# Where does this form of the induction law come from?

I am currently studying a eddy current disk break and came across this promising paper (PDF, princeton.edu) from 1942.

However, in the first formula a suspicious $$4\pi$$ appears before the current $$U(x,y),$$ stating

Let the eddy currents be confined to a finite region of the sheet which may or may not extend to infinity, and let us define the stream function $$U(x,y)$$ at any point in the sheet to be the current flowing through any cross section of the sheet extending from $$P$$ to its edge. The line integral of $$\mathbf B$$ or $$\mathbf H$$ over the closed path that bounds this section equals $$4\pi U.$$ From symmetry the contribution from the upper and lower halves of the path is the same so we may write \begin{align} 4\pi U = \oint \mathbf B\cdot \mathbf{ds} &= \pm 2 \int_x^\infty B_x~dx \\ &=\pm 2 \int_y^\infty B_y~dy\end{align} where the choice of sign depends on the side of the sheet chosen for the integration.

How can this be explained?

• Can you restate the formula here and provide context? The link might not work for future visitors. – The Photon Mar 19 '19 at 21:01
• @ThePhoton added. – CR Drost Mar 19 '19 at 21:28
• I retract my earlier comment: while the equation $\nabla \times E = -\dot B' - \dot B$ suggests from its lack of $c^{-1}$ that these are not CGS units, the author explicitly mentions on the last page gauss and dyne-cm units, suggesting a preference for such Gaussian units. – CR Drost Mar 19 '19 at 21:58