String-tension interpretation of BIon energy ($\Delta E=4\pi T_{3}c(X_{\mathrm{cut}}-X_{\infty })$)

For $X-X_{\infty }=c/r$, subtracting the planar D3-brane energy leaves $T_3\int |\nabla X{|}^2{d}^{3}x=4\pi T_3{c}^2/r_0$ outside a cutoff. Since $X_{\mathrm{cut}}-X_{\infty }=c/r_0$, its coefficient is the carried string tension. Electric charge quantization gives $c=\pi g_s|N|{\alpha }^{\prime }$ and coefficient $|N|T_{F1}$; magnetic charge gives $c=\pi |N|{\alpha }^{\prime }$ and coefficient $|N|T_{D1}$.