The surface integral of j over a surface S, followed by an integral over the time duration t1 to t2, gives the total amount of charge flowing through the surface in that time (t2 − t1):

$${\displaystyle q=\int _{t_{1}}^{t_{2}}\iint _{S}\mathbf {j} \cdot \mathbf {\hat {n}} \,{\rm {d}}A{\rm {d}}t.}{\displaystyle q=\int _{t_{1}}^{t_{2}}\iint _{S}\mathbf {j} \cdot \mathbf {\hat {n}} \,{\rm {d}}A{\rm {d}}t.}$$

I do not understand why this must be true, except if we are assuming this to be true. What I mean by that is, the continuity equation can be derived from Maxwell's equations. But let us say I do not have Maxwell's equations to use. That this equation must hold in the case of magnetostatics where there is no charge accumulation is not clear to me. Why? Because other than the heuristic explanation of how the flux of J through the surface is giving me charge going out of the surface, I have no real reason to believe this.

What is a rigorous proof that this equation is true?

  • $\begingroup$ I'm not sure what the question is. Is it how mathematically does this represent the idea that in order to leave a region of space, charge must pass through the boundary, or is it asking why this physical phenomenon is the case? $\endgroup$ – R. Romero Oct 21 '19 at 18:50
  • $\begingroup$ Charge conservation is an experimental fact. So any theory describing charge should mathematically imply charge conservation at some point of refinement. $\endgroup$ – my2cts Oct 21 '19 at 19:07

To derive the continuity equation the steps are:

  1. Clarify in your own mind what one is talking about, namely that if the amount of some stuff is changing in a given spatial region $R$, then it must be because stuff is flowing in or out through the boundary of $R$.

  2. The next step is to capture part 1 in mathematical terms. The changing amount of stuff is expressed by an integral over volume. The flow at the surface is expressed by a flux integrated over the surface. By setting these two equal one obtains an integral expression.

  3. One invokes the Gauss divergence theorem to convert the flux integral into a volume integral.

  4. One argues that the two integrals agree for any region of integration. It follows that the integrands must agree.

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  • $\begingroup$ Step 2 is exactly what my question is about. How do I know in the equation I mentioned in my question the total charge going out of a surface is equal to the integral of the current density flux? I understand step 1's process/idea, but how do I know the equation I mentioned is true? What is the proof? $\endgroup$ – childishsadbino Oct 21 '19 at 22:37
  • $\begingroup$ My question is simply this: What is the proof that the equation I have mentioned is true minus the statement "we are expressing charge conservation in mathematical terms" $\endgroup$ – childishsadbino Oct 21 '19 at 22:50
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    $\begingroup$ @childishsadbino If the quantity in question is not conserved then the two integrals need not be equal to one another. So their equality is not a purely mathematical relationship between integrals: it is a statement about a certain kind of flow. $\endgroup$ – Andrew Steane Oct 22 '19 at 8:32
  • $\begingroup$ I just had the biggest "Oh" moment, thank you so much! $\endgroup$ – childishsadbino Oct 22 '19 at 17:57

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