Understanding Ampere's Law, Gauss's Law, and the Continuity Equation

[touqra]

The Ampere's Law is $$\nabla \times B = \mu J$$ and Gauss's Law is $$\nabla \cdot E = \frac{1}{\epsilon} \rho$$

Since J is current density, is it right to say that, $$J = \frac{d}{dt} \rho$$ in general?
I am abit confused, since I know that a current four-vector, $$(\rho , J)$$ is similar to a spacetime four-vector $$(t, x)$$. But, x is not $$\frac{d}{dt} t$$

Also, does a non-zero J automatically implies a non-zero $$\rho$$ ?

 

[G01]

I think what you are looking for is the continuity equation in electrodynamics:

$$\vec{\nabla} \cdot \vec{J} = -\frac{\partial\rho}{\partial t}$$

In words, it states that a current must be caused by a change in the overall charge density of the system. So, a current implies a changing charge density.

 

[LURCH]

Sorry to butt in, but could you put that continuity equation into words a bit further? What does each symbol rerpesent? I just finished my associate degree and I'm taking at least one semester off, but I want to keep increasing my general knowledge while I'm not attending formal classes.

Thanks.

 

[mda]

In words the continuity equation means that positive divergence of current results in a negative rate of charge density, or (in integral form) the outward flux of current over a closed surface results in a reduction of charge contained within the surface.
As the OP says, conceptually current is movement of charge, so the continuity equation is fairly intuitive.

 

[blechman]

Since J is current density, is it right to say that, $$J = \frac{d}{dt} \rho$$ in general?
I am abit confused, since I know that a current four-vector, $$(\rho , J)$$ is similar to a spacetime four-vector $$(t, x)$$. But, x is not $$\frac{d}{dt} t$$

Also, does a non-zero J automatically implies a non-zero $$\rho$$ ?

In general:

$$\vec{J}=\rho\vec{v}$$

You can think of this as the "solution to the continuity equation" mentioned earlier. It also might explain your confusion about the current 4-vector.

So a nonzero current certainly implies nonzero charge density (how can you have a current without a charge?) but not vice versa (since a charge distribution at rest has no current).

 

[G01]

Sorry to butt in, but could you put that continuity equation into words a bit further? What does each symbol rerpesent? I just finished my associate degree and I'm taking at least one semester off, but I want to keep increasing my general knowledge while I'm not attending formal classes.

Thanks.

The continuity equation basically says that if you have some amount of current leaving some point, then the charge at that point must be decreasing. This makes sense since current is defined basically as movement of charge. If we have current leaving some point, then we have charge moving away from that point. If there is charge moving away from a point, then the charge at that point must be decreasing. The continuity equation is, in the end, another way to state conservation of charge.