Week1SectionProblems
Wednesday, February 4, 2015
12:01 AM
Section Answers Page 1
You can actually draw these equipotential lines as concentric and evenly spaced circles if you'd like. However, if you were to do this, my point is that the potential difference between the lines would fall off proportionally to 1/r . . . So the gaps between successive lines would represent DIFFERENT potential differences from one line to the next. This is why people often draw equipotential surfaces / lines as closer together near the point charge and spreading out farther away from the point charge. I have included a plot at the end of this set of notes from the online lab environment ‐‐ note how the potential differences keep changing when you look at the numbers even though the lines are evenly spaced. This is why showing evenly spaced concentric circles doesn't tend to represent the nature of a potential mountain (for +q) or a potential well (for ‐q) around a point charge very well. Since the charge is negative, potential is maximally negative near the charge and it approaches zero as you move farther away from the charge (so it can be said that "potential increases" as you move away from the negative charge). Equipotential lines NEVER cross one another. Think about what it would mean if two equipotential lines did intersect . . . It would mean that two different potentials had the same value ‐‐ so it would be like saying that 30 V = 40 V, which we know cannot be true. So these equipotential lines NEVER cross one another. See pg. 15 of the annotated class notes for an example of equipotential lines around like charges; see pg. 16 of the annotated class notes for an example of equipotential lines around unlike charges. It is also interesting that there are points of zero potential between like and unlike charges! Net force on a positive test charge situated at point A between like charges is equal to zero. However, net force on a positive (or negative) test charge at point A between unlike charges is not equal to zero. Section Answers Page 2
Remember: Add vectors tip‐to‐tail so that you are adding the tail of the second vector to the tip
Of the first. No! There is actually no point (that is not infinite distance away from both of these charges) where the net force is equal to zero. As you can see in the four examples (above) of where we could situate at third charge (Q3), there will always be some net force for any test charge (which is positive by convention but could certainly be negative) where there are unlike charges. If the test charge (Q3) were negative, we would do the same exercise and find a net force vector, but a negative test charge would be attracted to the positive charge (Q1) and repelled by the negative charge (Q2). Note how the existence of a net force and the existence of a zero potential point do NOT actually have to go together. Recall that we saw (on the previous page) that there could be a zero potential point between unlike charges but there would still be a net force acting on a test charge. Section Answers Page 3
Finally, it is important to note something that may seem quite obvious in view of this example ‐‐ i.e., there does NOT need to be a point charge situated at point A in order for there to be an electric potential at point A. There needs to be only a single charge anywhere for there to be an electric potential at any other point in space ‐‐ i.e., there doesn't even need to be a second charge anywhere. Remember, V = (kQ)/r . . . So for there to be an electric potential, there needs to be only a single point charge. However, where there are a plurality of point charges we use the principle of superposition as we have in this example to consider a plurality of point charges from a vantage point (point A in this example). Voltages are scalar values ‐‐ so adding them in the principle of superposition as scalars is just fine. It may help to think about the "principle of superposition" as a way to account for a plurality of charges. Section Answers Page 4
Plotting equipotential surfaces (lines)
Wednesday, February 11, 2015
10:32 AM
Note how the gap between successive concentric lines allows for a delta‐V (potential difference) that KEEPS CHANGING (getting smaller, asymptotically approaching zero). So there is a rapid falloff at first. This is why using concentric and evenly spaced circles is perhaps not the best way to represent equipotential surfaces because even though it can be done ‐‐ it doesn't represent the nature of the potential mountain (for +q) or well (for ‐q) very well at least in terms of showing how potential falls off over distance. Section Answers Page 5