# Coulomb's Law

Charles Augustin de Coulomb (1736-1806) formulated the inverse square law in 1785 that describes how the force between two point charges varies according to the separation between them. Joseph Priestly, who was not only one of the discoverers of oxygen, but who also wrote a comprehensive treatise on electricity, anticipated this result some two decades before Coulomb.

### Force

For a pair of points,

*i*and

*j*, whose partial charges are

*q*and

_{i}*q*respectively, and which are separated by a distance of

_{j}*r*, the magnitude of the force,

_{ij}*F*, between them is directly proportional to the product of the charges, and inversely proportional to the square of the distance:

*F*∝*q*_{i}*q*/_{j}*r*_{ij}^{2}

**in a vacuum**:

*F*=*q*_{i}*q*/ (4 π ε_{j}_{0}*r*_{ij}^{2})*F*is the force experienced by each point charge, in N

*q*and_{i}*q*are the charge at points_{j }*i*and*j*respectively, in C*r*is the magnitude of the distance between the two points_{ij }*i*and*j*, in m

- ε
_{0}is the permittivity of a vacuum, ε_{0}= 8.854 188 x10^{-12}J^{-1}C^{2}m^{-1}

*F*=*q*_{i}*q*/ (4 π ε_{j}_{}*r*_{ij}^{2})- ε = ε
_{r }ε_{0 }where ε_{r }is the relative permittivity (dielectric constant) of the medium. - Since ε
_{r }> 1 the strength of the electric field is reduced relative to the vacuum. -
Substance Dielectric Constant, ε _{r}Temperature (°C)

Methane

1.70 -173

Carbon tetrachloride

2.228

25

Cyclohexane

2.015

25

Benzene

2.274

25

Nitrobenzene

34.82

25

Methanol

32.63 25 Ethanol

24.30

25

Ammonia

16.9

25

Ammonia 22.4

-33

Hydrogen sulphide

9.26

-85

Water Ice (s)

88.00

0

Water (l) 80.37

20

Water (l)

78.54

25

Water (l) 55.33

100

Water Steam (g)

1.0126

110

Diamond (s)

5.5

17-22

Air (g)

1.000590

0

- Source of Dielectric Constants:
*Handbook of Chemistry and Physics*, Chemical Rubber Publishing Co.

Units and Dimensions:

- Force,
*F*, has SI units of N (Newton); 1 N = kg m / s^{2} - Force,
, is a vector quantity, and requires both a direction and a magnitude to be completely described.*F*

- Electric charge,
*q*, has SI units of C (Coulomb); 1 C = A s - The charge on an electron is -
*e*, where*e*, the charge on a proton = 1.602 19 x 10^{-19}C

- Distance,
*r*, has SI units of m (metre). - Position
vector is a vector quantity, and requires both a direction and a
magnitude to be completely described. The magnitude of the
position vector,
, is the distance,__r__*r*.

### "Electrical Work" or "Potential Energy", E_{coul}

Electrical work is the integral of -*F(r)*d

*r*, where

*F(r)*is the force opposing displacement through d

*r*. Therefore the work involved in bringing up a charge

*q*from infinity to a distance

_{i}*r*from charge

_{ij}*q*is:

_{j}*E*=_{coul}*q*_{i}*q*/ 4 π ε_{j}_{0}*r*_{ij}

*E*, of the system is raised by the same amount. We can thus express the potential energy as the product of the charge,

_{coul}*q*, and the electric potential,

_{i}*Φ*(

*r*):

_{ij}*E*=_{coul}*q*_{i}*Φ*(*r*)_{ij}- Note: 1 Joule = 1 Volt Coulomb

- Energy,
*E*, has SI units of J (Joule); 1 J = 1 Nm = 1 VC_{coul}

### "Electric Potential" or "Electrostatic Potential",* Φ*(*r *)

*Φ*(*r*)*= q*/ 4 π ε_{j}_{0}*r*

*Φ*(*r*) is the electrostatic potential in V (Volt); 1 V = 1 J C^{-1}s^{-1}

*q*_{j}is the magnitude of the electric point charge, in C_{}

*r*is the distance from the point charge, in m

Units and Dimensions:

- Electric (or Electrostatic) Potential,
*Φ*(*r*), has SI units of V (Volt); 1 V = Nm / As

### Dimensional Analysis

You can find out more about "dimensional analysis":dimensional on this page.