 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-v-resit/question-7/b/i/solution

Write the observation point as

(0, 0, h)

with

h > 0

, reserving source coordinates for the integral. For the straight segment,

d s^{'} = d x^{'}

and the separation is

x^{'2} + h^{2}

ϕ (h) = - G γ \int_{- a}^{a} \frac{d x ^{'}}{x ^{'2} + h ^{2}} = - 2 G γ [lo g (x^{'} + x^{'2} + h^{2})]_{0}^{a} = - 2 G γ lo g (\frac{a + a ^{2} + h ^{2}}{h}) .

Equivalently

ϕ (h) = - 2 G γ arsinh (a / h)

. At large

h

it approaches

- G (2 aγ) / h

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