The fact that charges exert forces on one another over a distance is explained by the idea of an electric field. An electric field is often referred to as an E-field, because of the symbol used to represent it: \(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} \). The little arrow over the “E” indicates that the electric field is a vector field—it has both a magnitude and a direction.
For example, a point charge (a theoretical charge which has zero size^{1}) creates an electric field which radiates outward from the charge in all directions. The magnitude of the electric field decreases with distance—the electric field gets weaker the farther you get away from it. Therefore, the electric field of a point charge has a direction which points straight out from the charge, and its magnitude decreases with the distance from the charge^{2}. The magnitude of the field is also dependent upon the strength of the charge—larger charges create stronger electric fields. The electric field induced by a point charge is illustrated in Fig. 1.
Mathematically, the electric field of a point charge is given, at least approximately, by^{3}:
(1.)
\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} \infty \hat R\frac{q}{{{R^2}}}\]Where the symbol “\(\infty \)” means “is proportional to” , “\(\hat R \)” provides the direction of the field (straight out from the charge), “q” is the amount of charge, and “ R” is the distance from the charge. Notice that this equation gives a handy summary of the previous discussion:
We are probably a little more familiar with gravitational fields than electric fields, so let's spend some time looking at electric fields in the context of gravity.
According to Newton's law, any two masses will have an attractive force between them, which tends to pull them toward one another. The attractive force is proportional to the product of the masses, and inversely proportional to the square of the distance between them. This force can be explained by hypothesizing a gravitational field which is produced by a mass. Like an electric field, a gravitational field has a magnitude and direction. The direction of the field is inward toward the mass, and its magnitude is proportional to the mass and inversely proportional to the square of the distance from the mass. Figure 2 illustrates at least the direction of the field, if nothing else.
Mathematically, the gravitational field, \(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \psi } \), is given by:
(2.)
\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \psi } {\rm{ }}\infty - \hat R\frac{m}{{{R^2}}}\](3.)
\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over F} {\rm{ = }}{m_2}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \psi } {\rm{ }}\infty - \hat R\frac{{m \cdot {m_2}}}{{{R^2}}}\]The similarity between equations (2) and (3) should be obvious. Mass takes the place of charge, and the sign on the field changes (a positive mass has a gravitational field that points toward it, while a positive charge has an electric field that points away from it).
The presence of the gravitational field helps us explain the attractive force between masses. If we place a second body in the gravitational field of the mass, \(m\), of Fig. 2 and equation (2); Newton's law tells us that the force attracting the two bodies will be the gravitational field of the first body times the mass of the second body. This means that the force on the second body, which we will claim has mass \({m_2}\), is shown in equation (3).
This matches the previous description we gave of Newton's law:
If we place a second mass in the region of the first mass, there will be a force which pulls the two masses together. A similar phenomenon results from the electric field around a charge, which we will discuss next.
Now let's develop the concept of the attractive or repulsive force between two charges more rigorously. Figure 3 provides a visual representation of what we are talking about; we have two charges, q_{1} and q_{2}, and we want to determine the attractive or repulsive force between them.
According to equation (1), the charge q_{1} creates an electric field:
(4.)
\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E } {\rm{ }}\infty {\rm{ }}\hat R\frac{{{q_1}}}{{{q_2}}}\](5.)
\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over F} {\rm{ }}\infty {\rm{ }}\hat R\frac{{{q_{1{\rm{ }}}}{q_2}}}{{{R_2}}}\]If we place charge q_{2} in this field, then the force between them will be q_{2} times this electric field (analogously to Newton's law presented above). This is shown mathematically in equation (5).
Equation (5) is (almost) Coulomb's law^{4}. It tells us that the magnitude of the force between two charges is proportional to the product of the charges and inversely proportional to the square of the distance between them. Equation (5) also gives the direction of the force. If the charges q _{1} and q_{2} have the same sign (if they are both positive or both negative), equation (5) says that the force on q_{2} will be pointing away from q_{1}, and q_{2} will be repelled by q_{1}. (Recall that \(\hat R\) is pointing away from q_{1}, per equation (1) and Fig. 1.) Contrariwise, if q_{1} and q_{2} have opposite signs, their product will be negative, and the sign on the force changes, so that the force on q_{2} is now toward q_{1} and the charges are attracted to one another (-\(\hat R\) points toward q_{1}, analogously to equation (2) and Fig. 2).