One runs into trouble when trying to apply Ampre's law to the magnetic field of a moving

One runs into trouble when trying to apply Ampère's law to the magnetic field of a moving charge. For instance, what is the current enclosed by an Ampèrian path that encircles the path of the particle? Because of the discrete nature of the charge, any sensible denition is zero most of the time and very large for a brief instant. The magnetic field does not have a similarly abrupt time dependence. One way out is to integrate over time. Show that the magnetic field \(\vec{B}\) produced by a charge \(q\) moving in a straight line at a constant velocity obeys \(\int_{-\infty}^{\infty}(\oint \vec{B} \cdot d \vec{\ell}) d t=\mu_{0} q\). Use a circular Ampèrian path in the plane perpendicular to and centred on the path of the charge.