Total flux through a spherical surface enclosing an off-centered source of uniformly distributed lines
Consider a point source \(Q\) in space from which \(N\) number of lines emanate uniformly in all directions. This source is enclosed in a spherical shell of radius \(R\) and is placed a distance \(d\) from the center of the sphere. Figure 1 (a) illustrates this arrangement. Figure 1 Let the symbol \(S\) represent the surface of the spherical shell. The flux through an infinitesimal area of the surface of the sphere is given by \[\vec{\rho}(\vec{r}) \cdot \vec{dA} \] where \(\vec{\rho}(\vec{r})\) is the density of line at a point \(\vec{r}\) on \(S\) and \(\vec{dA}\) is the infinitesimal area vector at \(\vec{r}\) on \(S\). It can be shown that the total outward flux of lines through the sphere is \[ \oint_{S} \vec{\rho}(\vec{r}) \cdot \vec{dA} = N\] Figure 1(b) illustrates the geometry of the problem, manner of placement of the coordinate system and various symbols used to represent variables in the problem. Note that the coordinate system has been chosen such that bo