Coulomb's law

In physics, Coulomb's law describes the forces acting between electric point charges. The law was first given by Charles-Augustin de Coulomb. It is an inverse-square law for two electric charges very similar to Newton's gravitational law for two masses. An important difference between the two laws is that masses always attract each other, whereas charges may repel or attract. That is, charges may be positive or negative, while masses have the same sign (are always positive by convention).

Formulation

Coulomb discovered experimentally that the force between two small charged bodies separated in air a distance large compared to their dimensions

1. varies directly as the magnitude of each charge,
2. varies inversely as the square of the distance between them,
3. is directed along the line joining the charges,
4. is attractive if the bodies are oppositely charged and repulsive if the bodies have the same type of charge.

Further it was shown experimentally that the force is additive, i.e., the force on a test body exerted by a number of charges around it, is the vector sum of the individual two-body forces. Given two charges q and q' a distance r apart, the force F between the particles is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F= \frac{q\,q'}{4\pi \epsilon_0 \epsilon_r r^2} }

The quantities ε₀ and ε_r are the vacuum permittivity and the relative static permittivity (also known as relative dielectric constant) of the medium, respectively. The relative dielectric constant of air is: ε_r = 1.000536 (at ambient temperature and pressure). The formulation of Coulomb's law is in the rationalized SI system of units.

Electrostatic vector field

Coulomb's law. Electric field E at point P due to positive charge q.

The force Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec{\mathbf{F}}} divided by Δq on a test particle of charge Δq is the electrostatic field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec{\mathbf{E}}} —a vector field. The force originates from one or more charges acting on the test particle. The charge Δq of the test particle is taken infinitesimally small, that is, it is negligible with respect to the charges causing the field and hence the test particle does not influence the electric field. The direction of the electric field is by convention such that it points away from a positive charge and points toward a negative charge.

Coulomb's law gives the electric field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec{\mathbf{E}}} at the point P due to an electrostatic point charge q at position Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec{\mathbf{r}}} . The strength of the electric field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec{\mathbf{E}}} depends on the inverse-squared distance of P to q and is along the line from q to P, that is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{\mathbf{E}} = \frac{q(\vec{\mathbf{R}} - \vec{\mathbf{r}})}{4\pi \epsilon_0\epsilon_r|\vec{\mathbf{R}} - \vec{\mathbf{r}}|^3}. }

Note that although this may look like an inverse-cubed dependence, we must not forget that the denominator has dimension length. Indeed, defining a (dimensionless) unit vector by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{r}}_{qp} \equiv \frac{\vec{\mathbf{R}} - \vec{\mathbf{r}}}{|\vec{\mathbf{R}} - \vec{\mathbf{r}}|}, }

we see more clearly the inverse-squared dependence on distance:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{\mathbf{E}} = \frac{q \hat{\mathbf{r}}_{qp}}{4\pi \epsilon_0\epsilon_r|\vec{\mathbf{R}} - \vec{\mathbf{r}}|^2}. }

If q is positive the electric field points from q to P, if q is negative it points from P to q. The force on a particle of charge −|Q| is equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle -|Q|\vec{\mathbf{E}}} , that is, the force is antiparallel to the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec{\mathbf{E}}} . The force on a particle with charge |Q| is equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle |Q|\vec{\mathbf{E}}} , parallel to the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec{\mathbf{E}}} . In total, charges of opposite sign attract and of same sign repel.

The experimentally observed additivity of electrostatic forces gives the following electric field at P when there are N charges q_k at fixed positions r_k. The electric field is a vector field

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{\mathbf{E}}(\vec{\mathbf{R}}) = \sum_{k=1}^N \frac{q_k(\vec{\mathbf{R}} - \vec{\mathbf{r}}_k)}{4\pi \epsilon_0\epsilon_r|\vec{\mathbf{R}} - \vec{\mathbf{r}}_k|^3} }

depending on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \vec{\mathbf{R}}} . The sum stands for a vector sum.

Coulomb potential

From here on we denote vectors by bold letters dropping the arrows on top of the symbols.

When the electrostatic field is irrotational, i.e.,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol{\nabla} \times \mathbf{E} = \mathbf{0} }

one can define a potential Φ(r), such that,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{E}(\mathbf{R}) = - \boldsymbol{\nabla} \Phi(\mathbf{R}). }

Since

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial }{\partial Y} \frac{Z}{R^3} = \frac{\partial }{\partial Z}\frac{Y}{R^3},\quad \frac{\partial }{\partial X}\frac{Z}{R^3} = \frac{\partial }{\partial Z}\frac{X}{R^3},\quad \frac{\partial }{\partial Y}\frac{X}{R^3} = \frac{\partial }{\partial X}\frac{Y}{R^3}, }

it follows that R/R³ is irrotational, and so is (R-r)/|R-r|³. The Coulomb potential Φ exists. The following expression for Φ is consistent with E,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(\mathbf{R}) = \frac{q}{4\pi \epsilon_0 \epsilon_r |\mathbf{R}-\mathbf{r}|} \quad\Longrightarrow\quad \mathbf{E}(\mathbf{R}) = - \boldsymbol{\nabla} \frac{q}{4\pi \epsilon_0 \epsilon_r |\mathbf{R}-\mathbf{r}|} = \frac{q(\mathbf{R}-\mathbf{r})}{4\pi \epsilon_0 \epsilon_r |\mathbf{R}-\mathbf{r}|^3}. }

The Coulomb potential is determined up to a constant. By requiring that Φ vanishes for infinite R the constant becomes zero.

The Coulomb potential at the point R due to N charges at r_k is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(\mathbf{R}) = \sum_{k=1}^N \frac{q_k}{4\pi \epsilon_0\epsilon_r|\mathbf{R} - \mathbf{r}_k|}. }

For a continuous charge distribution ρ(r) this expression generalizes to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(\mathbf{R}) = \iiint \frac{\rho(\mathbf{r})}{4\pi \epsilon_0\epsilon_r|\mathbf{R} - \mathbf{r}|} d\mathbf{r}. }

This generalization follows easily if we first substitute the infinitesimal charge at position r,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta q(\mathbf{r}) = \rho(\mathbf{r}) d\mathbf{r} \equiv \rho(\mathbf{r}) dx dy dz , }

and then replace the sum by a volume integral.

Derivation from Maxwell's equation

(To be continued)

Reference

J. D. Jackson, Classical Electrodynamics, 2nd edition, John Wiley, New York (1975).