# Maximisation of Magnetic Flux

It is given in my text that a loop (made up of a flexible wire) of irregular shape carrying current located in a uniform external magnetic field changes it's shape to that of a circle in order to maximise the magnetic Flux. But I am having doubts on why actually the Flux needs to be maximized. Can anyone make it clear?

• Probably has something to do with Lenz's Law. Aug 4, 2017 at 9:56
• According to the isoperimetric inequality $A\le \frac{L^2}{4\pi}$ en.wikipedia.org/wiki/Isoperimetric_inequality where $A$ is the area and $L$ is the perimeter of the curve, equality iff the figure is a circle. Aug 4, 2017 at 13:44
• @uhoh Yes it does. Aug 4, 2017 at 14:02
• @uhoh Edited the question. Thanks for the suggestion btw :) Aug 4, 2017 at 14:53
• @uhoh Made the necessary change. Thanks for the suggestion :) Aug 4, 2017 at 14:58

Maximising the total flux minimises the mechanical energy of a loop carrying a given current, I, in a given externally-generated magnetic field, of which flux $\Phi_{ext}$ is linked with the loop.
It can be shown (for example by replacing the loop with a sheet of magnetic dipoles) that the loop energy is $$U=\ –I\ \Phi_{ext}.$$ To minimise the energy, forces and/or torques will act so that the loop tends to change shape and/or orientation to maximise $I\ \Phi_{ext}$. But the circuit-linked flux, $\Phi_{I}$ that the circuit itself produces passes through the circuit in a direction related to the sense of the current by a right hand rule. That was the same convention, applied to $\Phi_{ext}$, responsible for the minus sign in the first equation. So when $I\ \Phi_{ext}$ is maximised, $\Phi_{I}$ will re-inforce $\Phi_{ext}$, so the total flux will be maximised.