Dr. Chuanfu Chen

Dr. Chuanfu Chen

Ph.D. in Mathematical Physics

Atmospheric and climate scientist, Moscow State University. Research focuses on ionospheric modeling, GNSS signal propagation, and space weather effects. Experienced in GNSS/remote sensing applications and interdisciplinary collaborations.

Research Field

Maxwell’s Equations

1. Gauss’s Law for Electricity

  • Differential form \(\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}\) The divergence of the electric field equals the charge density over the permittivity of free space. Charges are the sources of the electric field.

  • Integral form \(\oint_{\partial V} \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{in}}{\varepsilon_0}\) The electric flux through a closed surface is proportional to the enclosed charge.

2. Gauss’s Law for Magnetism

  • Differential form \(\nabla \cdot \mathbf{B} = 0\) Magnetic fields have no divergence — there are no magnetic monopoles.

  • Integral form \(\oint_{\partial V} \mathbf{B} \cdot d\mathbf{A} = 0\) The net magnetic flux through a closed surface is always zero.

3. Faraday’s Law of Induction

  • Differential form \(\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}\) A changing magnetic field induces a circulating electric field.

  • Integral form \(\oint_{\partial S} \mathbf{E} \cdot d\mathbf{l} = - \frac{d}{dt} \int_S \mathbf{B} \cdot d\mathbf{A}\) The induced electromotive force equals the negative rate of change of magnetic flux.

4. Ampère–Maxwell Law

  • Differential form \(\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\) Currents and time-varying electric fields produce circulating magnetic fields.

  • Integral form \(\oint_{\partial S} \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{in} + \mu_0 \varepsilon_0 \frac{d}{dt} \int_S \mathbf{E} \cdot d\mathbf{A}\)

✨ Summary:

  • Charges produce electric fields (Gauss’s law for electricity).
  • No magnetic monopoles exist (Gauss’s law for magnetism).
  • Changing magnetic fields induce electric fields (Faraday’s law).
  • Currents and changing electric fields induce magnetic fields (Ampère–Maxwell law).

Together with the Lorentz force law \(\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}),\) they form the foundation of classical electromagnetism.