7: Magnetostatics
- Page ID
- 3943
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Magnetostatics is the theory of the magnetic field in conditions in which its behavior is independent of electric fields, including
- The magnetic field associated with various spatial distributions of steady current
- The energy associated with the magnetic field
- Inductance, which is the ability of a structure to store energy in a magnetic field
The word ending “-statics” refers to the fact that these aspects of electromagnetic theory can be developed by assuming that the sources of the magnetic field are time-invariant; we might say that magnetostatics is the study of the magnetic field at DC. However, many aspects of magnetostatics are applicable at “AC” as well.
- 7.1: Comparison of Electrostatics and Magnetostatics
- Students encountering magnetostatics for the first time have usually been exposed to electrostatics already. Electrostatics and magnetostatics exhibit many similarities. The technical term for these similarities is duality. Duality also exists between voltage and current in electrical circuit theory.
- 7.2: Gauss’ Law for Magnetic Fields - Integral Form
- Gauss’ Law for Magnetic Fields states that the flux of the magnetic field through a closed surface is zero.
- 7.3: Gauss’ Law for Magnetism - Differential Form
- Just as Gauss’s Law for electrostatics has both integral and differential forms, so too does Gauss’ Law for Magnetic Fields. Here we are interested in the differential form for the same reason. Given a differential equation and the boundary conditions imposed by structure and materials, we may then solve for the magnetic field in very complicated scenarios.
- 7.4: Ampere’s Circuital Law (Magnetostatics) - Integral Form
- The integral form of Amperes’ Circuital Law (ACL) for magnetostatics relates the magnetic field along a closed path to the total current flowing through any surface bounded by that path.
- 7.5: Magnetic Field of an Infinitely-Long Straight Current-Bearing Wire
- In this section, we use the magnetostatic form of Ampere’s Circuital Law (ACL) to determine the magnetic field due to a steady current (units of A) in an infinitely-long straight wire. The wire is an electrically-conducting circular cylinder of fixed radius. Since the wire is a cylinder, the problem is easiest to work in cylindrical coordinates with the wire aligned along the z axis.
- 7.6: Magnetic Field Inside a Straight Coil
- In this section, we use the magnetostatic integral form of Ampere’s Circuital Law to determine the magnetic field inside a straight coil in response to a steady (i.e., DC) current. The result has a number of applications, including the analysis and design of inductors, solenoids (coils that are used as magnets, typically as part of an actuator), and as a building block and source of insight for more complex problems.
- 7.7: Magnetic Field of a Toroidal Coil
- Toroidal coils are commonly used to form inductors and transformers. The principal advantage of toroidal coils over straight coils in these applications is magnetic field containment – as we shall see in this section, the magnetic field outside of a toroidal coil can be made negligibly small. This reduces concern about interactions between this field and other fields and structures in the vicinity.
- 7.8: Magnetic Field of an Infinite Current Sheet
- We now consider the magnetic field due to an infinite sheet of current. The solution to this problem is useful as a building block and source of insight in more complex problems, as well as being a useful approximation to some practical problems involving current sheets of finite extent including, for example, microstrip transmission line and ground plane currents in printed circuit boards.
- 7.9: Ampere’s Law (Magnetostatics) - Differential Form
- In this section, we derive the differential form of Amperes’ Circuital Law. In some applications, this differential equation, combined with boundary conditions associated with discontinuities in structure and materials, can be used to solve for the magnetic field in arbitrarily complicated scenarios. A more direct reason for seeking out this differential equation is that we gain a little more insight into the relationship between current and the magnetic field.
- 7.10: Boundary Conditions on the Magnetic Flux Density (B)
- In homogeneous media, electromagnetic quantities vary smoothly and continuously. At an interface between dissimilar media, however, it is possible for electromagnetic quantities to be discontinuous. Continuities and discontinuities in fields can be described mathematically by boundary conditions and used to constrain solutions for fields away from these interfaces.
- 7.11: Boundary Conditions on the Magnetic Field Intensity (H)
- In homogeneous media, electromagnetic quantities vary smoothly and continuously. At a boundary between dissimilar media, however, it is possible for electromagnetic quantities to be discontinuous. Continuities and discontinuities in fields can be described mathematically by boundary conditions and used to constrain solutions for fields away from these boundaries. In this section, we derive boundary conditions on the magnetic field intensity H .
- 7.12: Inductance
- Current creates a magnetic field, which subsequently exerts force on other current-bearing structures. For example, the current in each winding of a coil exerts a force on every other winding of the coil. If the windings are fixed in place, then this force is unable to do work (i.e., move the windings), so instead the coil stores potential energy. This potential energy can be released by turning off the external source.
- 7.16: Magnetic Materials
- Magnetic fields arise in the presence of moving charge (i.e., current) and in the presence of certain materials. In this section, we address these “magnetic materials.”