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--- 184_notes:gradient [2020/09/22 16:34] – dmcpadden
+++ 184_notes:gradient [2021/02/23 20:18] (current) – bartonmo
@@ Line 15: / Line 15: @@
   - 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.
   -  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).
-  - The last thing to notice here is the electric field arrows. We said already that the gradient is what creates the electric field field. **This means that the electric field arrows need to match your surface charge gradient and vice versa**. Remember from Week 1, we know the electric field should point away from positive charges and toward negative charges. This is still true here. We also see the electric field follows the charge gradient. On the left side of the wire, we see the electric field arrows point from areas with lots of positive charge toward areas with small amounts of positive charge. On the right side of the wire, we see the electric field arrows point from areas with small amounts of negative charge towards areas with large amounts of negative charge. Thus, **because the gradient is constantly changing, we get a constant electric field in the wire**. There is no particular reason why we drew the size of the arrows to be roughly 0.5 cm - we could have picked 1 cm arrows or 0.25 cm arrows. The important part is that the electric field arrows point in the direction that matches the gradient (from positive to negative) and that they are all the same size (the electric field is constant). The final note here is your electric field arrows should roughly match your gradient. For example, if your gradient is jumping by 4's, you should draw large electric field arrows since a large gradient creates a large electric field. If your gradient is jumping by 1's, you should draw small electric field arrows since a small gradient creates a small electric field. (This can be especially important if you have wires with different materials or if you have different sizes of wire.)
+  - The last thing to notice here is the electric field arrows. We said already that the gradient is what creates the electric field field. **This means that the electric field arrows need to match your surface charge gradient and vice versa**. Remember from [[184_notes:pc_efield|Week 1]], we know the electric field should point away from positive charges and toward negative charges. This is still true here. We also see the electric field follows the charge gradient. On the left side of the wire, we see the electric field arrows point from areas with lots of positive charge toward areas with small amounts of positive charge. On the right side of the wire, we see the electric field arrows point from areas with small amounts of negative charge towards areas with large amounts of negative charge. Thus, **because the gradient is constantly changing, we get a constant electric field in the wire**. These arrows aren't necessarily to scale, the important part is that the electric field arrows point in the direction that matches the gradient (from positive to negative) and that they are all the same size (the electric field is constant). The final note here is your electric field arrows should roughly match your gradient. For example, if your gradient is jumping by 4's, you should draw large electric field arrows since a large gradient creates a large electric field. If your gradient is jumping by 1's, you should draw small electric field arrows since a small gradient creates a small electric field. (This can be especially important if you have wires with different materials or if you have different sizes of wire.)
 In the next page of notes, we'll talk more about how the surface charge gradient and electric field create a current in the wire.