Force

For a pair of points, i and j, whose partial charges are q_i and q_j respectively, and which are separated by a distance of r_ij, 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:

• F ∝ q_i  q_j / r_ij²

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

• F = q_i  q_j / (4 π ε₀ r_ij²)
• F is the force experienced by each point charge, in N
  
• q_i and q_j  are the charge at points i and j respectively, in C
• r_ij 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 = q_i  q_j / (4 π ε r_ij²)
• ε = ε_r ε₀   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, 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", E_coul

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 q_i from infinity to a distance r_ij from charge q_j is:

• E_coul  =  q_i  q_j  /  4 π ε₀ r_ij

If we perform the amount of work calculated as above, the potential energy, E_coul, of the system is raised by the same amount.  We can thus express the potential energy as the product of the charge, q_i, and the electric potential, Φ(r_ij):

• E_coul  =  q_i  Φ(r_ij)
• Note: 1 Joule = 1 Volt Coulomb
  

Units and Dimensions:

• Energy, E_coul, has SI units of J (Joule); 1 J = 1 Nm = 1 VC

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

• Φ(r )  =  q_j  /  4 π ε₀ 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.