Past exam of the physics 2 paper of the University of Oxford / 2013 / 10

Let the wire lie along $z$, and observe at perpendicular distance $r$. In the Biot-Savart law, the azimuthal contributions from line elements add with magnitudeIndependently, cylindrical symmetry makes the field constant and azimuthal on a circular Amperian loop, so Ampère's law gives $2\pi rB={\mu }_0{I}_0$. Both yieldwith direction fixed by the right-hand rule.

For a thin current-carrying cylindrical shell, a coaxial Amperian loop encloses zero current inside and the entire current outside. HenceMore generally $B(r)=0$ for $r<R$ and $B(r)={\mu }_0{I}_0/(2\pi r)$ for $r>R$. The tangential field discontinuity is supported by the surface current.

Let the column radius be $b=\sqrt{A/\pi }$, and $J=I_0/A$ be its uniform axial current density. Treating it as concentric shells, only the current at radii below $r$ contributes to $B(r)$: a surrounding shell has zero interior field. ThusThe magnetic force density is radially inward, $\mathbf{J}\times \mathbf{B}=-({\mu }_0{J}^{2}r/2)\hat{\mathbf{r}}$. Radial mechanical equilibrium at a fixed height requiresConsequentlyThe pressure is higher at the centre: its outward force balances the inward magnetic pinch. Vertical hydrostatic variation is separate from this radial pressure difference.