The Paradox that Induced Electric Field has Energy in Maxwell’s Theory of Classical Electromagnetic Field is Shown and Solved

Abstract

Those who have studied electromagnetic field theory know that the energy density of the magnetic field is proportional to the square of the magnetic field strength. The energy density of the electric field is proportional to the square of intensity of the electric field. It is assumed that the dimensions of devices such as the inductor are negligible compared with the wavelength of AC, so electromagnetic radiation can be ignore. It is no problem to calculate the energy of the magnetic field according to the above method. However, the electric field has two parts, one is the electrostatic field, and the other is the induced electric field, which is related to the time derivative of the magnetic vector potential. It is also clear that the electrostatic field has energy. However, it is not clear whether the induced electric field has electric energy. According to Maxwell’s equation, it refers to the radiation electromagnetic field equation including displacement current, the energy of the electric field naturally includes the energy of the induced electric field. However, the induced electric field is an electromagnetic induction phenomenon, and the energy of the magnetic field has been increased in this process. It seems that the energy of the induced electric field itself should not be calculated again. On the other hand, according to the electric and magnetic quasi-static electromagnetic field equation, the induced electromagnetic field has no energy. The author believes that the electric and magnetic quasi-static electromagnetic field equation is correct, and the induced magnetic field should not have electric field energy. The author believes that this contradiction is due to the fact that Maxwell’s equation (including displacement current term) is not suitable for the case of electric and magnetic quasi-static fields. As the textbook tells us, Maxwell’s equations are accurate equations, and magnetic quasi-static or electric and magnetic quasi-static electromagnetic field equations are approximate equations of Maxwell’s equations. The author thinks that the Maxwell equation obtained by adding the displacement current term can deduce the result of electromagnetic wave, but it is still a problem equation. The main problem is that the electric field and magnetic field obtained by Maxwell equation are not the seamless extension of the electromagnetic field under the original electric and magnetic quasi-static condition. That is to say, the electric field and magnetic field obtained according to Maxwell’s equation actually do not have the properties of the original electric field and magnetic field. In particular, the electric field energy, magnetic field energy and Poynting vector formed by such electric and magnetic fields are unreliable. In the electric and magnetic quasi-static condition, the most unreliable is the energy of the induced electric field. The induced electric field should not have energy. If the induced electric field has energy, we know that the energy is a quadratic function, so the energy of the induced electric field and the electrostatic electric field will have a cross mixing part, which is even more strange. The author thinks that the Poynting theorem is still correct under the quasi-static condition of electric and magnetic field, but the Poynting theorem derived from Maxwell equation (including displacement current) is not reliable.