# Current in a supermagnetic loop after rotating through a magnetic field

A problem was given as follows:

A superconducting loop of radius R has self inductance L. A uniform and constant magnetic field B is applied perpendicular to the plane of the loop. Initially, current in this loop is zero. The loop is rotated by $$180^\circ$$. The current in the loop after rotation is equal to:

Now this problem was solved by putting $$Li = 2\pi R^2B$$ why essentially uses $$L\frac{di}{dt} = \frac{d\phi}{dt}$$ However I also read recently that a superconducter doesn't allow any magnetic flux to enter or escape and hence according to me even on changing the configuration of the loop there should not be any effective change in magnetic flux through the "superconducting" loop. Then shouldn't the final current be zero aswell? or is there something I am doing wrong here?

• The total flux thjrough the loop is the sum of the external flux and the flux $Li$ due to the current. The begining and end flux are both zero. What, then, is $i$? Jul 14 at 13:34
• @mikestone total is $0$ but since external is changing so is $Li$ right? Jul 14 at 19:01
• I don't understand your question "is $Li$ right." You ned to know whether the current was zero before or after the field $B$ was applied, and the problem as stated is quite ambiguous about this. Jul 14 at 19:08
• It states "initally the current was zero in the loop" and the field already exists. It's just that the loop is then rotated in that field and then current is asked which is possibly produced due to change in flux Jul 14 at 19:25
• If you read it that way, then the initial flux was $\Phi= B\pi R^2$ and the final flux would be the sum $2\Phi- iL$. This must equal $\Phi$ because the flux can't change. There is a "2" because the flux due to the external field has changed from $+\Phi$ to $-\Phi$. Jul 14 at 19:35

Ideal conductors have zero resistivity meaning that the total flux through one remains constant, or $$\Phi = \text{const.}$$ Indeed, this can be proved through Kirchoff's loop rule $$\frac{\text{d}\Phi}{\text{d}t} = -RI = 0$$. Physically, this means that whenever the flux is increased or decreased, currents within the conductor will change to reduce the flux immediately. This also means that the flux can be nonzero. Superconductors are ideal conductors, but with the property that the magnetic fields stored within them are zero. Thus, no matter what initial conditions occur, all external magnetic fields will be expelled from an ideal conductor once it transitions into a superconducter. This is also collectively known as the Meissner effect.