= Dipole line-integral identity
{title2=$\int_0^c\frac{\mathbf r-t\mathbf r_0}{|\mathbf r-t\mathbf r_0|^3}\,dt=\frac{c(P_c\mathbf r+r\mathbf P_c)}{rP_c(rP_c+\mathbf r\cdot\mathbf P_c)}$}
Here $r=|\mathbf r|$, $\mathbf P_c=\mathbf r-c\mathbf r_0$ and $P_c=|\mathbf P_c|$. The identity holds where the segment of source points $t\mathbf r_0$, $0\leq t\leq c$, avoids the observation point and the denominator is nonzero. Differentiating the right side with respect to $c$ gives the integrand and its value at $c=0$ is zero. Dotting with a <current dipole> moment evaluates a continuous line of dipole <electric potentials>.
Back to article page
