Ampère's Law
Theory and Physics
Ampère's Law
Professor, how is Ampère's Law different from the Biot-Savart Law?
They express the same physics in different forms. The line integral of the magnetic field along a closed path is equal to the total current enclosed.
$$ \oint_C \mathbf{H} \cdot d\mathbf{l} = I_{enc} $$
Differential form (static magnetic field version of Maxwell's 4th equation):
$$ \nabla \times \mathbf{H} = \mathbf{J} $$
Gauss's Law ($\nabla \cdot \mathbf{D} = \rho$) is the electric field counterpart of Ampère's Law.
For problems with high symmetry, can we solve directly using Ampère's Law?
Current Distribution Ampèrian Loop $H$ Infinite straight wire Concentric circle $I/(2\pi r)$ Inside solenoid Rectangle $nI$ Toroidal coil Central circle $NI/(2\pi r)$
Summary
- $\nabla \times \mathbf{H} = \mathbf{J}$ — Maxwell's 4th equation (static magnetic field)
- Governing equation for FEM static magnetic field analysis — Used in combination with vector potential
Coffee Break Trivia
Ampère himself was a "theorist who didn't experiment"
André-Marie Ampère, the discoverer of Ampère's Law, was a rare type of physicist who hardly conducted experiments himself. Shortly after Ørsted reported in 1820 that a magnetic field is generated around an electric current, Ampère constructed and formulated the theory in just a few weeks. Not from experiment to theory, but grasping the essence mathematically immediately upon hearing others' experimental reports—this super-fast abstraction ability gave birth to the concise single line: ∮H·dl = I_enc.
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) generates voltage in a nearby coil by rotating a magnet—a direct application of this law that a time-varying magnetic field induces an electric field. Induction cooking heaters 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 law. Current and displacement current generate a magnetic field. 【Everyday example】When current flows through a wire, a magnetic field is generated 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 $\partial D/\partial t$ cannot be ignored, describing electromagnetic wave radiation.
- Gauss's Law $\nabla \cdot \mathbf{D} = \rho_v$: Indicates that charge is the divergence source of electric flux. 【Everyday example】Rubbing hair with a plastic sheet causes static electricity and makes hair stand up—electric field lines radiate outward from the charged sheet (charge), exerting force on the light hair. In capacitor design, the electric field distribution between electrodes is calculated 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 an N pole or only an S pole—N and S poles always exist as a pair. This means magnetic field lines form "closed loops with no start or end points." In numerical analysis, to satisfy this condition, the formulation using vector potential $\mathbf{B} = \nabla \times \mathbf{A}$ is used, automatically guaranteeing magnetic flux conservation.
Assumptions and Applicability Limits
- Linear material assumption: Permeability and permittivity do not depend on 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): Effective when current direction is uniform and end effects can be ignored
- Isotropy assumption: For anisotropic materials (e.g., rolling direction of silicon steel sheets), 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 to 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-sectional area 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, how is Ampère's Law different from the Biot-Savart Law?
They express the same physics in different forms. The line integral of the magnetic field along a closed path is equal to the total current enclosed.
Differential form (static magnetic field version of Maxwell's 4th equation):
Gauss's Law ($\nabla \cdot \mathbf{D} = \rho$) is the electric field counterpart of Ampère's Law.
For problems with high symmetry, can we solve directly using Ampère's Law?
| Current Distribution | Ampèrian Loop | $H$ |
|---|---|---|
| Infinite straight wire | Concentric circle | $I/(2\pi r)$ |
| Inside solenoid | Rectangle | $nI$ |
| Toroidal coil | Central circle | $NI/(2\pi r)$ |
- $\nabla \times \mathbf{H} = \mathbf{J}$ — Maxwell's 4th equation (static magnetic field)
- Governing equation for FEM static magnetic field analysis — Used in combination with vector potential
Ampère himself was a "theorist who didn't experiment"
André-Marie Ampère, the discoverer of Ampère's Law, was a rare type of physicist who hardly conducted experiments himself. Shortly after Ørsted reported in 1820 that a magnetic field is generated around an electric current, Ampère constructed and formulated the theory in just a few weeks. Not from experiment to theory, but grasping the essence mathematically immediately upon hearing others' experimental reports—this super-fast abstraction ability gave birth to the concise single line: ∮H·dl = I_enc.
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) generates voltage in a nearby coil by rotating a magnet—a direct application of this law that a time-varying magnetic field induces an electric field. Induction cooking heaters 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 law. Current and displacement current generate a magnetic field. 【Everyday example】When current flows through a wire, a magnetic field is generated 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 $\partial D/\partial t$ cannot be ignored, describing electromagnetic wave radiation.
- Gauss's Law $\nabla \cdot \mathbf{D} = \rho_v$: Indicates that charge is the divergence source of electric flux. 【Everyday example】Rubbing hair with a plastic sheet causes static electricity and makes hair stand up—electric field lines radiate outward from the charged sheet (charge), exerting force on the light hair. In capacitor design, the electric field distribution between electrodes is calculated 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 an N pole or only an S pole—N and S poles always exist as a pair. This means magnetic field lines form "closed loops with no start or end points." In numerical analysis, to satisfy this condition, the formulation using vector potential $\mathbf{B} = \nabla \times \mathbf{A}$ is used, automatically guaranteeing magnetic flux conservation.
Assumptions and Applicability Limits
- Linear material assumption: Permeability and permittivity do not depend on 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): Effective when current direction is uniform and end effects can be ignored
- Isotropy assumption: For anisotropic materials (e.g., rolling direction of silicon steel sheets), 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 to 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-sectional area 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
Substitute $\mathbf{B} = \mu\mathbf{H}$ and $\mathbf{B} = \nabla \times \mathbf{A}$ into $\nabla \times \mathbf{H} = \mathbf{J}$:
$\nu = 1/\mu$: Magnetic reluctivity. For nonlinear materials (iron cores), $\nu = \nu(|\mathbf{B}|)$ and Newton-Raphson iteration is required.
2D vs. 3D
| Dimension | Unknowns | Element | Computational Cost |
|---|---|---|---|
| 2D | $A_z$ (scalar) | Nodal element | Small |
| 2D axisymmetric | $rA_\phi$ (scalar) | Nodal element | Small |
| 3D | $\mathbf{A}$ (vector) | Edge element | Large |
Summary
- 2D is a scalar problem — Good computational efficiency
- 3D uses edge elements — Gauge condition handling is required
- Nonlinear materials → Newton-Raphson iteration
Ampèrian integration path—"Where to cut" can change computational cost by 100 times
When numerically implementing Ampère's Law, the computational cost changes drastically depending on how the Ampèrian loop is set. For circular or rectangular loops utilizing symmetry, the closed integral can be solved analytically, but for coils with complex shapes, discretization along an arbitrary integration path is necessary. In practice, the technique of "identifying symmetry planes to reduce the model to 1/4" is standard, and experienced engineers develop a habit of instantly looking for symmetry axes when seeing a shape. Saving computational resources starts with method selection.
Edge elements (Nedelec elements)
Elements specialized for electromagnetic field analysis. Automatically guarantee continuity of tangential components and eliminate spurious modes. Standard for 3D high-frequency analysis.
Nodal elements
Used for scalar potential formulation. Effective for static magnetic field scalar potential methods and electrostatic field analysis.
FEM vs Related Topics
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