 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2008/k/k3/solution

For each case omit the dipole's own singular field when evaluating the surrounding complex velocity. With the convention

W^{'} = u - i v

this gives

d \overset{a}{ˉ} / d t = W_{image}^{'} (a)

, which is the conjugated equation requested in the source. In case (a), differentiating

\overset{m}{ˉ} z / (1 - \overset{a}{ˉ} z)

gives

\frac{d a ˉ}{d t} = \frac{m ˉ}{( 1 - ∣ a ∣ ^{2} ) ^{2}} .

For case (b), differentiate the three image terms:

\frac{d a ˉ}{d t} = - \frac{m ˉ}{( a - a ˉ ) ^{2}} + \frac{m ˉ}{( a + a ˉ ) ^{2}} + \frac{m}{4 a ^{2}} = \frac{m ˉ}{4 ( Im a ) ^{2}} + \frac{m ˉ}{4 ( Re a ) ^{2}} + \frac{m}{4 a ^{2}} .

For case (c), differentiation gives

\frac{d a ˉ}{d t} = \frac{i m ˉ}{( a + i a ˉ ) ^{2}} = \frac{m ˉ}{2 ( Re a + Im a ) ^{2}} .

The physical equation for

d a / d t

is the complex conjugate of the displayed equation. These expressions apply away from the walls, where the point-dipole model is nonsingular.

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