Electric Field

Consider a system of $N$ point electric charges. Let the $i$th charge have electric charge $q_i$ and displacement ${\bf r}_i$. As is the case for gravitational forces (see Section 1.8.1), it is an experimentally demonstrated fact that electrical forces are superposable; that is, the electrical force acting on a test charge whose electric charge is $q$ and whose displacement is ${\bf r}$ is simply the sum of all of the Coulomb-law forces exerted on it by each of the other $N$ charges taken in isolation. In other words, the electrical force exerted by the $i$th charge (say) on the test charge is the same as if all of the other charges were not present. Thus, generalizing Equation (2.2), the force acting on the test charge is given by

$${\bf f}({\bf r}) = q \sum_{i=1,N} \frac{q_i}{4\pi\,\epsilon_0} \frac{{\bf r}-{\bf r}_i}{\vert{\bf r}-{\bf r}_i\vert^3}.$$ (2.9)

It is helpful to introduce a vector field, ${\bf E}({\bf r})$, known as the electric field, which is defined as the force exerted on a test charge of unit electric charge whose displacement is ${\bf r}$. Thus, from the previous equation, the electrical force on a test charge $q$ whose displacement is ${\bf r}$ is written

$${\bf f}({\bf r}) = q\,{\bf E}({\bf r}),$$ (2.10)

where the electric field is given by

$${\bf E}({\bf r}) = \sum_{i=1,N}\frac{q_i}{4\pi\,\epsilon_0} \frac{{\bf r}- {\bf r}_i}{\vert{\bf r}-{\bf r}_i\vert^3}.$$ (2.11)

At this point, we have no reason to believe that the electric field has any real physical existence. It is just a useful device for calculating the electrical force that acts on test charges placed at various locations.

According to the previous equation, the electric field generated by a single point electric charge $q$ located at the origin is purely radial, is directed outward if the charge is positive, and inward if it is negative, and has magnitude

$$E_r (r) = \frac{q}{4\pi\,\epsilon_0\,r^2},$$ (2.12)

where $r$ is a spherical polar coordinate. Moreover, $E_r$ is the radial component of the field in spherical polar coordinates. The other components are zero. (See Section A.23.) We can represent an electric field by so-called field-lines. The direction of the lines indicates the direction of the local electric field, and the density of the lines perpendicular to this direction is proportional to the magnitude of the local electric field. It follows from Equation (2.12) that the number of field-lines crossing the surface of a sphere centered on a point charge (which is equal to $E_r$ times the area, $4\pi\,r^2$, of the surface) is independent of the radius of the sphere. Thus, the field of a point positive electric charge is represented by a group of equally-spaced, unbroken, straight-lines radiating from the charge. See Figure 2.1. Likewise, field of a point negative charge is represented by a group of equally-spaced, unbroken, straight-lines converging on the charge.

Figure 2.1: Electric field-lines generated by a positive charge.

Because electrical forces are superposable, it follows that electric fields are also superposable. In other words, the electric field generated by a collection of electric charges is simply the sum of the fields generated by each of the charges taken in isolation. Suppose that, instead of having a collection of discrete electric charges, we have a continuous distribution of charge represented by an electric charge density $\rho({\bf r})$. Thus, the electric charge at displacement ${\bf r}'$ is $\rho({\bf r}')\,dV'$, where $dV'$ is the volume element at ${\bf r}'$. It follows from a straight-forward extension of Equation (2.11) that the electric field generated by this charge distribution is

$${\bf E}({\bf r}) =\frac{1}{4\pi\,\epsilon_0} \int_{V'} \rho({\bf r}')\, \frac{{\bf r}- {\bf r}' } {\vert{\bf r} - {\bf r}'\vert^3} \,dV',$$ (2.13)

where the volume integral is over a volume, $V'$, that contains all of the charges.