Advanced Electromagnetic and Mathematic Concepts

[Michael Lin]

Hi All,

From electromagnetic theories, with the Lorentz gauge condition for the magnetic vector potential, I get the following wave equation:
1/csquared * d2A/dt2 + del2 A= u0 j.
in some literatures, they ignored the d2A/dt2 term and I don't know why they can do that. Is it becasue they assumed some quasi-stationary condition on the E field created by the magnetic field?
This simplication leads to a big simplification in which they can solve it as a Poisson Equation. I just want to know why they can ignore that term.

Thanks,
Micahel

 

[Physics Monkey]

Yes, it is typically called the quasistatic approximation. You keep the Faraday term $$\partial B/ \partial t$$ in Maxwell's equations, but drop the Maxwell term $$\partial E/ \partial t$$. This is an approximation which is valid in some limited circumstances where the displacement current (Maxwell term) is small compared to the real current. A typical scenario might be a wire loop in a magnetic field and the like.

More properly, by leaving out the Maxwell term you ignore the possibility of electromagnetic waves. To justify this approximation, the frequencies of interest in your system must correspond to electromagnetic waves of wavelength much larger than your system of interest. In other words, everything should be slowly varying or "quasistatic".

 

[astrokreng]

Hello Michael,

Thank you for bringing up this interesting topic. The wave equation you have mentioned is a fundamental equation in electromagnetic theory and is derived from Maxwell's equations. The d2A/dt2 term represents the time-varying component of the magnetic vector potential. In some cases, such as when dealing with steady-state or quasi-static conditions, this term can be neglected without significantly affecting the accuracy of the solution. This is because the time-varying component is much smaller compared to the static component of the magnetic vector potential. Therefore, for simplicity and ease of solving, some researchers may choose to ignore this term and solve the simplified Poisson's equation. However, it is important to note that neglecting this term may not always be appropriate, especially when dealing with rapidly changing electromagnetic fields or high-frequency phenomena. In those cases, the full wave equation must be considered for accurate results. I hope this helps to clarify the reason behind neglecting the d2A/dt2 term in some cases.