Temperature of Body under Newton's Law of Cooling
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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
- 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 $\S 1.2$
- Beware - the expression given is incorrect.