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# 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 qi and qj respectively, and which are separated by a distance of rij, the magnitude of the force, F, between them is directly proportional to the product of the charges, and inversely proportional to the square of the distance:

• Fqi  qj / rij2
A more precise formulation of Coulomb's law is as follows, for two charges in a vacuum:

• F = qi  qj / (4 π ε0 rij2)
• F is the force experienced by each point charge, in N
• qi and qare the charge at points i and j respectively, in C
• rij is the magnitude of the distance between the two points i and j, in m
• ε0 is the permittivity of a vacuum, ε0 = 8.854 188 x10-12  J-1 C2 m-1
When we are not in a vacuum, say in an organic solvent, a protein or water, we scale the permittivity of the vacuum by an appropropriate amount:

• F = qi  qj / (4 π ε rij2)
• ε = εr ε0   where   εr is the relative permittivity (dielectric constant) of the medium.
• Since εr > 1 the strength of the electric field is reduced relative to the vacuum.
• Substance Dielectric Constant, εr 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 / s2
• Force, F, is a vector quantity, and requires both a direction and a magnitude to be completely described.
• 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, r, is the distance, r.

### "Electrical Work" or "Potential Energy", Ecoul

Electrical work is the integral of -F(r) dr, where F(r) is the force opposing displacement through dr.  Therefore the work involved in bringing up a charge qi from infinity to a distance rij from charge qj is:
• Ecoul  =  qi  qj  /  4 π ε0 rij
If we perform the amount of work calculated as above, the potential energy, Ecoul, of the system is raised by the same amount.  We can thus express the potential energy as the product of the charge, qi, and the electric potential, Φ(rij):
• Ecoul  =  qi  Φ(rij)
• Note: 1 Joule = 1 Volt Coulomb
Units and Dimensions:
• Energy, Ecoul, has SI units of J (Joule); 1 J = 1 Nm = 1 VC

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

• Φ(r )  =  qj  /  4 π ε0 r
• Φ(r ) is the electrostatic potential in V (Volt); 1 V = 1 J C-1 s-1
• qj  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.

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