In the previous page of notes, we showed that there must be some charges along the surface of a wire and they must be arranged in a gradient. We call these charges along the surface of the wire surface charges. The surface charges create an electric field in the wire, which then pushes electrons through the wire to create a current. In this page of notes, we will dig a little deeper into what a gradient of surface charges actually is and how we represent that gradient on a drawing.

Color gradient from dark blue to white.

It would be impossible to actually count or draw every single electron along the surface of a wire. There are simply too many, and it would make our picture too messy to be useful. Instead, we draw a small representative amount of charges on our picture. For example, if we go back to the picture with the battery and surface charges:

Simple circuit with surface charges and electric field drawn.

There are a few things to notice:

1. We have drawn a small amount of charges on the surface of the wire. For example, in the top right corner, we have drawn a total of eight negative signs on the outside of the wire. This does not mean that there is exactly eight electrons on the wire. Instead, this just shows that we expect this area of the wire to have a large concentration of negative charge. There is no particular reason why we drew eight - we could have picked 12 or 6 or any other number to start with. The important part is that there are more negative signs in this area than any other part of the wire.
2. As we move along the wire on the right side, the number of negative signs decreases from 8 to 6 to 4 to 2. This represents the gradient of surface charges. As we move along the wire, we expect there to be a lot of negative surface charges near the battery, and as you get further away along the wire, there should be fewer and fewer negative surface charges. Remember, the change in the amount of charge is what makes it a gradient. Similarly, as we approach the positive side of the battery, we draw 2 to 4 to 6 to 8 positive signs to represent the increase in the amount of positive surface charges. Note that there is an area between the negative and positive signs where we have zero surface charges. Together, this means our gradient is continuous along the whole wire (it goes from negative to zero to positive). Again, there is no particular reason why we jumped by 2's in the gradient - we could have jumped by 3's or 4's or 1's - but it is important that the jump is consistent (at least for the same material/size/shape of wire).