A conducting sphere of radius 1 m is in air. To avoid breakdown, the maximum number of electrons on it is about $7\times10^n$, with integer n. Find n. Given breakdown field $3\times10^6$ V/m, $\epsilon_0=8.85\times10^{-12}$ F/m and electron charge $1.60\times10^{-19}$ C.
Surface field is $E=Ne/(4\pi\epsilon_0R^2)$, so the threshold is $N=4\pi\epsilon_0R^2E/e=2.085\times10^{15}$. The source’s fixed mantissa $7\times10^n$ does not match this result for an integer n: solving it formally gives n ≈ 14.47. The official key accepts the interval 14–15 to accommodate the order-of-magnitude request, and that grading is retained.
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