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I'm reading Griffith's text, and he starts by defining Electrostatics as requiring the source charges don't move. I've seen a few slightly different definitions of electrostatics and magnetostatics. If you wanted to start from the full Maxwell equations in a vacuum, how would you precisely define Electrostatics and Magnetostatics? Would Electrostatics be the condition that $\frac{\partial\vec{B}}{\partial t}=\vec{0}\implies\nabla\cdot\vec{E}=\frac{\rho}{\epsilon_o},$ and $\nabla\times\vec{E}=\vec{0}?$

And would Magnetostatics be the condition that $\frac{\partial\vec{E}}{\partial t}=\vec{0}\implies \nabla\cdot\vec{B}=0,$ and $\nabla\times\vec{B}=\vec{0}?$

If so, how would you conclude from the electrostatic equations that the source charges don't move? I can see if you add in the requirement that $\frac{\partial\vec{E}}{\partial t}=\vec{0},$ but if you're only given $\frac{\partial\vec{B}}{\partial t}=\vec{0},$ how do you see this?

One of the other definitions for magnetostatics I've seen is $\frac{\partial\rho}{\partial t}=0.$ If magnetostatics is the condition that $\frac{\partial\vec{E}}{\partial t}=\vec{0},$ then can't you see $\frac{\partial\rho}{\partial t}=0.$ from $\frac{\partial\rho}{\partial t}=\frac{\partial}{\partial t}(\nabla\cdot\vec{E})=\nabla\cdot\frac{\partial\vec{E}}{\partial t}=0?$

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I guess different authors use different definitions. For me, it is that the E- and B-fields do not have time derivatives, hence curl free, conservative E-fields and B-fields that can depend only on steady currents.

The condition that the divergence of $\partial {\bf E}/\partial t = 0$ is not the same thing. The E-field could be time variable and have this still be true - e.g. in a transverse electromagnetic wave! Clearly that is not a magnetostatic situation either.

The curl of the B-field does not have to be zero in magnetostatics; steady currents are allowed, which obviously means you have to have (uniformly) moving charges. As ${\bf J} = \rho {\bf v}$, then $\partial {\bf J}/\partial t = 0$ implies only that ${\bf v}\partial \rho /\partial t + \rho \partial{\bf v}/\partial t = 0$. So it might be possible to arrange static magnetic fields by having a non-zero rate of change of charge density balanced by accelerating charges to somehow keep the current density constant! The continuity equation, $\nabla \cdot {\bf J} + \partial \rho/\partial t =0$, tells you that a time-varying charge density would require a current density divergence.

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Static electromagnetic fields implies: $$ \frac{\partial\mathbf E}{\partial t} = 0 \quad\mbox{ and }\quad \frac{\partial\mathbf B}{\partial t} = 0 $$

This means for electrostatics: $$ \nabla\cdot\mathbf E = \frac{\rho}{\epsilon_0}, \quad \nabla\times\mathbf E = 0 \quad $$

And for magnetostatics: $$ \nabla\cdot\mathbf B = 0, \quad \nabla\times\mathbf B = \mu_0\mathbf J $$

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Electrostatic and magnetostatic are specific cases of the general electromagnetism. Defining a special case does not require to know a law/model that rules the phenomena.

I don't need maxwell equations to define electrostatics or magnetostatics. I only need them if I want to know that my choice of special case is clever or useless. For instance, I can imagine magnetosquaretics where all currents are square waves, but it won't help me to solve Maxwell's equations.

Electrostatics = no moving charges, magnetostatics = no time-dependant currents. You can also consider that [field]statics just means that the [field] is not function of time. Then you may use the assumption to simplify your equations.

If your question is about the equivalence between static field and time-independant sources, you should look instead to the Biot et Savart's or Coulomb's laws.

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Electrostatics is the physics of current free charge distributions. Magnetostatics is the physics of stationary (time independent) current distributions.

Usually magnetostatics is defined as the physics of stationary and "divergence free" current distributions, however, a zero divergence is a superfluous current condition that is not satisfied in case of many magnetostatics experiments in the past and present. Real magnetostatics experiments of 'divergence free' currents are almost impossible to do, such experiments do not include batteries. The reason to include this unnatural extra 'magnetostatics condition' is to obscure the fact that Grassmann's force law (that derives from the more general Lorentz force law) does not satisfy Newton's third principle of motion, in case a stationary current distribution is not divergence free. In other words, Maxwell's classical electrodynamics (that includes the Lorentz force law) is inconsistent with classical mechanics, in case the current distribution is stationary and not free of divergence. The most common definition of magnetostatics is absolutely misleading and unphysical from a practical point of view.

So Magnetostatics is not the same as time-independent magnetic AND time-independent electric fields. Only the magnetic field should be static, and the electric field (and the charge distribution) may vary in time. Hence, the Maxwell Ampere law for magnetostatics must include a displacement current term:

$$\nabla \times \vec B - \epsilon \mu \frac{\partial \vec E}{\partial t} = \mu \vec J$$ where $$\vec E = -\nabla \Phi$$ since $$\frac{\partial \vec A}{\partial t} = \vec 0$$

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