Electric Potential Distribution Analysis
Theory and Physics
Overview
Professor, how is electric potential related to the electric field?
The electric potential $\phi$ is a scalar field, and the electric field is its negative gradient.
Since a three-component vector field can be reduced to a single scalar field, solving for the potential is standard in FEM.
Reducing the number of unknowns to one-third is a big advantage.
The equation satisfied by the potential is Poisson's equation.
In charge-free regions, it becomes Laplace's equation $\nabla^2 \phi = 0$.
Boundary Conditions
What types of boundary conditions are there for potential analysis?
There are three basic types of boundary conditions.
Boundary Condition Mathematical Expression Physical Meaning Dirichlet $\phi = V_0$ Fixed potential (conductor surface) Neumann $\partial\phi/\partial n = -\sigma_s/\varepsilon$ Normal electric field specified Symmetric Boundary $\partial\phi/\partial n = 0$ Surface parallel to electric field lines
Conductors are equipotential, so $\phi = \text{const}$. In COMSOL, these can be set intuitively as "Electric Potential" and "Ground" boundary conditions.
Electrostatic Energy
How do you calculate electrostatic energy from the potential?
Electrostatic energy is obtained by volume integration of the square of the electric field.
$$ W_e = \frac{1}{2}\int_\Omega \varepsilon |\nabla\phi|^2\,d\Omega $$
This matches the capacitor energy $W = \frac{1}{2}CV^2$. It can be calculated directly using COMSOL's "Volume Integration".
Coffee Break Trivia Corner
"Potential is a scalar, so why is it more convenient than the vector electric field?"
The electric field $\mathbf{E}$ is a vector quantity (has x, y, z components), while the electric potential $\phi$ is a scalar quantity (a single number). When solving with FEM, solving for a scalar reduces the degrees of freedom to one-third, significantly lowering computational cost. It is overwhelmingly more efficient to "solve for the potential and then calculate the electric field via $\mathbf{E} = -\nabla\phi$" than to "solve for the electric field directly". This is why electric potential distribution analysis is the standard approach in CAE. On the other hand, in time-varying electromagnetic fields where potential cannot be defined (integral value changes depending on the path), "vector potential" is used instead of scalar potential. The choice between scalar or vector determines computational cost—this is a perspective worth keeping as a CAE engineer.
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 where a changing magnetic field induces an electric field. Induction cooking (IH) 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'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 $\partial D/\partial t$ 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 a plastic sheet on hair makes hair stand up due to static electricity—electric field lines radiate 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 a N pole or only a S pole—they 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 uses vector potential $\mathbf{B} = \nabla \times \mathbf{A}$, 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 edge effects can be ignored
- Isotropic 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
What types of boundary conditions are there for potential analysis?
There are three basic types of boundary conditions.
| Boundary Condition | Mathematical Expression | Physical Meaning |
|---|---|---|
| Dirichlet | $\phi = V_0$ | Fixed potential (conductor surface) |
| Neumann | $\partial\phi/\partial n = -\sigma_s/\varepsilon$ | Normal electric field specified |
| Symmetric Boundary | $\partial\phi/\partial n = 0$ | Surface parallel to electric field lines |
Conductors are equipotential, so $\phi = \text{const}$. In COMSOL, these can be set intuitively as "Electric Potential" and "Ground" boundary conditions.
How do you calculate electrostatic energy from the potential?
Electrostatic energy is obtained by volume integration of the square of the electric field.
This matches the capacitor energy $W = \frac{1}{2}CV^2$. It can be calculated directly using COMSOL's "Volume Integration".
"Potential is a scalar, so why is it more convenient than the vector electric field?"
The electric field $\mathbf{E}$ is a vector quantity (has x, y, z components), while the electric potential $\phi$ is a scalar quantity (a single number). When solving with FEM, solving for a scalar reduces the degrees of freedom to one-third, significantly lowering computational cost. It is overwhelmingly more efficient to "solve for the potential and then calculate the electric field via $\mathbf{E} = -\nabla\phi$" than to "solve for the electric field directly". This is why electric potential distribution analysis is the standard approach in CAE. On the other hand, in time-varying electromagnetic fields where potential cannot be defined (integral value changes depending on the path), "vector potential" is used instead of scalar potential. The choice between scalar or vector determines computational cost—this is a perspective worth keeping as a CAE engineer.
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 where a changing magnetic field induces an electric field. Induction cooking (IH) 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'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 $\partial D/\partial t$ 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 a plastic sheet on hair makes hair stand up due to static electricity—electric field lines radiate 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 a N pole or only a S pole—they 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 uses vector potential $\mathbf{B} = \nabla \times \mathbf{A}$, 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 edge effects can be ignored
- Isotropic 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
Details of Numerical Methods
Could you explain the FEM formulation for potential in detail?
Discretize the weak form of Poisson's equation. Approximating the potential with shape functions $N_i$, the element stiffness matrix is
It's positive definite symmetric, so the CG method is optimal. AMG is effective for preconditioning. COMSOL automatically selects MUMPS and AMG.
Improving Accuracy of Electric Field from Potential
Are there any tips for improving electric field accuracy?
The potential is continuous, but $\mathbf{E} = -\nabla\phi$ can be discontinuous at element boundaries.
- SPR Method: Reconstructs electric field on a patch basis from gradient values at superconvergent points
- Quadratic Elements: If potential is quadratic, electric field varies linearly, improving accuracy
- Smoothing Processing: Available as a post-processing option in COMSOL
Using quadratic elements is the minimum requirement in practice, right?
Exactly. With linear elements, the electric field becomes constant within an element, failing to correctly capture field concentrations. Ansys Maxwell's automatic adaptive meshing also operates based on quadratic elements.
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