{"id":713,"date":"2026-08-06T14:29:03","date_gmt":"2026-08-06T14:29:03","guid":{"rendered":"https:\/\/physicsfundamentalsinfo.com\/blog\/?p=713"},"modified":"2026-08-06T15:24:34","modified_gmt":"2026-08-06T15:24:34","slug":"fouriers-law","status":"publish","type":"post","link":"https:\/\/physicsfundamentalsinfo.com\/blog\/thermodynamics\/fouriers-law\/","title":{"rendered":"Fourier’s Law of Thermal Conduction (Q\/t = kA\u0394T\/d)"},"content":{"rendered":"\n
Definition<\/div>

\nFourier’s law states that the rate of heat conduction through a material is proportional to its thermal conductivity, its cross-sectional area and the temperature difference across it, and inversely proportional to its thickness. In one dimension it is written Q\/t = kA\u0394T\/d, giving the heat-flow rate in watts.\n<\/p><\/div>\n\n

Rest your hand on a metal railing on a January morning and it bites. Rest it on the wooden bench beside it and nothing much happens. Same air, same temperature, wildly different sensation \u2014 and the entire difference is one number in one equation.<\/p>\n\n

That number is thermal conductivity, and the equation is Fourier’s law. It is the piece of physics that decides how much heat leaks out of your walls tonight, how fast a saucepan browns an onion, and whether a processor survives its own workload. Get comfortable with it and you can put a watt figure on almost anything warm.<\/p>\n\n

What Is Fourier’s Law?<\/h2>\n\n

Fourier’s law is the rule that fixes how fast heat flows by conduction through a solid material. It says the flow rate rises with the material’s conductivity, with the area heat can cross, and with the temperature difference driving it \u2014 and falls as the material gets thicker.<\/p>\n\n

Think of it as an electrical circuit. Temperature difference is the voltage that pushes; thickness divided by conductivity is the resistance that holds back; the heat-flow rate is the current that results. Physicists lean on that analogy constantly, and once you see it you cannot unsee it.<\/p>\n\n

Joseph Fourier first set the law out in an 1807 memoir to the Institut de France, then published it in mature form in his 1822 Th\u00e9orie analytique de la chaleur<\/em>. To crack it he had to invent a new branch of mathematics along the way \u2014 Fourier series \u2014 which now underpins everything from MP3 compression to MRI scanners.<\/p>\n\n

One thing to be clear about from the start: this law describes conduction only<\/strong>. It is silent on heat carried by moving fluids or beamed across empty space.<\/p>\n\n

The Fourier’s Law Formula<\/h2>\n\n

The steady-state, one-dimensional form of Fourier’s law is written like this:<\/p>\n\n

Q\/t = k \u00b7 A \u00b7 \u0394T \/ d<\/div>\n\n

Every symbol, with its SI unit:<\/p>\n\n

\n\n\n\n\n\n\n\n\n\n
Symbol<\/th>\nQuantity<\/th>\nSI unit<\/th>\nNotes<\/th>\n<\/tr>\n<\/thead>\n
Q\/t<\/strong><\/td>\nRate of heat transfer<\/td>\nwatt (W) = J\/s<\/td>\nOften written as P<\/em>; it is a power, not an energy<\/td>\n<\/tr>\n
k<\/strong><\/td>\nThermal conductivity<\/td>\nW\/(m\u00b7K)<\/td>\nA property of the material, not the object<\/td>\n<\/tr>\n
A<\/strong><\/td>\nCross-sectional area<\/td>\nm\u00b2<\/td>\nMeasured perpendicular to the heat flow<\/td>\n<\/tr>\n
\u0394T<\/strong><\/td>\nTemperature difference<\/td>\nK (or \u00b0C)<\/td>\nHot face minus cold face; a difference, so K and \u00b0C match<\/td>\n<\/tr>\n
d<\/strong><\/td>\nThickness<\/td>\nm<\/td>\nMeasured along the direction of heat flow<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n

Three of those four inputs scale the answer directly: double the area, double the loss. Only thickness works the other way, sitting in the denominator where it divides the loss down.<\/p>\n\n

Once the physics is clear, the arithmetic is the boring part \u2014 you can hand the numbers to our Thermal Conduction (Fourier’s Law) Calculator<\/a>, which also rearranges the formula to solve for k, A, \u0394T or d without you doing the algebra.<\/p>\n\n

Rearranging the equation<\/h3>\n\n

Exam questions rarely hand you the variable you want. The four rearrangements worth memorising:<\/p>\n\n