There are two cases for which we know what the result should be. First, let x become really small. As the point in Figure 23.2 approaches the origin, the fields $\vec{E}{}_1$ and $\vec{E}{}_3$ become opposite to each other and cancel. Thus as $x\to 0$, the field should be that of the single point charge q at the origin, a field we already know. Is it? Notice that

Thus $E_{\text{net}}\to q\text{/}4\pi {ϵ}_0{x}^2$ as $x\to 0$, the expected field of a single point charge.

Now consider the opposite situation, where x becomes extremely large. From very far away, the three source charges will seem to merge into a single charge of size 3q, just as three very distant lightbulbs appear to be a single light. Thus the field for $x\gg d$ should be that of a point charge 3q. Is it?

The field is zero in the limit $x\to \infty$. That doesn’t tell us much, so we don’t want to go that far away. We simply want x to be very large in comparison to the spacing d between the source charges. If $x\gg d$, then the denominator of the second term of $\vec{E}{}_{\text{net}}$ is well approximated by ${(x^2+d^2)}^{3\text{/}2}\approx {\left(x^2\right)}^{3\text{/}2}=x^3$. Thus

Consequently, the net electric field far from the source charges is

As expected, this is the field of a point charge 3q. These checks of limiting cases provide confidence in the result of the calculation.

FIGURE 23.3 is a graph of the field strength $E_{\text{net}}$ for the three charges of Example 23.1. Although we don’t have any numerical values, we can specify x as a multiple of the charge separation d. Notice how the graph matches the field of a single point charge when $x\ll d$ and matches the field of a point charge 3q when $x\gg d$.

FIGURE 23.3 The electric field strength along a line perpendicular to three equal point charges.