Understanding Continuity Equation & Conservation Laws

[Pushoam]

TL;DR Summary

It is about why $\frac{ d\rho} {dt} = - \nabla \cdot \vec J$ is called continuity equation.

I understand that from local conservation of charge, we get eqn. 8.4. I don't get why it is called continuity eqn. What is continuous in it?

Conservation of momentum gives us equation, $\frac {d\vec p }{dt} = \vec F$. This equation is not called continuity equation. Can we get a continuity equation from every conservation law?
The images are taken from Griffith's Electrodynamics, 4ed.

 

[Dale]

I don't get why it is called continuity eqn. What is continuous in it?

Nomenclature like this doesn’t matter. It makes no difference why it is called the continuity equation. The important thing is what it says. I would not waste time asking why it is called that.

Can we get a continuity equation from every conservation law?

Not global conservation laws. The continuity equation applies for locally conserved quantities.

 

[Pushoam]

Thanks.
Could you please give me an example of something which is globally conserved, but not locally?

 

[Chestermiller]

Well, if charge is conserved locally, it is certainly conserved globally.

 

[anorlunda]

Thanks.
Could you please give me an example of something which is globally conserved, but not locally?

Energy. In "classical" circumstances energy is conserved. But on the scale of the universe, it is not necessarily conserved. Also at at the quantum mechanics level, we have time-energy uncertainty.

 

[Pushoam]

Thanks to all.

 

[Dale]

Thanks.
Could you please give me an example of something which is globally conserved, but not locally?

I think that the fundamental laws all involve locally conserved quantities, but you can easily write a useful Lagrangian where something that is locally conserved in the fundamental laws is only globally conserved in your Lagrangian. I think that energy and angular momentum are examples in classical orbital mechanics.

 

[Pushoam]

Thanks @Dale