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Definition/Summary
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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: [itex]\mathbf{F}_{12}\ \propto\ Q_1Q_2\,\mathbf{\hat{r}}_{12}/r_{12}^2[/itex]
That constant (Coulomb's constant) is the same in any material, and is [itex]1/4\,\pi\,\varepsilon_0[/itex], where [itex]\varepsilon_0[/itex] is the electric constant (the permittivity of the vacuum, with dimensions of chargeČ/force.area, and measured in units of farads/metre), and [itex]4\,\pi[/itex] is the ratio between the surface area of a sphere and the radius squared.
If the charges have the same sign, then [itex]Q_1Q_2[/itex] is positive, and the force vector points outward (the force is repulsive); if they have opposite signs, then [itex]Q_1Q_2[/itex] 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. |
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Equations
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Force on stationary charge 2 from stationary charge 1:
[tex]\mathbf{F}_{12}\ =\ \frac{Q_1\,Q_2}{4\,\pi\,\varepsilon_0\,r_{12}^2}\,\mathbf{\hat{r}}_{12}[/tex]
Electric field of charge 1 at position of charge 2 (from Lorentz force equation):
[tex]\mathbf{E}_{12}\ =\ \frac{\mathbf{F}_{12}}{Q_2}\ =\ \frac{Q_1}{4\,\pi\,\varepsilon_0\,r_{12}^2}\,\mathbf{\hat{r}}_{12}[/tex]
Since this is independent of the magnitude of charge 2, it may be rewritten:
[tex]\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}}[/tex]
where [itex]A(r)[/itex] is the surface area of the sphere [itex]S(r)[/itex] through [itex]\mathbf{r}[/itex] with charge 1 at its centre.
Obviously, the divergence of [itex]\mathbf{E}_1[/itex] at any point other than the position of charge 1 is zero (differential form of Gauss' law for zero charge density [itex]\rho[/itex]):
[tex]\nabla\cdot\mathbf{E}_1\ =\ 0\ \ \text{if}\ \ \rho\ =\ 0[/tex]
And the flux of [itex]\mathbf{E}_1[/itex] through the sphere [itex]S(r)[/itex] is [itex]Q_1/\varepsilon_0[/itex]:
[tex]\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}[/tex]
and so, from Stoke's theorem, the flux of [itex]\mathbf{E}_1[/itex] through any closed surface S containing charge 1 is [itex]Q_1/\varepsilon_0[/itex] and through any other closed surface is zero:
[tex]\oint_S \, \mathbf{E}_1 \cdot(\mathbf{\hat{n}} \, dA)\ =\ \left\{\begin{array}{cc}
Q_1/\varepsilon_0 & \text{if S contains charge 1}\\
0 & \text{if S does not contain charge 1}\end{array}\right.[/tex] |
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Recent forum threads on Coulomb's law
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Breakdown
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Physics
> Electromagnetism
>> Electrostatics
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