Magnetic Vector Potential
Theory and Physics
Vector Potential$\mathbf{A}$
Professor, what is the magnetic vector potential?
The "potential" for magnetic flux density $\mathbf{B}$. Since $\mathbf{B}$ has zero divergence ($\nabla \cdot \mathbf{B} = 0$), it can always be expressed as the curl of some vector field $\mathbf{A}$:
$$ \mathbf{B} = \nabla \times \mathbf{A} $$
It corresponds to the electric potential $\phi$ in electrostatic fields.
Correspondence:
Electrostatic Field Magnetostatic Field $\mathbf{E} = -\nabla\phi$ $\mathbf{B} = \nabla \times \mathbf{A}$ $\phi$: Scalar $\mathbf{A}$: Vector (3 components) Poisson: $\nabla^2\phi = -\rho/\varepsilon$ $\nabla^2\mathbf{A} = -\mu\mathbf{J}$ (Coulomb gauge)
Gauge Condition
$\mathbf{A}$ is not uniquely determined (adding the gradient of any scalar field $\psi$ does not change $\mathbf{B}$). To ensure uniqueness, impose a gauge condition:
- Coulomb Gauge: $\nabla \cdot \mathbf{A} = 0$ — Standard for magnetostatics
- Lorenz Gauge: $\nabla \cdot \mathbf{A} + \mu\varepsilon \partial\phi/\partial t = 0$ — For dynamic problems
2D $A_z$
In 2D problems (uniform in the $z$ direction), $\mathbf{A} = A_z(x,y) \hat{z}$ becomes a scalar problem. Magnetic field lines are contours of $A_z$. This is the greatest advantage of 2D magnetic field FEM.
Summary
- $\mathbf{B} = \nabla \times \mathbf{A}$ — The basis of magnetic field FEM
- Uniqueness guaranteed by gauge condition — Coulomb gauge is standard
- Scalar problem for $A_z$ in 2D — Magnetic field lines = $A_z$ contours
Coffee Break Trivia Corner
Magnetic Vector Potential—Why a "Quantity Without Physical Substance" Supports FEM
The magnetic vector potential A (B=rot A) is sometimes called a "shadow quantity" of electric and magnetic fields; it cannot be measured directly but is indispensable for computation. The reason FEM magnetic field analysis uses A, not B, as the unknown is because automatically satisfying div B=0 (magnetic flux continuity) is more important than numerically handling rot B=μ₀J. In the A formulation, div B=0 is automatically guaranteed as rot(rot A)=0. In the 1970s, Oliver C. Zienkiewicz and others applied the A formulation to FEM, and it now forms the foundation of all major magnetic field FEM solvers.
Physical Meaning of Each Term
- Electric field term $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$: Faraday's law of electromagnetic induction. A time-varying magnetic flux density generates an electromotive force. 【Everyday Example】A bicycle dynamo (generator) produces voltage in a nearby coil by rotating a magnet—a direct application of this law stating that a time-varying magnetic field induces an electric field. Induction heating (IH) cookers also use the same principle, where high-frequency magnetic field changes induce eddy currents in the pot bottom, heating it via Joule heat.
- Magnetic field term $\nabla \times \mathbf{H} = \mathbf{J} + \partial \mathbf{D}/\partial t$: Ampère-Maxwell's law. Current and displacement current generate a magnetic field. 【Everyday Example】When current flows through a wire, a magnetic field is created around it—this is Ampère's law. Electromagnets operate on this principle, passing current through a coil to create a strong magnetic field. Smartphone speakers also apply this law: current → magnetic field → force on the diaphragm. At high frequencies (e.g., GHz-band antennas), the displacement current $\partial D/\partial t$ becomes non-negligible, describing electromagnetic wave radiation.
- Gauss's Law $\nabla \cdot \mathbf{D} = \rho_v$: Indicates that electric charge is the divergence source of electric flux. 【Everyday Example】Rubbing a plastic sheet against hair makes hair stand up due to static electricity—the charged sheet (electric charge) radiates electric field lines outward, exerting force on the light hair. Capacitor design calculates the electric field distribution between electrodes using this law. ESD (electrostatic discharge) countermeasures are also based on electric field analysis following Gauss's law.
- Magnetic flux conservation $\nabla \cdot \mathbf{B} = 0$: Expresses the non-existence of magnetic monopoles. 【Everyday Example】Even if you cut a bar magnet in half, you cannot create a magnet with only a N pole or only a S pole—they always exist as a N-S pair. This means magnetic field lines form "closed loops with no start or end points." In numerical analysis, to satisfy this condition, the formulation $\mathbf{B} = \nabla \times \mathbf{A}$ (vector potential) is used, automatically guaranteeing magnetic flux conservation.
Assumptions and Applicability Limits
- Linear material assumption: Permeability and permittivity are independent of magnetic/electric field strength (nonlinear B-H curve needed in saturation region)
- Quasi-static approximation (low frequency): Displacement current term can be ignored ($\omega \varepsilon \ll \sigma$). Common in eddy current analysis
- 2D assumption (cross-section analysis): Valid when current direction is uniform and end effects can be ignored
- Isotropy assumption: For anisotropic materials (e.g., silicon steel rolling direction), direction-specific property definitions are needed
- Non-applicable cases: Additional constitutive laws are needed for plasma (ionized gas), superconductors, nonlinear optical materials
Dimensional Analysis and Unit Systems
Variable SI Unit Notes / Conversion Memo Magnetic Flux Density $B$ T (Tesla) 1T = 1 Wb/m². Permanent magnets: 0.2–1.4T Magnetic Field Strength $H$ A/m Horizontal axis of B-H curve. Conversion from CGS Oe (Oersted): 1 Oe = 79.577 A/m Current Density $J$ A/m² Calculated from conductor cross-section and total current. Note non-uniform distribution due to skin effect Permeability $\mu$ H/m $\mu = \mu_0 \mu_r$. In vacuum $\mu_0 = 4\pi \times 10^{-7}$ H/m Electrical Conductivity $\sigma$ S/m Copper: approx. 5.96×10⁷ S/m. Decreases with temperature rise
Professor, what is the magnetic vector potential?
The "potential" for magnetic flux density $\mathbf{B}$. Since $\mathbf{B}$ has zero divergence ($\nabla \cdot \mathbf{B} = 0$), it can always be expressed as the curl of some vector field $\mathbf{A}$:
It corresponds to the electric potential $\phi$ in electrostatic fields.
Correspondence:
| Electrostatic Field | Magnetostatic Field |
|---|---|
| $\mathbf{E} = -\nabla\phi$ | $\mathbf{B} = \nabla \times \mathbf{A}$ |
| $\phi$: Scalar | $\mathbf{A}$: Vector (3 components) |
| Poisson: $\nabla^2\phi = -\rho/\varepsilon$ | $\nabla^2\mathbf{A} = -\mu\mathbf{J}$ (Coulomb gauge) |
$\mathbf{A}$ is not uniquely determined (adding the gradient of any scalar field $\psi$ does not change $\mathbf{B}$). To ensure uniqueness, impose a gauge condition:
- Coulomb Gauge: $\nabla \cdot \mathbf{A} = 0$ — Standard for magnetostatics
- Lorenz Gauge: $\nabla \cdot \mathbf{A} + \mu\varepsilon \partial\phi/\partial t = 0$ — For dynamic problems
2D $A_z$
In 2D problems (uniform in the $z$ direction), $\mathbf{A} = A_z(x,y) \hat{z}$ becomes a scalar problem. Magnetic field lines are contours of $A_z$. This is the greatest advantage of 2D magnetic field FEM.
Summary
- $\mathbf{B} = \nabla \times \mathbf{A}$ — The basis of magnetic field FEM
- Uniqueness guaranteed by gauge condition — Coulomb gauge is standard
- Scalar problem for $A_z$ in 2D — Magnetic field lines = $A_z$ contours
Magnetic Vector Potential—Why a "Quantity Without Physical Substance" Supports FEM
The magnetic vector potential A (B=rot A) is sometimes called a "shadow quantity" of electric and magnetic fields; it cannot be measured directly but is indispensable for computation. The reason FEM magnetic field analysis uses A, not B, as the unknown is because automatically satisfying div B=0 (magnetic flux continuity) is more important than numerically handling rot B=μ₀J. In the A formulation, div B=0 is automatically guaranteed as rot(rot A)=0. In the 1970s, Oliver C. Zienkiewicz and others applied the A formulation to FEM, and it now forms the foundation of all major magnetic field FEM solvers.
Physical Meaning of Each Term
- Electric field term $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$: Faraday's law of electromagnetic induction. A time-varying magnetic flux density generates an electromotive force. 【Everyday Example】A bicycle dynamo (generator) produces voltage in a nearby coil by rotating a magnet—a direct application of this law stating that a time-varying magnetic field induces an electric field. Induction heating (IH) cookers also use the same principle, where high-frequency magnetic field changes induce eddy currents in the pot bottom, heating it via Joule heat.
- Magnetic field term $\nabla \times \mathbf{H} = \mathbf{J} + \partial \mathbf{D}/\partial t$: Ampère-Maxwell's law. Current and displacement current generate a magnetic field. 【Everyday Example】When current flows through a wire, a magnetic field is created around it—this is Ampère's law. Electromagnets operate on this principle, passing current through a coil to create a strong magnetic field. Smartphone speakers also apply this law: current → magnetic field → force on the diaphragm. At high frequencies (e.g., GHz-band antennas), the displacement current $\partial D/\partial t$ becomes non-negligible, describing electromagnetic wave radiation.
- Gauss's Law $\nabla \cdot \mathbf{D} = \rho_v$: Indicates that electric charge is the divergence source of electric flux. 【Everyday Example】Rubbing a plastic sheet against hair makes hair stand up due to static electricity—the charged sheet (electric charge) radiates electric field lines outward, exerting force on the light hair. Capacitor design calculates the electric field distribution between electrodes using this law. ESD (electrostatic discharge) countermeasures are also based on electric field analysis following Gauss's law.
- Magnetic flux conservation $\nabla \cdot \mathbf{B} = 0$: Expresses the non-existence of magnetic monopoles. 【Everyday Example】Even if you cut a bar magnet in half, you cannot create a magnet with only a N pole or only a S pole—they always exist as a N-S pair. This means magnetic field lines form "closed loops with no start or end points." In numerical analysis, to satisfy this condition, the formulation $\mathbf{B} = \nabla \times \mathbf{A}$ (vector potential) is used, automatically guaranteeing magnetic flux conservation.
Assumptions and Applicability Limits
- Linear material assumption: Permeability and permittivity are independent of magnetic/electric field strength (nonlinear B-H curve needed in saturation region)
- Quasi-static approximation (low frequency): Displacement current term can be ignored ($\omega \varepsilon \ll \sigma$). Common in eddy current analysis
- 2D assumption (cross-section analysis): Valid when current direction is uniform and end effects can be ignored
- Isotropy assumption: For anisotropic materials (e.g., silicon steel rolling direction), direction-specific property definitions are needed
- Non-applicable cases: Additional constitutive laws are needed for plasma (ionized gas), superconductors, nonlinear optical materials
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Magnetic Flux Density $B$ | T (Tesla) | 1T = 1 Wb/m². Permanent magnets: 0.2–1.4T |
| Magnetic Field Strength $H$ | A/m | Horizontal axis of B-H curve. Conversion from CGS Oe (Oersted): 1 Oe = 79.577 A/m |
| Current Density $J$ | A/m² | Calculated from conductor cross-section and total current. Note non-uniform distribution due to skin effect |
| Permeability $\mu$ | H/m | $\mu = \mu_0 \mu_r$. In vacuum $\mu_0 = 4\pi \times 10^{-7}$ H/m |
| Electrical Conductivity $\sigma$ | S/m | Copper: approx. 5.96×10⁷ S/m. Decreases with temperature rise |
Numerical Methods and Implementation
FEM Formulation of the A-Method
$\mathbf{N}_i$: Shape function for edge elements (Nédélec elements).
2D Formulation
In 2D, discretize $A_z$ using nodal elements:
Exactly the same form as the Poisson equation for electrostatics. $\varepsilon \to \nu$, $\phi \to A_z$, $\rho \to J_z$.
Summary
- 3D: Edge elements (Nédélec) — Naturally handles gauge conditions
- 2D: Nodal elements for $A_z$ — Same form as Poisson equation
Gauge Fixing—Numerical Techniques to Resolve "Non-Uniqueness" in the AV Formulation
Magnetic vector poten...
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