Coulomb's law

Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was studied and first published in 1783 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism. Nevertheless, the dependence of the electric force with distance had been proposed previously by Joseph Priestley[1] and the dependence with both distance and charge had been discovered, but not published, by Henry Cavendish, prior to Coulomb's works.

Basic equation

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 law states that: "The magnitude of the Electrostatics force of interaction between two point charges is directly proportional to the scalar multiplication of the magnitudes of charges and inversely proportional to the square of the distances between them."
The scalar form of Coulomb's law is an expression for the magnitude and sign of the electrostatic force between two idealized point charges, small in size compared to their separation. This force (F) acting simultaneously on point charges (q1) and (q2), is given by
F = k_\mathrm{e} \frac{q_1q_2}{r^2}
where r is the separation distance and ke is a proportionality constant. A positive force implies it is repulsive, while a negative force implies it is attractive.[2] The proportionality constant ke, called the Coulomb constant (sometimes called the Coulomb force constant), is related to defined properties of space and can be calculated based on knowledge of empirical measurements of the speed of light:[3]
k_\mathrm{e} &= \frac{1}{4 \pi \varepsilon_0} = \frac{c^2 \ \mu_0}{4 \pi} = c^2 \cdot 10^{-7} \ \mathrm{H} \cdot \mathrm{m}^{-1}\\
               &= 8.987\ 551\ 787\ 368\ 176\ 4 \times 10^9 \ \mathrm{N  \cdot m^2 / C^{2}}
In SI units, the meter is defined such that the speed of light in vacuum (or electromagnetic waves, in general), denoted c,[4] is exactly 299,792,458 m·s−1[5], and the magnetic constant (μ0) is set at 4π × 10−7 H·m−1.[6] In agreement with electromagnetic theory, requiring that
{1 \over \mu _0 \varepsilon _0 }= c^2 ,
the value for the electric constant (ε0) is derived to be ε0 = 1/(μ0c2) ≈ 8.85418782×10−12 F·m−1.[7] In electrostatic units and Gaussian units, the unit charge (esu or statcoulomb) is defined in such a way that the Coulomb constant is 1 and dimensionless.
In the more useful vector-form statement, the force in the equation is a vector force acting on either point charge, so directed as to push it away from the other point charge; the right-hand side of the equation, in this case, must have an additional product term of a unit vector pointing in one of two opposite directions, e.g., from q1 to q2 if the force is acting on q2; the charges may have either sign and the sign of their product determines the ultimate direction of that force. Thus, the vector force pushing the charges away from each other (pulling towards each other if negative) is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The square of the distance part arises from the fact that the force field due to an isolated point charge is uniform in all directions and gets "diluted" with distance as much as the area of a sphere centered on the point charge expands with its radius.
The law of superposition allows this law to be extended to include any number of point charges, to derive the force on any one point charge by a vector addition of these individual forces acting alone on that point charge. The resulting vector happens to be parallel to the electric field vector at that point, with that point charge (or "test charge") removed.
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

Vector form

In order to obtain both the magnitude and direction of the force on a charge, q1 at position \mathbf{r}_1, experiencing a field due to the presence of another charge, q2 at position \mathbf{r}_2, the full vector form of Coulomb's law is required.
\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},
where r is the separation of the two charges. This is simply the scalar definition of Coulomb's law with the direction given by the unit vector, \mathbf{\hat{r}}_{21}, parallel with the line from charge q2 to charge q1.[8]
If both charges have the same sign (like charges) then the product q1q2 is positive and the direction of the force on q1 is given by \mathbf{\hat{r}}_{21}; the charges repel each other. If the charges have opposite signs then the product q1q2 is negative and the direction of the force on q1 is given by -\mathbf{\hat{r}}_{21}; the charges attract each other.

System of discrete charges

The principle of linear superposition may be used to calculate the force on a small test charge, q, due to a system of N discrete charges:
\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,
where qi and \mathbf{r}_i are the magnitude and position respectively of the ith charge, \mathbf{\hat{R}}_{i} is a unit vector in the direction of \mathbf{R}_{i} = \mathbf{r} - \mathbf{r}_i (a vector pointing from charge qi to charge q), and Ri is the magnitude of \mathbf{R}_{i} (the separation between charges qi and q).[8]

Continuous charge distribution

For a charge distribution an integral over the region containing the charge is equivalent to an infinite summation, treating each infinitesimal element of space as a point charge dq.
For a linear charge distribution (a good approximation for charge in a wire) where \lambda(\mathbf{r^\prime}) gives the charge per unit length at position \mathbf{r^\prime}, and dl^\prime is an infinitesimal element of length,
dq = \lambda(\mathbf{r^\prime})dl^\prime.[9]
For a surface charge distribution (a good approximation for charge on a plate in a parallel plate capacitor) where \sigma(\mathbf{r^\prime}) gives the charge per unit area at position \mathbf{r^\prime}, and dA^\prime is an infinitesimal element of area,
dq = \sigma(\mathbf{r^\prime})\,dA^\prime.\,
For a volume charge distribution (such as charge within a bulk metal) where \rho(\mathbf{r^\prime}) gives the charge per unit volume at position \mathbf{r^\prime}, and dV^\prime is an infinitesimal element of volume,
dq = \rho(\mathbf{r^\prime})\,dV^\prime.[8]
The force on a small test charge q^\prime at position \mathbf{r} is given by
\mathbf{F} = {q^\prime \over 4\pi\varepsilon_0}\int dq {\mathbf{r} - \mathbf{r^\prime} \over |\mathbf{r} - \mathbf{r^\prime}|^3}.

Graphical representation

Below is a graphical representation of Coulomb's law, when q1q2 > 0. The vector \mathbf{F}_1 is the force experienced by q1. The vector \mathbf{F}_2 is the force experienced by q2. Their magnitudes will always be equal. The vector \mathbf{r}_{21} is the displacement vector between two charges (q1 and q2).
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

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


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