Electrodynamics/Coulombs Law

Coulomb's Law for Point Charges

The repulsive or attractive electrostatic force between two point charges is determined by an equation called Coulomb's law. Consider the situation where we have two charges, labeled for convenience q, and Q ("q" or "Q" are common variables to describe a point charge, and we will use these letters to describe charges throughtout this book). The following picture shows charge q at a certain point with charge Q at a distance of r away from it. The presence of Q causes an electrostatic force to be exerted on q. The distance vector between Q and q is r.

Using Coulomb's Law, we can find the strength of the electric force between these two charges:

[Coulomb's Law]

$F=k_{\mathrm {e} }{\frac {qQ}{r^{2}}}$

Where: $k_{\mathrm {e} }$ = Coulomb's constant = 8.9875 × 10⁹ N·m²/C² in free space.

Students who have studied Gravity may recognize that Coulomb's Law looks very similar in form to Newton's Law of Universal Gravitation.

The magnitude of the electrostatic force F, on charge q, due to charge Q, equals Coulomb's constant times the product of the two charges (in coulombs) divided by the square of the distance r, between the charges q and Q.

Dielectrics

The value of Coulomb's constant given here is such that the preceding Coulomb's Law equation will work if both q and Q are given in units of coulombs, r in meters, and F in newtons and there is no dielectric material between the charges. A dielectric material is one that reduces the electrostatic force when placed between charges. Furthermore, Coulomb's constant can be given by:

[Coulomb's Constant]

$k={\frac {1}{4\pi \epsilon }}$

where ε = permittivity. When there is no dielectric material between the charges (for example, in free space or a vacuum),

[Permittivity of free space]

$\epsilon _{0}=8.85419\times 10^{-12}{\frac {C^{2}}{(N{\dot {m}}^{2})}}$

Air is only very weakly dielectric and the value above for ε₀ will work well enough with air between the charges. If a dielectric material is present, then:

$\epsilon =\kappa \epsilon _{0}\,$

where κ is the dielectric constant which depends on the dielectric material. In a vacuum (free space), κ = 1 and thus ε = ε₀. For air, κ = 1.0006. Typically, solid insulating materials have values of κ > 1 and will reduce electric force between charges. The dielectric constant can also be called relative permittivity, symbolized as ε_r in Wikipedia.

This shows that the electric force between the charges decreases as the charges are located further from each other by the square of the distance between them. As the charges become located further enough apart, their effect on each other becomes negligible.

Force Vectors

Any force on an object is a vector quantity. In a force vector, the direction is the one in which the force pulls the object. The symbol F is used here for the electric force vector. If charges q and Q are either both positive or both negative, then they will repel each other. This means the direction of the electric force F on q due to Q is away from Q in exactly the opposite direction, as shown by the red arrow in the preceding diagram. If one of the charges is positive and the other negative, then they will attract each other. This means that the direction of F on q due to Q is exactly in the direction towards Q, as shown by the blue arrow in the preceding diagram. The Coulomb's equation shown above will give a magnitude for a repulsive force away from the Q charge. If the magnitude given by the above equation is negative due to opposite charges, the direction of the resulting force will be towards Q, an attractive force. In other sources, different variations of Coulombs' Law are given, including vector formulas in some cases (see Wikipedia link and reference(s) below).

Vector Form

We can vectorize Coulomb's Law to use position and force vectors instead of scalar quantities. We can express the location of charge q as r_q, and the location of charge Q as r_Q. In this way we can know both how strong the electric force is on a charge, but also what direction that force is directed in. Coulomb's Law using vectors can be written as:

[Coulomb's Law (vectors)]

$\mathbf {F} ={\frac {kqQ(\mathbf {r} _{q}-\mathbf {r} _{Q})}{|\mathbf {r} _{q}-\mathbf {r} _{Q}|^{3}}}$

If we have r = r_q - r_Q, and r = |r|, then we can rewrite this equation as:

$\mathbf {F} ={\frac {kqQ\mathbf {r} }{r^{3}}}$

The vector F is a force vector that shows the direction and the magnitude of the force.

n Charges

In many situations, there may be many charges, Q₁, Q₂, Q₃, through Q_n, on the charge q in question. Each of the Q₁ through Q_n charges will exert an electric force on q. The direction of the force depends on the location of the surrounding charges. A Coulomb's Law calculation between q and a corresponding Q_i charge would give the magnitude of the electric force exerted by each of the Q_i charges for i = 1 through n, but the direction of each of the component forces must also be used to determine the individual force vectors, F₁, F₂, F₃, ...., F_n. To determine the total electric force on q, the electric force contributions from each of these charges add up as vector quantities, not just like ordinary (or scalar) numbers.

$\mathbf {F} _{q}=\mathbf {F} _{1}+\mathbf {F} _{2}+\mathbf {F} _{3}+.....+\mathbf {F} _{n}$

The total electric force on q can be added to any other forces affecting it as a vector quantity to obtain the total force vector on the charged object q. In many cases, there are billions of electrons or other charges present, so that geometrical distributions of charges are used with equations stemming from Coulomb's Law.

We can re-write Coulomb's law as a sum of n charges:

[Coulomb's Law (n charges)]

$\mathbf {F_{n}} =\sum _{i\neq n}{\frac {q_{n}q_{i}(\mathbf {r} _{n}-\mathbf {r} _{i})}{4\pi \epsilon _{0}|\mathbf {r} _{n}-\mathbf {r} _{i}|^{3}}}$