7.1: Comparison of Electrostatics and Magnetostatics
- Page ID
- 24281
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 |