Themes > Science > Physics > Electromagnetism > Magnetostatics > Magnetic Field > Charge Conservation

Conservation of charge is implied by Maxwell's equations. Taking the divergence of Ampère's law gives $\bigtriangledown .(\bigtriangledown \times {\bf B}) - \frac{1}{g c^2 } \frac{\p... ...l t}\bigtriangledown .{\bf E} = \frac{4\pi k}{g c^2 } \bigtriangledown .{\bf j}$. However $\bigtriangledown .(\bigtriangledown \times {\bf B})=0$ (do Problem 3.1) and using the electric Gauss law we have $-\frac{4\pi k}{g c^2 } \frac{\partial \rho }{\partial t}= \frac{4\pi k}{g c^2 } \bigtriangledown .{\bf j}$. We see that the constants cancel leaving  
 \begin{displaymath} \bigtriangledown .{\bf j}+ \frac{\partial \rho }{\partial t} = 0\end{displaymath} (129)

which is the continuity equation, or conservation of charge in differential form. Because the constants all cancelled, this equation is the same in all systems of units. This equation is a local conservation law in that it tells us how charge is conserved locally. That is if the charge density increases in some region locally (yielding a non-zero $\frac{\partial \rho }{\partial t}$), then this is caused by current flowing into the local region by the amount $\frac{\partial \rho }{\partial t} = - \bigtriangledown .{\bf j}$. If the charge decreases in a local region (a negative $- \frac{\partial \rho }{\partial t}$), then this is due to current flowing out of that region by the amount $- \frac{\partial \rho }{\partial t} = \bigtriangledown .{\bf j}$. The divergence here is positive corresponding to ${\bf j}$ spreading outwards (see Fig. 2.1).

Contrast this with the global conservation law obtained by integrating over the volume of the whole global universe. $\frac{\partial }{\partial t} \int \rho d\tau = \frac{\partial q }{\partial t}$ where q is the charge and $\int \bigtriangledown .{\bf j} d\tau = \oint {\bf j}. d {\bf A} $ according to Gauss' divergence theorem. We are integrating over the whole universe and so $\oint d{\bf A}$ covers the 'surface area' of the universe at infinity. But by the time we reach infinity all local currents will have died off to zero and so $\oint {\bf j}. d {\bf A} = 0 $ yielding  
 \begin{displaymath} \frac{\partial q }{\partial t} = 0\end{displaymath} (130)

which is the global conservation of charge law. It says that the total charge of the universe is constant.

Finally, let's go back and look at Ampère's law. The original form of Ampère's law didn't have the second term. It actually read $\bigtriangledown \times {\bf B} = \frac{4\pi k}{g c^2 } {\bf j}$. From our above discussion this would have lead to $ \bigtriangledown .{\bf j} = 0$. Thus the original form of Ampère's law violated charge conservation [Guidry p.74]. Maxwell added the term $- \frac{\partial {\bf E}}{\partial t}$, which is now called Maxwell's displacement current. Writing $ {\bf j}_D \equiv \frac{1}{4 \pi k} \frac{\partial {\bf E}}{\partial t}$, Ampère's law is $\bigtriangledown \times {\bf B} = \frac{4\pi k}{g c^2 } ({\bf j} + {\bf j}_D )$, or in integral form $\oint {\bf B}.d {\bf l} = \frac{4\pi k}{gc^2}(i + i_D)$ Maxwell's addition of the displacement current made Ampère's law agree with conservation of charge.


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