Past exam of the physics B course of the University of Cambridge / 2018 / Paper 1 / Section B / B7

The field law. In the Biot-Savart law, $d\mathbf{B}$ is the contribution to the magnetic flux density at the observation point from a directed wire element $d\mathbf{\ell }$ carrying steady conventional electric current $I$. The vector $\mathbf{R}$ points from that element to the observation point, $R=|\mathbf{R}|$, and $\hat{\mathbf{R}}=\mathbf{R}/R$. The constant ${\mu }_0$ is the vacuum permeability; the cross product sets the direction by the right-hand rule. Integrating the elements gives the total field, in teslas.

The moment of the shown loop. The arrows in the original PDF run counterclockwise viewed from $+z$. Thus the magnetic dipole moment isIts units are $\mathrm{A}{\mathrm{m}}^2$. Reversing the current reverses both this moment and the field.

Use source coordinates $(x^{\prime },y^{\prime },0)$ and observation point $(0,0,z)$. The axial numerator in the Biot-Savart law isFor $|z|$ much larger than the loop dimensions, replace the denominator $[z^2+x^{\prime 2}+y^{\prime 2}{]}^{3/2}$ by $|z{|}^3$ to leading order. Transverse components cancel between opposite sides. The loop integral is twice its oriented area, soThereforeIt points along $\mathbf{m}$ on both halves of the dipole axis, not along the signed radial unit vector.

At $(x,0,0)$ the field is perpendicular to the loop plane. Its Biot-Savart numerator isThe leading denominator variation must be kept:The integral of $-xd{y}^{\prime }$ is zero, but its product with $3x^{\prime }/x$ is not. Keeping all terms of the leading nonzero order givesFor the counterclockwise rectangle, $\oint x^{\prime }d{y}^{\prime }=\Delta x\Delta y$ and $\oint y^{\prime }d{x}^{\prime }=-\Delta x\Delta y$. HenceThis transverse field is opposite to the magnetic dipole moment. Replacing every denominator by a constant before examining the term proportional to $x$ would give the wrong answer.

Why the shape is immaterial far away. Partition an arbitrary small planar loop into small rectangular loops with the same oriented current. Currents on internal shared edges cancel, leaving only the original boundary. Their magnetic dipole moments add to $\mathbf{m}=I\mathcal{A}\mathbf{n}$, with $\mathcal{A}$ the enclosed area. Far enough away, the leading fields add as dipoles at the same position. The general magnetic dipole field isHigher spatial multipoles contain shape-dependent corrections. The axial and transverse results above are the special directions of this expression.

The aligned terrestrial model. Take the moment to point toward geographic south. At northern latitude $\lambda$, the magnetic dipole inclination versus latitude gives local northward and downward componentsUsing the specified $\lambda =53^{\circ }$ and $B_{\mathrm{eq}}=35\mu \mathrm{T}$,The direction is geographic north and $69.4^{\circ }$ below horizontal, with no eastward component in this ideal model. These values use the latitude printed in the paper.