 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2021/10/iii/solution

\nabla \cdot G = 3 x^{2} + 3 y^{2} + 3 z^{2} = 3 r^{2} .

\iint_{S} G \cdot n d S = ∭_{V} 3 r^{2} d V = 12 π \int_{0}^{a} r^{4} d r = \frac{12 π a ^{5}}{5} .

The

a^{5}

dependence also follows by combining the cubic field magnitude with the quadratic scaling of surface area.

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