Heat transfer — conduction (Fourier's law k), convection, radiation (Stefan-Boltzmann σT⁴)

1. Conduction — Fourier's Law

WHAT we want: the rate of heat flow $\dfrac{dQ}{dt}$ (the heat current $H$ , units W) through a slab.

HOW we build the formula (derivation from scratch):

Think of a slab of cross-section $A$ , thickness $\Delta x$ , with hot face at $T_1$ and cold face at $T_2$ ( $T_1>T_2$ ).

Heat current must grow with area $A$ — twice the wall, twice the flow. So $H\propto A$ .
It must grow with the temperature difference $\Delta T = T_1-T_2$ — steeper drop pushes more heat. So $H\propto \Delta T$ .
It must shrink with thickness $\Delta x$ — a thicker wall slows it. So $H\propto 1/\Delta x$ .

Combine and insert a material constant $k$ to fix the units:

$H = \frac{dQ}{dt} = kA\,\frac{\Delta T}{\Delta x}$

In the limit of a continuous gradient this is the proper Fourier's law:

2. Convection

Natural convection: driven by buoyancy (density differences) alone.
Forced convection: fluid pushed by a fan/pump.

Newton's law of cooling (engineering model): $H = hA(T_s - T_{\text{fluid}})$ where $h$ is the convection coefficient (W m⁻² K⁻¹), $T_s$ the surface temperature.

3. Radiation — Stefan–Boltzmann Law

WHY the 4th power: integrating Planck's blackbody spectrum over all frequencies gives a total emitted power $\propto T^4$ . The key consequence: doubling absolute temperature multiplies radiated power by $2^4 = 16$ .

Figure — Heat transfer — conduction (Fourier's law k), convection, radiation (Stefan-Boltzmann σT⁴)

Common Mistakes (Steel-man + Fix)

Active Recall

#flashcards/physics

State Fourier's law of conduction with the meaning of every symbol.

H=-kA\,dT/dx

;

H

=heat current (W),

k

=thermal conductivity,

A

=area,

dT/dx

=temperature gradient; minus = flow toward cold.

What are the three modes of heat transfer and what carries energy in each?

Conduction (atomic collisions/electrons, no bulk motion), convection (moving fluid carries it), radiation (EM waves, no medium needed).

Why does conduction not occur with bulk motion of matter but convection does?

Conduction = energy passed between fixed/touching molecules; convection = hot fluid physically flows to a new place.

Define thermal resistance and how it combines in series vs parallel.

R=\Delta x/(kA)

; series → resistances add; parallel → conductances (

1/R

) add.

State the Stefan–Boltzmann law and the value of σ.

P=e\sigma A T^4

\sigma=5.67\times10^{-8}

W m⁻² K⁻⁴,

T

in kelvin.

Why is total radiated power proportional to

T^4

Integrating Planck's blackbody spectrum over all wavelengths gives total power ∝

T^4

Write the NET radiation exchange formula and justify it.

P_{net}=e\sigma A(T^4-T_0^4)

; body emits

e\sigma AT^4

, absorbs

e\sigma AT_0^4

from surroundings (Kirchhoff: same

e

If absolute temperature triples, radiated power changes by what factor?

3^4=81

times.

Why must radiation use kelvin but Fourier's law can use °C?

T^4

is absolute (power of T); Fourier uses only a difference

\Delta T

, which is identical in °C and K.

State Newton's law of cooling for convection.

H=hA(T_s-T_{fluid})

h

=convection coefficient.

Recall Feynman: explain to a 12-year-old

Imagine a hot bowl of soup. Conduction is like a line of kids holding hands passing a hot potato hand to hand — nobody moves, the heat hops along. Convection is one kid grabbing the hot potato and running to the cold side — the carrier itself moves. Radiation is the potato so hot it glows, and you feel its warmth on your face from across the room — that warmth flew through empty air as invisible light. The glowing one follows a sneaky rule: make it twice as hot (in true kelvin), and it shines sixteen times brighter.

Connections

Thermal conductivity k — material property table (metals high, gases low)
Second law of thermodynamics — heat spontaneously flows hot→cold
Black body radiation & Planck's law — origin of the $T^4$ law
Wien's displacement law — peak wavelength shifts with $T$
Newton's law of cooling — exponential cooling derived from convection
Electrical resistance Ohm's law — same math as thermal resistance
Greenhouse effect — radiation balance application

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, heat transfer ke sirf teen tareeke hain. Conduction mein heat solid ke andar atom-se-atom collision se travel karti hai — matter khud move nahi karta, bas energy aage badhti hai (jaise garam chamach). Iska rule Fourier's law hai: $H=kA\,\Delta T/\Delta x$ . Matlab zyada area, zyada temperature difference, aur patli wall → zyada heat flow. Yahan $k$ material ka apna property hai (metal ka high, gas/air ka bahut low — isiliye air gap achha insulator hai).

Convection mein fluid (liquid ya gas) khud bahta hai aur heat ko le jaata hai — garam fluid halka hoke upar uthta hai, thanda neeche aata hai, ek loop ban jaata hai. Yeh solid mein nahi ho sakta kyunki solid bah nahi sakta. Engineering formula: $H=hA(T_s-T_{fluid})$ .

Radiation sabse mast hai — yeh EM waves se jaati hai, koi medium chahiye hi nahi! Isi liye Sun ki garmi vacuum cross karke Earth tak aati hai. Iska law Stefan–Boltzmann: $P=e\sigma A T^4$ . Sabse important baat: yahan $T$ hamesha kelvin mein lo, kyunki yeh $T$ ka fourth power hai — agar temperature double karoge to power $2^4=16$ guna ho jaati hai! Yeh chhoti si galti (°C use karna) exam mein sabse zyada marks katwaati hai.

Yaad rakho: jab bhi $T$ ka power lena ho (radiation), kelvin; jab sirf difference $\Delta T$ chahiye (conduction), °C aur K dono same kaam karte hain. Bas yahi 80/20 trick hai poora chapter samajhne ke liye.

Heat transfer — conduction (Fourier's law k), convection, radiation (Stefan-Boltzmann σT⁴)

1. Conduction — Fourier's Law

2. Convection

3. Radiation — Stefan–Boltzmann Law

Common Mistakes (Steel-man + Fix)

Active Recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Thermodynamics

Connections