Newton's Law of Cooling
Exponential decay to ambient temperature.
Key Notes
Newton's law of cooling: rate of cooling of a hot body in air is proportional to its excess temperature over surroundings.
dT/dt = −k·(T − T_s), where T_s = surroundings, k = cooling constant.
Solution: T(t) = T_s + (T₀ − T_s)·e^(−kt) — exponential approach to T_s.
Valid for SMALL temperature differences (in linear regime).
For large ΔT, convective/radiative cooling is more strongly nonlinear (radiation ∝ T⁴).
Cooling constant k depends on size, shape, and surface — larger surfaces cool faster.
Used in: cooling curves of liquids, forensic temperature analysis (post-mortem time of death).
Hot soup cools faster in a wide bowl (more surface area) than in a tall cup.
Formulas
Newton's law of cooling
Cooling rate proportional to excess T.
Solution
Exponential approach to ambient.
Half-cooling time
Time for excess T to drop by half.
Average rate over small ΔT
Approximation when computing finite-time cooling.
Important Points
Cooling is EXPONENTIAL — never reaches T_s exactly, but asymptotically approaches.
Half the cooling happens in t = ln 2 / k.
Larger surface area ⇒ larger k ⇒ faster cooling. Why coffee in a wide cup cools faster.
Higher initial T ⇒ initial cooling rate larger (proportional to excess T).
Newton's law breaks down for large ΔT (radiation ∝ T⁴ dominates).
Forensic application: estimating time of death by body cooling rate (body's k from anatomy, mass, environment).
Newton's Law of Cooling notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.