2017-15.md

course	course_year	question_number	tags	title	year
Electromagnetism	IB	15	IB 2017 Electromagnetism	Paper 4 , Section I, $7 \mathrm{C}$	2017

A thin wire, in the form of a closed curve $C$, carries a constant current $I$. Using either the Biot-Savart law or the magnetic vector potential, show that the magnetic field far from the loop is of the approximate form

$$\mathbf{B}(\mathbf{r}) \approx \frac{\mu_{0}}{4 \pi}\left[\frac{3(\mathbf{m} \cdot \mathbf{r}) \mathbf{r}-\mathbf{m}|\mathbf{r}|^{2}}{|\mathbf{r}|^{5}}\right]$$

where $\mathbf{m}$ is the magnetic dipole moment of the loop. Derive an expression for $\mathbf{m}$ in terms of $I$ and the vector area spanned by the curve $C$.