7.1: Comparison of Electrostatics and Magnetostatics
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Students encountering magnetostatics for the first time have usually been exposed to electrostatics already. Electrostatics and magnetostatics exhibit many similarities. These are summarized in Table \(\PageIndex{1}\). The elements of magnetostatics presented in this table are all formally introduced in other sections; the sole purpose of this table is to point out the similarities. The technical term for these similarities is duality . Duality also exists between voltage and current in electrical circuit theory.
| electrostatics | magnetostatics | |
|---|---|---|
| Sources | static charge | steady current, magnetizable material |
| Field intensity | \({\bf E}\) (V/m) | \({\bf H}\) (A/m) |
| Flux density | \({\bf D}\) (C/m\(^2\)) | \({\bf B}\) (Wb/m\(^2\)=T) |
| Material relations | \({\bf D}=\epsilon{\bf E}\) | \({\bf B}=\mu{\bf H}\) |
| \({\bf J}=\sigma{\bf E}\) | ||
| Force on charge \(q\) | \({\bf F}=q{\bf E}\) | \({\bf F}=q{\bf v} \times {\bf B}\) |
|
Maxwell’s Eqs.
(integral) |
\(\oint_{\mathcal S}{\bf D}\cdot d{\bf s} = Q_{encl}\) | \(\oint_{\mathcal S}{\bf B}\cdot d{\bf s} = 0\) |
| \(\oint_{\mathcal C}{\bf E}\cdot d{\bf l} = 0\) | \(\oint_{\mathcal C}{\bf H}\cdot d{\bf l} = I_{encl}\) | |
|
Maxwell’s Eqs.
(differential) |
\(\nabla\cdot{\bf D}=\rho_v\) | \(\nabla\cdot{\bf B}=0\) |
| \(\nabla\times{\bf E}=0\) | \(\nabla\times{\bf H}={\bf J}\) | |
| Boundary Conditions | \(\hat{\bf n} \times \left[{\bf E}_1-{\bf E}_2\right] = 0\) | \(\hat{\bf n} \times \left[{\bf H}_1-{\bf H}_2\right] = {\bf J}_s\) |
| \(\hat{\bf n} \cdot \left[{\bf D}_1-{\bf D}_2\right] = \rho_s\) | \(\hat{\bf n} \cdot \left[{\bf B}_1-{\bf B}_2\right] = 0\) | |
| Energy storage | Capacitance | Inductance |
| Energy density | \(w_e=\frac{1}{2}\epsilon E^2\) | \(w_m=\frac{1}{2}\mu H^2\) |
| Energy dissipation | Resistance |