The divergence of 2xa, + 2yay + 2za, is
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We want now to solve these equations mathematically in a general way, that is, without requiring any special symmetry or intuitive guessing. In electrostatics, we found that there was a straightforward procedure for finding the field when the positions of all electric charges are known: One simply works out the scalar potential ϕ by taking an integral over the charges—as in Eq. (4.25). Then if one wants the electric field, it is obtained from the derivatives of ϕ. We will now show that there is a corresponding procedure for finding the magnetic field B if we know the current density j of all moving charges.
In electrostatics we saw that (because the curl of E was always zero) it was possible to represent E as the gradient of a scalar field ϕ. Now the curl of B is not always zero, so it is not possible, in general, to represent it as a gradient. However, the divergence of B is always zero, and this means that we can always represent B as the curl of another vector field. For, as we saw in Section 2–7, the divergence of a curl is always zero. Thus we can always relate B to a field we will call A by