/sci/ - Science & Math

>>11478429
Because the magnetic flux through a solenoid [math]p:S \rightarrow S^1[/math] is proportional to the winding number and the mutual inductance [math]M_{12}[/math] between two slenoids [math]S_2,S_2[/math] is proportional to their linking number. Let [math]L[/math] be a line bundle on [math]S^1[/math] with the [math]U(1)[/math]-connection [math]A[/math], such that magnetic flux through it reads [math]\Phi = \oint_{S^1}A[/math]. By Chern-Weil theory the curvature [math]B= dA[/math] lies in the integral cohomology class [math]H^2(D^2,\mathbb{Z})[/math] where [math]\partial D^2 = S^1[/math], hence [math]\Phi[/math] through the solenoid actually computes the first Chern number [math]c_1(p^*L) = nc_1(L) \in n\mathbb{Z}[/math] where [math]n = \operatorname{deg}p[/math] is the degree of the cover [math]p: S\rightarrow S^1[/math].
Now given [math]L_{1,2}[/math] line bundles on [math]S^1_{1,2}[/math] around the solenoids [math]p_{1,2}:S_{1,2}\rightarrow S^1_{1,2}[/math], Poincare duality [math]H^1(D,\mathbb{Z}) \cong H_1(D,\mathbb{Z})[/math] allows us to define the non-degenerate intersection pairing [math]\langle \operatorname{PD}A_1,A_2\rangle = \operatorname{lk}_{12} \in \mathbb{Q}[/math], and we are able to compute [math]M_{12} \propto \langle \operatorname{PD}p_1^*A_1,p_2^*A_2\rangle = n_1n_2\operatorname{lk}_{12}[/math]. This shows why the fluxes, and hence the EMFs, are quantized in terms of the number of turns on the solenoid.
>>11478445
Yep my campus is closed too, but I'm able to hold online meetings with my supervisor through Zoom.