Coulomb's law Did You Mean Coulomb's?

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In physics, Coulomb's law is an inverse-square law indicating the magnitude and direction of electrical force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another.

Coulomb's Law may be stated as follows:

"The magnitude of the electrostatic force between two point charges is directly proportional to the magnitudes of each charge and inversely proportional to the square of the distance between the charges."

When one is interested only in the magnitude of the force (and not in its direction), it may be easiest to consider a simplified, scalar version of the law

$F = k \frac{\left|q_1\right| \left|q_2\right|}{r^2}$

where (in SI units):

$F \$ is the magnitude of the force exerted, measured in newtons

$q_1 \$ is the charge on one body, measured in coulombs

$q_2 \$ is the charge on the other body, also measured in coulombs

$r \$ is the distance between them measured in metres

$k \$ is the electrostatic constant or Coulomb force constant, often written as $\frac{1}{ 4 \pi \epsilon_0}$ where $\epsilon_0 \$ is a physical constant, the permittivity of free space. $k \$ ≈ 8 987 742 438 F⁻¹·m or C⁻²·N·m², and $\epsilon_0 \$ ≈ 8.854 × 10⁻¹² F·m⁻¹ or C²·N⁻¹·m⁻². In cgs units, the unit charge, esu of charge or statcoulomb, is defined so that this Coulomb force constant is 1.

Note that $\frac{1}{\mu_0\epsilon_0}=c^2$ , where $\mu_0 \$ is the permeability of vacuum and $c \$ is the speed of light.

This formula says that the magnitude of the force is directly proportional to the magnitude of the charges of each object and inversely proportional to the square of the distance between them. Because, when measured in units people commonly use (such as MKS), the Coulomb force constant, $k \$ , is numerically much much larger than the universal gravitational constant $G \$ , which means that for objects with charge that is of the order of a unit charge (C) and mass of the order of a unit mass (kg), that the electrostatic forces will be so much larger than the gravitational forces that the latter force can be ignored. This is not the case when Planck units are used and both charge and mass are of the order of the unit charge and unit mass. However, charged elementary particles have mass that is far less than the Planck mass while their charge is about the Planck charge so that, again, gravitational forces can be ignored.

The force $F \$ acts on the line connecting the two charged objects. Charged objects of the same polarity repel each other along this line and charged objects of opposite polarity attact each other along this line connecting them.

For calculating the direction and magnitude of the force simultaneously, one will wish to consult the full-blown vector version of the Law

$\mathbf{F} = \frac{1}{ 4 \pi \epsilon_0} \frac{q_1 q_2 \mathbf{r}}{ \left|\mathbf{r}\right|^3}$

where

$\mathbf{F}$ is the electrostatic force vector,

$\mathbf{r}$ is the vector between the two charges, such that

$\mathbf{r}=\mathbf{r_1}-\mathbf{r_2}$

where

$\mathbf{r_1} \$ is vector indicating the position of the charge on which the force acts

$\mathbf{r_2} \$ is the vector indicating the position of the other charge.

This vector equation indicates that opposite charges attract, and like charges repel. When $q_1 q_2 \$ is negative, the force is attractive. When positive, the force is repulsive. $|\mathbf{r}| \$ has been raised to the third power instead of the second in the denominator in order to normalize the length of the $\mathbf{r} \$ vector in the numerator to 1.

Below is a graphical representation of Coulomb's law. $\mathbf{F_2}$ is the force experienced by $\mathbf{Q_2}$ . $\mathbf{R_{12}}$ is the vector between two charges ( $\mathbf{Q_1}$ and $\mathbf{Q_2}$ ).

A graphical representation of Coulomb's law.

In either formulation, Coulomb's law is fully accurate only when the objects are stationary, and remains approximately correct only for slow movement. When movement takes place, magnetic fields are produced that alter the force on the two objects. The force resulting from magnetic field between moving charges can be thought of as a manifestation of the force from the electrostatic field but with Einstein's theory of relativity taken into consideration.

References

Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.), Prentice Hall. ISBN 013805326X
Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.), W. H. Freeman. ISBN 0716708108

External links

MISN-0-114 Coulomb's Law (PDF file) by J. Kovacs for Project PHYSNET.

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