Coulomb's law

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Coulomb's law, developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form as follows:

The magnitude of the electrostatic force between two point electric charges is directly proportional to the product of the magnitudes of each charge and inversely proportional to the square of the distance between the charges.

Contents

Scalar form

Diagram describing the basic mechanism of Coulomb's law; like charges repel each other and opposite charges attract each other. Diagram describing the basic mechanism of Coulomb's law; like charges repel each other and opposite charges attract each other. Coulomb's torsion balance Coulomb's torsion balance \scriptstyle{q_1}
F = {1 \over 4\pi\varepsilon_0}\frac{q_1q_2}{r^2}
\scriptstyle{r}\scriptstyle{k_e}

\begin{align}
k_e &= \frac{1}{4\pi\varepsilon_0} = \frac{\mu_0\ {c_0}^2}{4 \pi} = 10^{-7}\ {c_0}^2 \\
&= 8.987\ 551\ 787\ \times 10^9 \\
\end{align}
\approx 9 \times 10^9

In SI units the speed of light in vacuum c0 is defined as the numerical value c0 = 299 792 458 m s−1 (See c0) and the magnetic constant μ0 is defined as 4π x 10−7 H · m−1 (See μ0), leading to the definition for the electric constant of ε0 = 1/(μ0c02) ≈ 8.854 187 817 x 10−12 F m−1 (See NIST ε0). In cgs units, the unit charge, esu of charge or statcoulomb, is defined so that this Coulomb force constant is 1.

This formula says that the magnitude of the force is directly proportional to the magnitude of the charges of each object and inversely proportional to the square of the distance between them. The exponent in Coulomb's Law has been found to differ from −2 by less than one in a billion.

\scriptstyle{k_e}

Coulomb's law can also be interpreted in terms of atomic units with the force expressed in Hartrees per Bohr radius, the charge in terms of the elementary charge, and the distances in terms of the Bohr radius.

Electric field

Main article: Electric field
\scriptstyle{\mathbf{E}}
E = {1 \over 4\pi\varepsilon_0}\frac{q}{r^2}
\scriptstyle{q}

Vector form

\scriptstyle{q_1}
\mathbf{F} = {1 \over 4\pi\varepsilon_0}{q_1q_2(\mathbf{r}_1 - \mathbf{r}_2) \over |\mathbf{r}_1 - \mathbf{r}_2|^3} = {1 \over 4\pi\varepsilon_0}{q_1q_2 \over r^2}\mathbf{\hat{r}}_{21}
\scriptstyle{r}\scriptstyle{q_1q_2}

System of discrete charges

\scriptstyle{q}
\mathbf{F}(\mathbf{r}) = {q \over 4\pi\varepsilon_0}\sum_{i=1}^N {q_i(\mathbf{r} - \mathbf{r}_i) \over |\mathbf{r} - \mathbf{r}_i|^3} = {q \over 4\pi\varepsilon_0}\sum_{i=1}^N {q_i \over R_{i}^2}\mathbf{\hat{R}}_{i}
\scriptstyle{q_i}

Continuous charge distribution

\scriptstyle{dq}\scriptstyle{\lambda(\mathbf{r^\prime})}
dq = \lambda(\mathbf{r^\prime})dl^\prime
\scriptstyle{\sigma(\mathbf{r^\prime})}
dq = \sigma(\mathbf{r^\prime})dA^\prime
\scriptstyle{\rho(\mathbf{r^\prime})}
dq = \rho(\mathbf{r^\prime})dV^\prime
\scriptstyle{q^\prime}
\mathbf{F} = q^\prime\int dq {\mathbf{r} - \mathbf{r^\prime} \over |\mathbf{r} - \mathbf{r^\prime}|^3}

Graphical representation

\scriptstyle{q_1q_2 > 0} A graphical representation of Coulomb's law.A graphical representation of Coulomb's law.

Electrostatic approximation

In either formulation, Coulomb's law is fully accurate only when the objects are stationary, and remains approximately correct only for slow movement. These conditions are collectively known as the electrostatic approximation. When movement takes place, magnetic fields are produced which alter the force on the two objects. The magnetic interaction between moving charges may be thought of as a manifestation of the force from the electrostatic field but with Einstein's theory of relativity taken into consideration.

Table of derived quantities

\mathbf{F}_{12}= q_1 \mathbf{E}_{12}\mathbf{F}_{12}=-\mathbf{\nabla}U_{12}\mathbf{E}_{12}=-\mathbf{\nabla}V_{12}U_{12}=q_1 V_{12} \
Particle property Relationship Field property
Vector quantity \mathbf{F}_{12}= {1 \over 4\pi\varepsilon_0}{q_1 q_2 \over r^2}\mathbf{\hat{r}}_{21} \
Force (on 1 by 2)
\mathbf{E}_{12}= {1 \over 4\pi\varepsilon_0}{q_2 \over r^2}\mathbf{\hat{r}}_{21} \
Electric field (at 1 by 2)
Relationship
Scalar quantity U_{12}={1 \over 4\pi\varepsilon_0}{q_1 q_2 \over r} \
Potential energy (at 1 by 2)
V_{12}={1 \over 4\pi\varepsilon_0}{q_2 \over r}
Potential (at 1 by 2)