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

This page builds every symbol the parent note leans on, starting from a reader who has never seen a single one. Each entry gives you three things: plain meaning → the picture → why the topic needs it. Read top to bottom; each rung of the ladder rests on the one below it.

0. What even is heat? (the ground floor)

Picture a jar of tiny balls (atoms) that never stop trembling. In a cold jar they barely quiver; in a hot jar they shake violently.

• Plain meaning: $T$ measures average jiggle energy per atom.
• The picture: left jar = slow, small arrows; right jar = fast, big arrows.
• Why the topic needs it: every single formula here is driven by a temperature difference. Without $T$ there is no "hot" and "cold", and heat has no direction to flow.

1. Two ways to measure temperature: Celsius and Kelvin

$T_{\text{K}}=T_{{}^{\circ }\text{C}}+273.15$

• The picture: one vertical thermometer with two rulers glued side by side; the tick marks line up but the zero of each is in a different place. The spacing between ticks is identical.
• Why the topic needs it — the crucial subtlety: because the ticks are the same size, a difference is the same on both scales: $\Delta T=20^{\circ }\text{C}-0^{\circ }\text{C}=20\text{ K}.$ But a single value is not: $20^{\circ }\text{C}=293.15\text{ K}$, not $20$ K.

2. Rate of change: what $\frac{dQ}{dt}$ means

Why this tool and not just "$Q$"? A wall doesn't care about total joules — it cares about flow speed. Asking "how many joules per second?" is exactly what a rate answers, and the physics symbol for "per second, instant by instant" is the derivative $\frac{d(\cdot )}{dt}$.

• The picture: water leaving a tap. $Q$ is the total litres in the bucket; $\frac{dQ}{dt}$ is how fast the stream is running right now.
• This rate has a name in this topic: the heat current $H$.

3. The temperature gradient $\frac{dT}{dx}$

• The picture: a wall drawn as a ramp of temperature — hot face high on the left, cold face low on the right. The slope of that ramp is $\frac{dT}{dx}$. A steep ramp = big gradient = strong push for heat.
• Why the sign is negative: as you walk in the $+x$ direction (into the cold), $T$ goes down, so $\frac{dT}{dx}$ is a negative number. Heat flows the opposite way — toward $+x$ — which is why Fourier's law carries a $-$ to keep $H$ positive.
• Why the topic needs it: conduction is driven by steepness of the drop, not by the raw temperatures. The gradient is that steepness.

4. Area $A$ and thickness $\Delta x$ — the shape of the wall

• The picture: a slab (a brick). $A$ = the flat front you'd paint; $\Delta x$ = how deep the brick goes front-to-back.
• Why the topic needs both: a wider face ($A$) lets more heat through in parallel; a deeper brick ($\Delta x$) makes heat travel further, slowing it. These are the two geometry knobs in Fourier's law.

5. Material constants: $k$, $h$, $e$, $\sigma$

Each mode of transfer hides its "how easily?" inside a constant. Meet them.

6. Power $P$ and the 4th power $T^4$

• The picture: four curves $T$, $T^2$, $T^3$, $T^4$ climbing out of the origin — $T^4$ rockets up far above the others. That steepness is why a small rise in temperature causes a huge jump in radiated power.
• Why the topic needs it: radiation obeys $P\propto T^4$, so it becomes the dominant loss channel at high temperature (a filament, the Sun) even though it's tiny at room temperature.

7. The thermal ↔ electrical analogy

The parent maps heat onto electricity. Line the pictures up:

• Why the topic borrows this: you already know series/parallel resistor rules; reusing them for composite walls saves inventing new maths. Heat "flowing downhill" from high $\Delta T$ is exactly current flowing from high voltage.

Prerequisite map

Equipment checklist

Say each answer out loud before revealing.