 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/13/d/solution

With

β = 0

, integrating

Φ_{x} = 2 αx + α^{2} y + α z

gives

Φ = α x^{2} + α^{2} x y + αx z + K (y, z) .

The other two partial derivatives require

K_{y} = K_{z} = 0

. The origin condition fixes the remaining constant to zero. Hence the scalar potential is

Φ (x, y, z) = α x^{2} + α^{2} x y + αx z .

Its value at

(1, 1, 1)

gives

α^{2} + 2 α = 1

, so

α = - 1 \pm 2 .

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