Students encountering time-varying electromagnetic fields for the first time have usually been exposed to electrostatics and magnetostatics already. These disciplines exhibit many similarities as summarized in Table 8.1. The principles of time-varying electromagnetics presented in this table are all formally introduced in other sections; the sole purpose of this table is to point out the differences. We can summarize the differences as follows:
Maxwell’s Equations in the general (time-varying) case include extra terms that do not appear in the equations describing electrostatics and magnetostatics. These terms involve time derivatives of fields and describe coupling between electric and magnetic fields.
| Electrostatics / | Time-Varying | |
| Magnetostatics | (Dynamic) | |
| Electric & magnetic | independent | possibly coupled |
| fields are... | ||
| Maxwell’s Eqns. | \(\oint_{\mathcal S}{{\bf D}\cdot d{\bf s}} = Q_{encl}\) | \(\oint_{\mathcal S}{{\bf D}\cdot d{\bf s}} = Q_{encl}\) |
| (integral) | \(\oint_{\mathcal C}{{\bf E}\cdot d{\bf l}} = 0\) | \(\oint_{\mathcal C}{{\bf E}\cdot d{\bf l}} = {\color{blue}-\frac{\partial}{\partial t}\int_{\mathcal S}{\bf B}\cdot d{\bf s}}\) |
| \(\oint_{\mathcal S}{{\bf B}\cdot d{\bf s}} = 0\) | \(\oint_{\mathcal S}{{\bf B}\cdot d{\bf s}} = 0\) | |
| \(\oint_{\mathcal C}{{\bf H}\cdot d{\bf s}} = I_{encl}\) | \(\oint_{\mathcal C}{{\bf H}\cdot d{\bf l}} = I_{encl} {\color{blue} + \int_{\mathcal S}\frac{\partial}{\partial t}{\bf D}\cdot d{\bf s}}\) | |
| Maxwell’s Eqns. | \(\nabla\cdot{\bf D}=\rho_v\) | \(\nabla\cdot{\bf D}=\rho_v\) |
| (differential) | \(\nabla\times{\bf E}=0\) | \(\nabla\times{\bf E}={\color{blue}-\frac{\partial}{\partial t}{\bf B}}\) |
| \(\nabla\cdot{\bf B}=0\) | \(\nabla\cdot{\bf B}=0\) | |
| \(\nabla\times{\bf H}={\bf J}\) | \(\nabla\times{\bf H}={\bf J} {\color{blue}+ \frac{\partial}{\partial t}{\bf D}}\) |
The coupling between electric and magnetic fields in the time-varying case has one profound consequence in particular. It becomes possible for fields to continue to exist even after their sources – i.e., charges and currents – are turned off. What kind of field can continue to exist in the absence of a source? Such a field is commonly called a wave. Examples of waves include signals in transmission lines and signals propagating away from an antenna.
Additional Reading:
“Maxwell’s Equations” on Wikipedia.