Coulomb's Law
Theory and Physics
Coulomb's Law
Professor, Coulomb's Law is the starting point of electromagnetism, right?
It is the most fundamental law describing the force acting between two point charges.
$\varepsilon_0 = 8.854 \times 10^{-12}$ F/m (vacuum permittivity). Like signs cause repulsion, opposite signs cause attraction.
Inversely proportional to the square of the distance... it has the same form as gravity.
Correct. However, the electric force is $10^{36}$ times stronger than gravity. Comparing the electric force and gravity between two electrons, the electric force is overwhelmingly stronger. The reason we don't feel electric forces in daily life is because positive and negative charges almost completely cancel each other out.
Electric Field (Electric Field)
Electric field created by a point charge $q$:
$$ \mathbf{E} = \frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\hat{\mathbf{r}} $$
The electric field is "force per unit charge". If the electric field is known, the force acting on any charge $Q$ can be calculated as $\mathbf{F} = Q\mathbf{E}$.
Superposition Principle
When multiple charges are present, the total electric field is the vector sum of the fields created by each charge. Linearity holds.
$$ \mathbf{E}_{total} = \sum_i \frac{1}{4\pi\varepsilon_0}\frac{q_i}{r_i^2}\hat{\mathbf{r}}_i $$
For continuous distributions, this is replaced by an integral. This is the foundation for solving electrostatic fields with FEM.
Summary
- $F \propto q_1 q_2 / r^2$ — Inversely proportional to the square of the distance
- Electric field $\mathbf{E}$ — Force per unit charge
- Superposition principle — Linearity is the foundation of FEM
Coffee Break Yomoyama Talk
Coulomb's Torsion Balance—The Astonishing Accuracy of 1785
Coulomb demonstrated the inverse-square law in 1785. The tool he used was a "torsion balance", a device that measures force from the twist angle of a thin metal wire. Using this analog method of reading the force between charged spheres from their minute twist angles, he achieved results of surprisingly high accuracy even compared to modern measuring instruments. At that time, neither Ohm's law nor Ampère's law existed, and even the "strength of electric force" was not quantitatively known. By deriving the inverse-square relationship from meticulous experiments, Coulomb enabled the later theoretical framework of electromagnetism. Behind the $F = kq_1q_2/r^2$ we use so casually in CAE lies such a craftsman-like experiment.
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 (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 hair with a plastic sheet creates static electricity, making hair stand up—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 an N pole or only an S pole—they always exist as an 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 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 edge 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 relations 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
Electric field created by a point charge $q$:
The electric field is "force per unit charge". If the electric field is known, the force acting on any charge $Q$ can be calculated as $\mathbf{F} = Q\mathbf{E}$.
When multiple charges are present, the total electric field is the vector sum of the fields created by each charge. Linearity holds.
For continuous distributions, this is replaced by an integral. This is the foundation for solving electrostatic fields with FEM.
- $F \propto q_1 q_2 / r^2$ — Inversely proportional to the square of the distance
- Electric field $\mathbf{E}$ — Force per unit charge
- Superposition principle — Linearity is the foundation of FEM
Coulomb's Torsion Balance—The Astonishing Accuracy of 1785
Coulomb demonstrated the inverse-square law in 1785. The tool he used was a "torsion balance", a device that measures force from the twist angle of a thin metal wire. Using this analog method of reading the force between charged spheres from their minute twist angles, he achieved results of surprisingly high accuracy even compared to modern measuring instruments. At that time, neither Ohm's law nor Ampère's law existed, and even the "strength of electric force" was not quantitatively known. By deriving the inverse-square relationship from meticulous experiments, Coulomb enabled the later theoretical framework of electromagnetism. Behind the $F = kq_1q_2/r^2$ we use so casually in CAE lies such a craftsman-like experiment.
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 (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 hair with a plastic sheet creates static electricity, making hair stand up—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 an N pole or only an S pole—they always exist as an 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 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 edge 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 relations 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 for Electrostatic Fields
How do you solve Coulomb's Law with FEM?
Rather than calculating Coulomb's law directly, we reduce it to the Poisson equation.
$\phi$: Electric potential, $\rho_v$: Charge density. Solve for $\phi$ using FEM, then calculate the electric field as $\mathbf{E} = -\nabla\phi$.
So solving the Poisson equation is more efficient.
Direct Coulomb force calculation is $O(N^2)$, but with FEM you only need to solve a sparse matrix. BEM (Boundary Element Method) is also effective for open-space problems.
Summary
- Poisson equation — Differential form of Coulomb's law
- Solve for potential $\phi$ with FEM → $\mathbf{E} = -\nabla\phi$ — Standard method
- BEM — Effective for open-space electrostatic fields
Problems Solvable and Unsolvable with Point Charge Models
The biggest pitfall in the numerical implementation of Coulomb's law is "the vicinity of a point charge". As the distance $r \to 0$, the force diverges to infinity, so in FEM or BEM, charges must be treated as finite-sized charged bodies, or singularity handling must be introduced. A common practical problem is "electric field concentration at the tip of a needle electrode causing simulation divergence". In practice, a trick used is to model the tip as a tiny sphere to avoid the singularity. Even though the theoretical formula is simple, it's interesting that the idealization of a "point charge" cannot be used as-is in numerical implementation.
Edge Elements (Nedelec Elements)
Elements specialized for electromagnetic field analysis. Automatically guarantee tangential component continuity and eliminate spurious modes. Standard for 3D high-frequency analysis.
Nodal Elements
Used for scalar potential formulations. Effective for scalar potential methods in magnetostatics and electrostatic field analysis.
FEM vs Related Topics
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