If the Maxwell-Ampère equation is written in the form, $$\begin{align}\nabla \times H = \dfrac{\partial D}{\partial t} + J\end{align}$$ that the both sides of the equation will be zero. So the Maxwell-Ampère equation should be unuseful outside the region of current. If the area does not approach zero when the curl of the magnetic field is calculated, the equation becomes meaningful. But that violates the definition of curl.

The questions:

Is the Maxwell-Ampère equation in differential form is unuseful outside the region of the current?

Is the integral form more general than the differential form? However, the integral form misses the information of the direction of the vectors.

  • $\begingroup$ It takes a lot of guts to question the usefulness of the Maxwell-Ampère equation, to put it mildly. $\endgroup$ – my2cts Sep 30 '18 at 8:54
  • $\begingroup$ I am sorry. I am poor at English skill. Could you give me an example? $\endgroup$ – IvanaGyro Oct 1 '18 at 2:05

Maxwell's equations apply at points in space (and time). I am unclear why you think the right hand side is zero in free space, unless you are including the the rate of change of displacement field in your definition of "current [density]", which would prohibit electromagnetic waves!

In any case, even if the RHS is zero, the fact that the curl of the magnetic field strength is zero is not meaningless. It indicates that the field is conservative and could therefore be expressed in terms of the gradient of a (magnetostatic) potential.

To answer your final question - neither form of Maxwell's equations is more fundamental, since one can be derived from the other via Stokes' and Gauss's theorems. The integral forms are more useful in problems with high degrees of symmetry, but the differential forms deal with the fields at a point and have more general applicability (e.g. in numerical work). For example is is not easy to show that a field is conservative using the integral forms.

  • $\begingroup$ Why we can derive that the field is conservative if the LHS is zero when the RHS is zero? $\endgroup$ – IvanaGyro Oct 1 '18 at 2:21
  • $\begingroup$ @IvenCJ7 if the curl of a field is zero (everywhere), then it is conservative. If the curl is zero in a particular region then it can be expressed as the gradient of a scalar potential, since the curl of a gradient is always zero. $\endgroup$ – Rob Jeffries Oct 1 '18 at 6:01

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