Temperature of Body under Newton's Law of Cooling

Theorem

Let $B$ be a body in an environment whose ambient temperature is $H_a$.

Let $H$ be the temperature of $B$ at time $t$.

Let $H_0$ be the temperature of $B$ at time $t = 0$.

Then:

$H = H_a - \paren {H_0 - H_a} e^{-k t}$

where $k$ is some positive constant.

Proof

By Newton's Law of Cooling:

The rate at which a hot body loses heat is proportional to the difference in temperature between it and its surroundings.

We have the differential equation:

$\dfrac {\d H} {\d t} \propto - \paren {H - H_a}$

That is:

$\dfrac {\d H} {\d t} = - k \paren {H - H_a}$

where $k$ is some positive constant.

This is an instance of the Decay Equation, and so has a solution:

$H = H_a + \paren {H_0 - H_a} e^{-k t}$

$\blacksquare$

Source of Name

This entry was named for Isaac Newton.

Historical Note

Isaac Newton applied this law to make an estimate of the temperature of a red-hot iron ball.

Although this approximation was somewhat crude, it was better than anything else up till then.

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 4$: Growth, Decay and Chemical Reactions: Problem $5$
• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Chapter $1$: Mathematical Models: $\S 1.2$: Mathematical Models: Example $\S 1.2$

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