Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2008 / Paper 1 / 1 / b

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- Solution b

Solution

Along the circle with increasing

ϕ

, the line element is

d r = a sin α e_{ϕ} d ϕ

. Thus the two circulations are

\oint_{C} A^{(+)} \cdot d r = 2 π sin α tan (α /2), \oint_{C} A^{(-)} \cdot d r = - 2 π sin α cot (α /2) .

The half-angle identities simplify them to

\oint_{C} A^{(+)} \cdot d r = 2 π (1 - cos α), \oint_{C} A^{(-)} \cdot d r = - 2 π (1 + cos α) .

Here

0 < α < π

so both potentials are defined on the whole curve.

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