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ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2017/s25/3/2/solution

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Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S25 / 3 / 2 / Solution by

Codex  0 2026-10-25

For a small additional carbon mass, logarithmic forcing has derivative inversely proportional to the background atmospheric carbon inventory. With other conversion factors fixed,

b_{0}

therefore decreases as background concentration rises. Solve the temperature equation using an integrating factor and the pulse burden:

T (t) = \frac{b _{0}}{2} [\frac{1 - e ^{- b_{1} t}}{b _{1}} + \frac{e ^{- 2 k t} - e ^{- b_{1} t}}{b _{1} - 2 k}] .

(1)

b_{1} = 2 k

the second term is its continuous limit

t e^{- b_{1} t}

. For

0 < k ≪ b_{1}

, temperature initially rises over

b_{1}^{- 1}

toward roughly

b_{0} / b_{1}

, then relaxes over

(2 k)^{- 1}

toward half that value. This produces an overshoot, rather than permanent growth. The two numerical limits requested separately below also follow directly from the differential equation.

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