James Clerk Maxwell

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# 3.3.1 An expanding sphere of charge

First consider an expanding spherically-symmetric ball of positive charge. This is not an implausible state of affairs. If the charges in the distribution are not held in place, their mutual repulsion leads to a spherically-symmetric expansion and a spherically-symmetric outward flow of current. Any spherically-symmetric distribution of current is magnetically silent – that is, it produces no magnetic field. This is true both outside and inside the current distribution. We will now show that this rather surprising result is fully consistent with the Ampère–Maxwell law.

Figure 3 An expanding sphere of charge.

Using a spherical coordinate system with its origin at the centre of the charge distribution, we consider a point P with radial coordinate r (Figure 3). Because the charge distribution is spherically-symmetric, the electric field at P is

where Qin is the total charge inside a sphere of radius r (the dashed sphere in Figure 3). The outward current through the surface of the dashed sphere is equal to the rate of decrease of charge inside it, so we have

where S is the surface of the dashed sphere and J = Jr(r) er is the current density on the surface of this sphere. It follows that the Maxwell term at point P on S is

Combining this equation with the Ampère–Maxwell law (Equation 7.10), we finally obtain

which is consistent with B = 0. Note that the Maxwell term is essential for this cancellation. Ampère's law would wrongly imply that curl B0 at points where J0.

(Incidentally, div B is also equal to zero, by virtue of the no-monopole law. Although we shall not prove it, the fact that both curl B and div B vanish everywhere, and the natural assumption that B tends to zero at infinity, turns out to be sufficient to guarantee that B = 0 everywhere.)

SMT359_2