# Can we depict the electric field lines between three or more charges?

Last year we studied in school electric field that was created around charges and we were showed how to depict the electric field lines for one charge or between two charges. Then i wondered how would it be like between three (or more) charges so i asked. However the answer was rather startling. He said we cant because its too complicated! I wondered..we have so powerful computers and we cant? But i accepted it.

However last night i started thinking about it (don't know why) and it bothered me. So i checked it online and for a long time all i got was two charges till i found a wolfram demonstration. So my question is : Is it possible to depict the electric field lines between three or more charges? Is this demonstration accurate?

• Sure it's numerically possible to get the field of arbitrary charge configuration, just perhaps not analytically. – ACuriousMind Mar 17 '15 at 14:41

An electric charge has an electrostatic potential associated with it. This potential is a scalar, so it has no direction associated with it, and its value at any distance $r$ from the charge is simply:

$$V = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r}$$

If we graph the potential created by the charge as a 2D plot then we get something like this:

and if we plot this as a contour map it looks something like:

(all my own work this time)

The field lines are just the lines at right angles to the contours. I've drawn a few sample field lines, and it should be obvious that for a single charge the field lines are just radially outwards/inwards (outwards if it's a positive charge and inwards if it's a negative charge).

Now, the cool thing about potentials is that if you have more than one charge all the potentials just add up. So at a point that is a distance $r_a$ from a charge $Q_a$, $r_b$ from a charge $Q_b$ and $r_c$ from a charge $Q_c$, the potential is just:

$$V = \frac{1}{4\pi\varepsilon_0}\left(\frac{Q_a}{r_a} + \frac{Q_b}{r_b} + \frac{Q_c}{r_c}\right)$$

remembering that the $Q$s can be positive or negative so the potentials can add or subtract.

If you have a complicated collection of charges then the potential can get a bit involved to calculate, but at the end of the day we can draw a contour map and as above the field lines are just the lines at right angles to the contours. So they really aren't that big a deal to calculate. Mathematically, we just take the gradient of the potential field.

But, your teacher might have a point if they are asking for the equation that describes a field line i.e. what is the equation for the $(x, y)$ curve that follows the field line. For all but symmetric arrangements of charges there isn't going to be a simple equation and we'd have to be satisfied with calculating it numerically on a computer. But you wouldn't need a very powerful computer. You could probably do it on your phone.