# Deriving Magnostatics equations from steady currents

• FG_313
In summary, the conversation discusses the relationship between Maxwell's equations and the magnetostatics equations. The main question is whether it is possible to derive the magnetostatics equations by assuming steady currents and using Maxwell's equations. The motivation for this question is that the condition for magnetostatics is steady currents. The conversation also mentions the Jefimenko equations and the use of retarded potentials for static sources. Finally, the conversation acknowledges the difficulty of applying the formulas to real cases.
FG_313
I've always heard that maxwell's equations contains essentially all of eletromagnetic theory. However, there's one thing I'm having trouble doing for myself: deriving the magnestatics equations from the maxwell's equations. Of course: it's clear that if you put ∂[t]E=∂[t]B=0 (partial derivative in relation to time), you get the magnostatic equations. What I'm wondering is: can we get that ∂[t]E=∂[t]B=0 by assuming steady currents (∂[t]ρ=0) and using the maxwell's equations? The motivation for this question is that the condition for being in magnostatics is steady currents, so the Maxwell's equations should reflect:
(Steady currents)⇒(Magnostatics equations). I believe I've proved the other direction of the implication, but not the one above, that is the most interesting for me: it would mean that, given the maxwell's equation, we can never have non-static situations coming from a "static" current distribution. Please contribute.

Well, you can solve the Maxwell equations for given sources ##(\rho,\vec{j})##, in terms of the socalled Jefimenko equations (althought I never understood why these equations are named after Jefimenko, because they are known much longer in terms of the retarded potentials, to my knowledge first written down by Ludvig Lorenz in the 1860ies, but that's a minor historical detail)

https://en.wikipedia.org/wiki/Jefimenko's_equations

Now consider stationary sources ##\partial_t \rho=0##, ##\partial_t \vec{j}=0## and see what you get!

Dale
Supporting the comments by @vanhees71 the retarded potentials are particularly easy to evaluate for static sources since the sources are the same at every retarded time, so you don't actually need to calculate the correct retarded time.

## 1. What are Magnostatics equations?

Magnostatics equations are a set of equations that describe the relationship between steady currents and their associated magnetic fields. They are derived from Maxwell's equations and are used to analyze the behavior of electromagnetic systems.

## 2. How are Magnostatics equations derived from steady currents?

Magnostatics equations are derived by applying Maxwell's equations to steady currents. This involves simplifying the original equations and making assumptions about the behavior of the currents, such as assuming that they are constant over time. The resulting equations describe the magnetic field produced by the steady currents.

## 3. What is the significance of deriving Magnostatics equations from steady currents?

Deriving Magnostatics equations from steady currents allows us to understand and predict the behavior of electromagnetic systems that involve steady currents. These equations are essential for studying the effects of magnetic fields on materials and for designing devices such as motors and generators.

## 4. Are there any limitations to using Magnostatics equations?

Yes, there are limitations to using Magnostatics equations. These equations are only applicable to systems with steady currents and cannot be used to analyze time-varying currents. Additionally, they make assumptions about the behavior of the currents and may not accurately describe complex systems.

## 5. Can Magnostatics equations be used to study the behavior of all types of electromagnetic systems?

No, Magnostatics equations are only applicable to systems with steady currents. They cannot be used to analyze systems with time-varying currents, such as those found in high-frequency circuits. In these cases, other equations, such as the time-dependent Maxwell's equations, must be used.

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