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

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- Solution c

Solution

For an axisymmetric azimuthal vector field, the curl in spherical coordinates has components

(\nabla \times A)_{r} = \frac{1}{r s i n θ} \partial_{θ} (sin θ A_{ϕ}), (\nabla \times A)_{θ} = - \frac{1}{r} \partial_{r} (r A_{ϕ}), (\nabla \times A)_{ϕ} = 0.

The angular expressions are

sin θ A_{ϕ}^{(+)} = (1 - cos θ) / r

and

sin θ A_{ϕ}^{(-)} = - (1 + cos θ) / r

, each with

θ

sin θ / r

. Also

r A_{ϕ}

is independent of

r

, so the second component vanishes. Consequently, on each potential's regular patch,

\nabla \times A^{(+)} = \nabla \times A^{(-)} = \frac{e _{r}}{r ^{2}} = B .

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