Coulomb's Law: Definition & Summary

[Greg Bernhardt]

Definition/Summary

Coulomb's law is an inverse-square law stating that the force vector between two stationary charges is a constant times the unit vector between them and times the product of the magnitudes of the charges divided by the square of the distance between them: $\mathbf{F}_{12}\ \propto\ Q_1Q_2\,\mathbf{\hat{r}}_{12}/r_{12}^2$

That constant (Coulomb's constant) is the same in any material, and is $1/4\,\pi\,\varepsilon_0$, where $\varepsilon_0$ is the electric constant (the permittivity of the vacuum, with dimensions of charge²/force.area, and measured in units of farads/metre), and $4\,\pi$ is the ratio between the surface area of a sphere and the radius squared.

If the charges have the same sign, then $Q_1Q_2$ is positive, and the force vector points outward (the force is repulsive); if they have opposite signs, then $Q_1Q_2$ is negative, and the force vector points inward (the force is attractive).

Gauss' law (one of Maxwell's equations) may be derived from Coulomb's law.

Equations

Force on stationary charge 2 from stationary charge 1:

$$\mathbf{F}_{12}\ =\ \frac{Q_1\,Q_2}{4\,\pi\,\varepsilon_0\,r_{12}^2}\,\mathbf{\hat{r}}_{12}$$

Electric field of charge 1 at position of charge 2 (from Lorentz force equation):

$$\mathbf{E}_{12}\ =\ \frac{\mathbf{F}_{12}}{Q_2}\ =\ \frac{Q_1}{4\,\pi\,\varepsilon_0\,r_{12}^2}\,\mathbf{\hat{r}}_{12}$$

Since this is independent of the magnitude of charge 2, it may be rewritten:

$$\mathbf{E}_1(\mathbf{r})\ =\ \frac{Q_1}{4\,\pi\,\varepsilon_0\,r^2}\,\mathbf{\hat{r}}\ =\ \frac{Q_1}{\varepsilon_0\,A(r)}\,\mathbf{\hat{r}}$$

where $A(r)$ is the surface area of the sphere $S(r)$ through $\mathbf{r}$ with charge 1 at its centre.

Obviously, the divergence of $\mathbf{E}_1$ at any point other than the position of charge 1 is zero (differential form of Gauss' law for zero charge density $\rho$):

$$\nabla\cdot\mathbf{E}_1\ =\ 0\ \ \text{if}\ \ \rho\ =\ 0$$

And the flux of $\mathbf{E}_1$ through the sphere $S(r)$ is $Q_1/\varepsilon_0$:

$$\oint_{S(r)}\,\mathbf{E}_1\cdot(\mathbf{\hat{r}}\,dA)\ =\ \frac{Q_1}{\varepsilon_0 \,A(r)}\,\oint_{S(r)}\,\mathbf{\hat{r}}\cdot\mathbf{\hat{r}}\,dA\ =\ \frac{Q_1}{\varepsilon_0 \,A(r)}\,\oint_{S(r)}dA\ =\ \frac{Q_1}{\varepsilon_0}$$

and so, from Stoke's theorem, the flux of $\mathbf{E}_1$ through any closed surface S containing charge 1 is $Q_1/\varepsilon_0$ and through any other closed surface is zero:

$$\oint_S \, \mathbf{E}_1 \cdot(\mathbf{\hat{n}} \, dA)\ =\ \left\{\begin{array}{cc}<br /> Q_1/\varepsilon_0 & \text{if S contains charge 1}\\<br /> 0 & \text{if S does not contain charge 1}\end{array}\right.$$

Extended explanation



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[AaronS]

Coulomb's law is an important concept in electrical engineering and physics. It states that the force between two stationary charges is proportional to the product of the magnitudes of the charges divided by the square of the distance between them. This law can be used to calculate electric fields and derive Gauss' law, which is one of Maxwell's equations. It is also important to note that if the charges have the same sign, the force vector points outward and is repulsive, while if they have opposite signs, the force vector points inward and is attractive.