9: Magnetic Potential
- Page ID
- 5472
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 9.1: Introduction to Magnetic Potential
- The force on a charge q in a magnetic field is qv×B . This force (the Lorentz force) does not depend only on the position of the particle, but also on its velocity (speed and direction). Thus the force is not conservative. This suggests that perhaps we cannot express the magnetic field merely as the gradient of a scalar potential function – and this is correct; we cannot.
- 9.2: The Magnetic Vector Potential
- This page introduces the magnetic vector potential, \(\textbf{A}\), whose curl defines the magnetic field \(\textbf{B}\). It emphasizes that \(\textbf{A}\) is not unique, akin to the scalar potential \(V\) in electrostatics, allowing scalar additions without changing \(\textbf{B}\). The page also covers the relationship between current elements and \(\textbf{A}\), discussed through the Biot-Savart law, providing expressions for \(\textbf{dA}\) derived from circuit elements.
- 9.3: Long, Straight, Current-carrying Conductor
- This page details the calculation of the magnetic vector potential near a long, straight, current-carrying wire using cylindrical coordinates. It derives the potential from a small wire segment, highlighting the difference between infinite and finite wire segments. It emphasizes the derivation and also shows that the magnetic field can be derived from the curl of the vector potential, confirming established results for the field around an infinite wire.
- 9.4: Long Solenoid
- This page covers the magnetic field and vector potential of an infinitely long solenoid with current. Inside the solenoid, the uniform magnetic field is \(\mu n I \hat{\textbf{z}}\), while it is zero outside. The vector potential \(\textbf{A}\) has a \(\phi\)-component, given by \(\textbf{A}=\frac{1}{2}\mu n r I \hat{\boldsymbol{\phi}}\) inside, and \(\textbf{A}=\frac{\mu na^2 I}{2r}\hat{\boldsymbol{\phi}}\) outside the solenoid.
- 9.5: Divergence
- Like the magnetic field itself, the lines of magnetic vector potential form closed loops (except in the case of the infinitely long straight conducting wire, in which case they are infinitely long straight lines). That is to say A has no sources or sinks, or, in other words, its divergence is everywhere zero.