C²
F = (9.0 × 10^9) × (3.0 × 10^-6) × (5.0 × 10^-6) / (0.30)²
F = (9.0 × 10^9) × (15.0 × 10^-12) / 0.09
F = 135 × 10^-3 / 0.09
F = 1.5 N
Since the charges are opposite, the force is ATTRACTIVE.
EXAMPLE 2: Finding Number of Electrons
A body has a charge of -4.8 × 10^-17 C. How many excess electrons does it have?
Solution:
n = Q / e = (-4.8 × 10^-17) / (-1.6 × 10^-19)
n = 300 electrons
EXAMPLE 3: Charge Redistribution
Two identical metal spheres have charges +6.0 μC and -2.0 μC. They are touched together and then separated. What is the final charge on each?
Solution:
Q_final = (+6.0 + (-2.0)) / 2 = +4.0 / 2 = +2.0 μC
Each sphere will have +2.0 μC after separation.
12. APPLICATION: LIGHTNING CONDUCTOR
Principle:
A lightning conductor protects buildings by utilizing the "power of points" (high charge density at sharp points).
How It Works:
The sharp metal rod at the top of the building has a very high charge density
This creates a strong electric field that ionizes the air around the point
The ionized air provides a conducting path for charge to leak slowly into the atmosphere
During a lightning strike, the conductor provides a low-resistance path to the ground
The thick copper strip safely conducts the massive current into the earth
Key Points:
Sharp points allow gradual discharge (preventing buildup)
Thick conductor provides safe path for large currents
The building is protected because charges are safely diverted to ground
13. QUICK REFERENCE TABLE
Table
Concept Formula/Value Unit
Elementary charge (e) 1.602 × 10^-19 C C
Charge on electron -1.602 × 10^-19 C C
Charge on proton +1.602 × 10^-19 C C
Coulomb's constant (k) 8.99 × 10^9 N·m²/C² N·m²/C²
Permittivity of free space (ε₀) 8.854 × 10^-12 C²/(N·m²) C²/(N·m²)
1 Coulomb 6.24 × 10^18 electrons C
Charge quantization Q = n·e C
Coulomb's Law F = k·q₁·q₂/r² N
Electric field E = F/q = k·Q/r² N/C
Surface charge density σ = Q/A C/m²
Charge from current Q = I·t C
Capacitance C = Q/V F (Farad)
14. SYLLABUS CHECKLIST
WAEC SSCE / UTME (JAMB) Requirements:
Define electric charge and electrostatics
Identify positive and negative charges
Describe methods of producing charges (friction, conduction, induction)
State the fundamental law of electrostatics
Distinguish between conductors and insulators
Explain charge distribution on conductors
Describe devices for storing charge (Leyden jar, capacitor, electrophorus)
Explain the principle of lightning conductors
Solve problems using Coulomb's Law
Cambridge IGCSE (0625) Requirements:
State there are positive and negative charges
State like charges repel, unlike charges attract
Describe charging by friction experiments
Explain that friction charging involves transfer of negative charge (electrons)
Distinguish conductors and insulators using electron model
Know charge is measured in coulombs (Supplement)
Describe electric fields (Supplement)
Draw electric field patterns (Supplement)
AP Physics Requirements:
Understand charge as a fundamental property
Apply Coulomb's Law quantitatively
Understand electric field and potential
Apply superposition principle
Understand charge distribution on conductors
Apply Gauss's Law (AP Physics C)
15. COMMON EXAM QUESTIONS & ANSWERS
Q1: Why does a charged body eventually lose its charge?
A: Charges leak away through the air due to ionization of air molecules, especially in humid conditions.
Q2: Why is the charge density highest at sharp points?
A: Because the same amount of charge concentrates on a smaller surface area, and repulsive forces between like charges push them to regions of greatest curvature.
Q3: Can charge exist inside a hollow conductor?
A: No. All charge resides on the outer surface. The electric field inside is zero.
Q4: What happens when an uncharged conductor is brought near a positively charged rod?
A: Negative charges in the conductor are attracted toward the rod; positive charges are repelled. The near side becomes negative, the far side becomes positive.
Q5: Why does rubbing not create charge?
A: Rubbing only TRANSFERS existing electrons from one material to another. Charge is conserved.
16. SUMMARY
Electric charge is a fundamental property of matter, carried by electrons (-) and protons (+)
Charges are produced by FRICTION, CONDUCTION, and INDUCTION
Like charges repel; unlike charges attract
Charge is quantized: Q = n·e
Charge is conserved in any isolated system
On conductors, charge resides on the outer surface, with highest density at sharp points
Charges are stored in Leyden jars, electrophorus, and capacitors
Coulomb's Law: F = k·q₁·q₂/r²
SI unit: Coulomb (C); Dimensional formula: [M⁰ L⁰ T¹ I¹]
END OF NOTE

17/06/2026

16/06/2026

HEAT TRANSFER — COMPREHENSIVE PHYSICS NOTES
Aligned to WAEC, JAMB UTME, AP Physics, & IGCSE (0625) Syllabi
================================================================================ TABLE OF CONTENTS
Fundamental Definitions
Conduction
Convection
Radiation
Comparative Summary
Applications
Thermos Flask (Vacuum Flask)
Sea Breeze & Land Breeze
Solved Numerical Problems
Exam Tips & Common Pitfalls
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FUNDAMENTAL DEFINITIONS
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Term Symbol Definition SI Unit Dimensional Formula
Heat (Q) Q Energy transferred between bodies due to Joule (J) [M L^2 T^-2]
temperature difference
Temperature (T) T Measure of the average kinetic energy of particles Kelvin (K) [Theta]
Temperature Gradient dT/dx Rate of change of temperature with distance K/m or K.m^-1 [Theta L^-1]
Thermal Conductivity k or λ Measure of a material's ability to conduct heat W.m^-1.K^-1 [M L T^-3 Theta^-1]
Heat Flux q or Φ Rate of heat transfer per unit area W.m^-2 [M T^-3]
Specific Heat Capacity c Heat required to raise unit mass by unit temp J.kg^-1.K^-1 [L^2 T^-2 Theta^-1]
Thermal Resistance R Opposition to heat flow K.W^-1 [M^-1 L^-2 T^3 Theta]
Key Fact: Heat always flows from a region of HIGHER TEMPERATURE to LOWER TEMPERATURE until thermal equilibrium is reached (Zeroth Law of Thermodynamics).
================================================================================ 2. CONDUCTION
2.1 DEFINITION
Conduction is the transfer of heat through a material WITHOUT the bulk movement of the material itself. It occurs primarily in SOLIDS, and requires physical contact between particles.
2.2 MECHANISM
Material Type Mechanism
Metals Free electron movement + lattice vibrations
Non-metals (Insulators) Lattice vibrations only
Liquids & Gases Very poor — particles too far apart
Why metals conduct best: Metals contain FREE (delocalized) electrons that gain kinetic energy when heated and rapidly transfer energy through collisions with atoms. This is far more efficient than vibration alone.
2.3 FOURIER'S LAW OF HEAT CONDUCTION
The rate of heat flow through a material is given by:
Q = kA(T2 - T1)t / d
Or in differential form:
dQ/dt = -kA(dT/dx)
Where:
Q = Heat energy transferred (J)
k = Thermal conductivity (W.m^-1.K^-1)
A = Cross-sectional area (m^2)
(T2 - T1) = Temperature difference (K)
t = Time (s)
d = Thickness/distance (m)
dQ/dt = Rate of heat flow (W)
Negative sign in differential form indicates heat flows from high to low temperature (opposite to temperature gradient).
2.4 THERMAL CONDUCTIVITY VALUES (COMMON MATERIALS)
Material Thermal Conductivity k (W.m^-1.K^-1) Classification
Silver 430 Excellent conductor
Copper 400 Excellent conductor
Aluminum 237 Good conductor
Iron 80 Moderate conductor
Glass 1.0 Poor conductor (insulator)
Water 0.6 Poor conductor
Air 0.026 Very poor conductor
Wool/Felt 0.04 Good insulator
Styrofoam 0.033 Excellent insulator
Exam Tip (WAEC/JAMB): Remember that GASES are the POOREST CONDUCTORS because their particles are far apart. This is why air gaps are used for insulation.
2.5 FACTORS AFFECTING RATE OF CONDUCTION
Temperature Difference (ΔT): Greater ΔT → Faster conduction
Cross-sectional Area (A): Larger A → More heat flow
Thickness (d): Greater d → Slower conduction
Material (k): Higher k → Better conduction
2.6 DIMENSIONAL PROOF OF k
From Fourier's Law: k = Q.d / (A.ΔT.t)
[k] = [M L^2 T^-2] . [L] / ([L^2] . [Theta] . [T])
[k] = [M L T^-3 Theta^-1]
================================================================================ 3. CONVECTION
3.1 DEFINITION
Convection is the transfer of heat in FLUIDS (liquids and gases) by the ACTUAL MOVEMENT of the fluid itself, forming circulating currents.
3.2 MECHANISM (STEP-BY-STEP)
Step 1: Fluid near heat source gains heat → expands
Step 2: Expansion → density decreases
Step 3: Less dense fluid rises (buoyancy)
Step 4: Fluid moves away, cools down, contracts
Step 5: Density increases → sinks back
Step 6: Cycle repeats → CONVECTION CURRENT
3.3 TYPES OF CONVECTION
Type Description Example
Natural (Free) Driven by density differences from temp Boiling water, sea breeze
Convection gradients
Forced Convection Driven by external means (pump, fan) Car radiator, air conditioning
3.4 KEY EQUATIONS
Newton's Law of Cooling (related to convection):
dQ/dt = hA(Ts - Tf)
Where:
h = Convective heat transfer coefficient (W.m^-2.K^-1)
Ts = Surface temperature (K)
Tf = Fluid temperature (K)
Note: Unlike conduction, convection does NOT have a single universal equation like Fourier's Law. The coefficient h depends on fluid properties, flow type, and geometry.
3.5 FACTORS AFFECTING CONVECTION
Temperature difference between fluid and surface
Fluid velocity (forced convection)
Surface area exposed to fluid
Fluid properties (viscosity, density, specific heat)
Orientation of surface (horizontal/vertical)
================================================================================ 4. RADIATION
4.1 DEFINITION
Radiation is the transfer of heat by ELECTROMAGNETIC WAVES (primarily infrared) WITHOUT requiring any medium. It can travel through a vacuum.
4.2 KEY PROPERTIES
Property Description
Medium None required (vacuum OK)
Speed Speed of light (c ≈ 3 x 10^8 m/s)
Nature Electromagnetic waves (IR spectrum)
Absorption/Emission Depends on surface nature
4.3 STEFAN-BOLTZMANN LAW
The total power radiated by a black body:
P = σ A T^4
Where:
P = Power radiated (W)
σ (sigma) = Stefan-Boltzmann constant = 5.67 x 10^-8 W.m^-2.K^-4
A = Surface area (m^2)
T = Absolute temperature (K)
4.4 NET RADIATION HEAT TRANSFER
For a body at temperature T in surroundings at T0:
P_net = σ A ε (T^4 - T0^4)
Where ε (epsilon) = emissivity (0 ≤ ε ≤ 1)
Surface Type Emissivity (ε)
Perfect black body 1.0
Matte black paint 0.97
Polished metal 0.05 – 0.10
Human skin 0.98
Water 0.96
4.5 WIEN'S DISPLACEMENT LAW
λ_max = b / T
Where:
λ_max = Wavelength of maximum emission (m)
b = Wien's constant = 2.898 x 10^-3 m.K
T = Absolute temperature (K)
Example: The Sun (T ≈ 5800 K) emits peak radiation at λ_max ≈ 500 nm (visible light). The Earth (T ≈ 300 K) emits peak at ~10 μm (infrared).
4.6 EFFECT OF SURFACE NATURE
Surface Color/Texture Emission Absorption Reflection
Dull/Matte Black Best emitter Best absorber Poor reflector
Shiny/Polished Silver Poor emitter Poor absorber Best reflector
White Poor emitter Poor absorber Good reflector
IGCSE/WAEC Experiment: Place thermometer bulbs coated with different paints (black, white, silver) under identical heat lamps. The black-coated thermometer shows the highest temperature rise.
================================================================================ 5. COMPARATIVE SUMMARY
Feature Conduction Convection Radiation
Medium Solid (mainly) Fluid (liquid/gas) No medium needed
Particle Movement Vibrations only Bulk fluid movement Electromagnetic waves
Speed Slow Moderate Fastest (speed of light)
Direction Any direction Upward (natural) All directions
Key Law Fourier's Law Newton's Cooling Stefan-Boltzmann
SI Unit of Coefficient W.m^-1.K^-1 W.m^-2.K^-1 W.m^-2.K^-4
Exam Example Metal spoon in hot soup Boiling water Heat from Sun
================================================================================ 6. APPLICATIONS
6.1 THERMOS FLASK (VACUUM FLASK)
PURPOSE:
To keep liquids hot OR cold by MINIMIZING all three modes of heat transfer.
COMPONENT PARTS & WORKING PRINCIPLE:
Part Function Heat Transfer Prevented
Double-walled glass vessel Creates vacuum between walls Conduction & Convection
(vacuum = no particles)
Silvered inner surfaces Reflects infrared radiation Radiation (reflects heat back)
Cork/Plastic stopper Seals opening, poor conductor Conduction
Plastic/cork outer casing Protects glass, insulates Conduction
Vacuum between walls Eliminates air molecules Conduction & Convection
HOW IT WORKS (WAEC/JAMB STYLE ANSWER):
"The thermos flask minimizes heat transfer by:
Conduction & Convection: The vacuum between double glass walls removes air molecules, preventing both conduction (no particles to vibrate) and convection (no fluid to circulate).
Radiation: The silvered surfaces reflect infrared radiation back into the flask, minimizing radiative heat loss.
Conduction at opening: The cork/plastic stopper is a poor conductor, reducing heat transfer at the neck."
WHY GLASS?
Glass is a poor conductor, and the vacuum seal prevents the two surfaces from touching.
6.2 SEA BREEZE & LAND BREEZE
PRINCIPLE:
Caused by the DIFFERENCE IN SPECIFIC HEAT CAPACITIES of land and water, leading to differential heating and convection currents.
SPECIFIC HEAT VALUES:
Water: c = 4200 J.kg^-1.K^-1
Land (soil/rock): c ≈ 800–2000 J.kg^-1.K^-1
Water has a much higher specific heat capacity than land, so it heats up and cools down more slowly.
A. SEA BREEZE (Daytime)
plain
COOL AIR (from sea)
↓
←←←←←←←←←←←←←←←←←←←
↑ ↓
LAND (hot) SEA (cool)
↑ ↓
←←←←←←←←←←←←←←←←←←←
↑
WARM AIR (rises from land)
EXPLANATION:
During the day, the Sun heats both land and sea.
Land heats up FASTER (lower specific heat capacity).
Air above land becomes warm, expands, becomes less dense, and RISES.
Cooler air from over the sea moves inland to replace it.
Result: SEA BREEZE (cool wind blows from sea to land).
B. LAND BREEZE (Nighttime)
plain
WARM AIR (from land, rises)
↑
→→→→→→→→→→→→→→→→→→→
↓ ↑
LAND (cool) SEA (warm)
↓ ↑
→→→→→→→→→→→→→→→→→→→
↓
COOL AIR (from land)
EXPLANATION:
At night, both land and sea lose heat.
Land cools down FASTER (lower specific heat capacity).
Sea remains relatively warmer.
Air above sea becomes warm, rises.
Cooler air from land moves seaward to replace it.
Result: LAND BREEZE (cool wind blows from land to sea).
MNEMONIC: "Sea breeze by day, Land breeze by night — the water's always RIGHT!" (Water takes longer to heat/cool)
================================================================================ 7. SOLVED NUMERICAL PROBLEMS
PROBLEM 1: Conduction (WAEC/JAMB Style)
A copper rod of length 0.5 m and cross-sectional area 2.0 x 10^-4 m^2 has one end in boiling water (100°C) and the other in melting ice (0°C). Calculate the rate of heat flow. [k_copper = 400 W.m^-1.K^-1]
SOLUTION:
Given:
k = 400 W.m^-1.K^-1
A = 2.0 x 10^-4 m^2
ΔT = 100 – 0 = 100 K
d = 0.5 m
Using Fourier's Law:
dQ/dt = kAΔT / d
dQ/dt = (400 x 2.0 x 10^-4 x 100) / 0.5
dQ/dt = 8 / 0.5 = 16 W
ANSWER: Rate of heat flow = 16 W (or 16 J/s)
PROBLEM 2: Thermal Resistance
Calculate the thermal resistance of a glass window of thickness 5 mm and area 2 m^2. [k_glass = 1.0 W.m^-1.K^-1]
SOLUTION:
Thermal resistance:
R = d / (kA)
R = (5 x 10^-3) / (1.0 x 2) = 0.005 / 2 = 2.5 x 10^-3 K.W^-1
ANSWER: R = 2.5 x 10^-3 K.W^-1
PROBLEM 3: Radiation (Stefan-Boltzmann)
A black body of surface area 0.1 m^2 is at temperature 727°C. Calculate the power radiated. [σ = 5.67 x 10^-8 W.m^-2.K^-4]
SOLUTION:
First, convert to Kelvin:
T = 727 + 273 = 1000 K
Using Stefan-Boltzmann Law:
P = σ A T^4
P = 5.67 x 10^-8 x 0.1 x (1000)^4
P = 5.67 x 10^-8 x 0.1 x 10^12
P = 5.67 x 10^3 = 5670 W
ANSWER: Power radiated = 5670 W (or 5.67 kW)
PROBLEM 4: Temperature Gradient
The two ends of a metal rod are maintained at 150°C and 50°C. If the rod is 2 m long, calculate the temperature gradient.
SOLUTION:
Temperature gradient = ΔT / Δx = (150 - 50) / 2 = 100 / 2 = 50 K/m
ANSWER: Temperature gradient = 50 K.m^-1
================================================================================ 8. EXAM TIPS & COMMON PITFALLS
CALCULATION TIPS:
Tip Explanation
Always convert to Kelvin for radiation T(K) = T(°C) + 273
calculations
Check units carefully k in W.m^-1.K^-1, area in m^2, thickness in m
Temperature difference is same in K or °C ΔT(K) = ΔT(°C)
For composite walls, use thermal resistance R_total = R1 + R2 + R3
in series
For parallel conduction 1/R_total = 1/R1 + 1/R2
COMMON MISTAKES TO AVOID:
Using °C in Stefan-Boltzmann Law (MUST use Kelvin)
Confusing thermal conductivity (k) with heat transfer coefficient (h)
Forgetting that convection requires a fluid medium
Assuming shiny surfaces are good emitters (they're actually POOR emitters)
Mixing up sea breeze (day) and land breeze (night)
SYLLABUS-SPECIFIC FOCUS AREAS:
Exam Board Key Focus Areas
WAEC Thermos flask working, land/sea breeze, thermal conductivity problems, comparing conductivities
JAMB UTME Temperature gradient, heat flux calculations, Stefan-Boltzmann law, engine principles
AP Physics 1/2 Dimensional analysis, Stefan-Boltzmann derivation, thermal resistance networks
IGCSE (0625) Experimental demonstrations, good vs. bad conductors, surface effects on radiation
================================================================================ QUICK REFERENCE FORMULA SHEET
Q = mcΔT → Heat gained/lost (specific heat)
Q = mL → Latent heat
Q = kA(ΔT)t/d → Conduction (Fourier's Law)
dQ/dt = -kA(dT/dx) → Differential form
P = σAT^4 → Stefan-Boltzmann Law
P_net = σAε(T^4-T0^4) → Net radiation
λ_max = b/T → Wien's Displacement Law
R = d/(kA) → Thermal Resistance
T(K) = T(°C) + 273 → Temperature conversion
================================================================================ IMPORTANT FACTS TO MEMORIZE
Best conductor: Silver (k ≈ 430 W.m^-1.K^-1)
Best insulator: Air (k ≈ 0.026 W.m^-1.K^-1) — trapped air insulates
Stefan-Boltzmann constant: σ = 5.67 x 10^-8 W.m^-2.K^-4
Wien's constant: b = 2.898 x 10^-3 m.K
Water specific heat: c = 4200 J.kg^-1.K^-1
Black bodies: Perfect absorbers AND perfect emitters (ε = 1)
Polished metals: Poor absorbers, poor emitters, good reflectors
Convection currents: Driven by density differences, NOT by "heat rising" alone
================================================================================ SYLLABUS CROSS-REFERENCE
Topic WAEC JAMB AP Physics IGCSE
Conduction mechanism ✓ ✓ ✓ ✓
Convection currents ✓ ✓ ✓ ✓
Radiation & surface effects ✓ ✓ ✓ ✓
Thermal conductivity (k) ✓ ✓ ✓ ✓
Temperature gradient ✓ ✓ ✓ —
Heat flux ✓ ✓ ✓ —
Thermos flask ✓ ✓ — ✓
Sea/land breeze ✓ ✓ — ✓
Stefan-Boltzmann Law ✓ ✓ ✓ —
Wien's Displacement Law — — ✓ —
================================================================================
Sources: WAEC Physics Syllabus 2026/2027, JAMB UTME Physics Syllabus, Cambridge IGCSE 0625 Syllabus 2026–2028, CBSE Senior Secondary Physics 2025–26
End of Notes — Best of luck in your examinations!