Video Transcript
The diagram shows two identical circuits. Each circuit’s conducting wire has had a section of it greatly magnified to show the ions that the conducting wire is comprised of and the free electrons that move between these ions. The conducting wires are both made of the same substance. Which of the following statements correctly states how the resistivities of the sections of conducting wire in diagram (a) and diagram (b) compare to each other? (A) The resistivity of the section in diagram (a) is greater than the resistivity of the section in diagram (b). (B) The resistivity of the section in diagram (b) is greater than the resistivity of the section in diagram (a). (C) The resistivities are the same for both sections.
In our two diagrams (a) and (b), we’re told that we have identical circuits. Everything about these circuits is the same, but they don’t look the same because we have this larger magnified section in diagram (a) than in diagram (b). We can keep in mind though that this difference is just because we’re magnifying a larger region of wire in diagram (a) than we are in diagram (b). The total length of wire in the circuit in both cases is the same. In these magnified sections then, we’re looking at two unequal lengths of identical wire.
In this part of our question, we want to relate the resistivity of this section in diagram (a) to that in diagram (b). We can be helped in doing this by recalling that resistivity is not the same as electrical resistance. The resistance of a wire, for example, increases the longer the wire gets, while that wire’s resistivity, so long as it’s always made of the same material, stays constant. Resistivity, we could say, is a length corrected measurement of the opposition of a given material to the flow of charge.
In our problem statement, we’re told that the wires in both of these circuits are made of the same material. Therefore, the resistivity of the wire in diagram (a) is the same as that in diagram (b). And because the resistivity of a substance is independent of the amount of substance being considered, the fact that the highlighted section in diagram (a) is longer than that in diagram (b) doesn’t change the fact that the resistivity of these sections of conducting wire are equal. Because the wires in both circuits are the same, the resistivities in both circuits are the same. This is true even when we consider unequal lengths of wire across the two circuits. We choose answer option (C): the resistivities are the same for both sections.
Let’s look now at part two of this question.
Which of the following statements correctly states how the cross-sectional areas of the conducting wires compare to each other? (A) The cross-sectional area of the wire in diagram (a) is greater than the cross-sectional area of the wire in diagram (b). (B) The cross-sectional area of the wire in diagram (b) is greater than the cross-sectional area of the wire in diagram (a). (C) The cross-sectional areas of the wires are the same.
The key to answering this part of our question is to recall that these two circuits depicted in our diagrams are identical. Not only are the wires in each circuit made of the same substance, but the circuits are the same in every way. Therefore, even though the amount of wire section highlighted in diagram (a) is greater than the amount of wire section highlighted in diagram (b), the cross-sectional areas of those wires, coming as they do from identical circuits, are the same. We choose answer option (C).
Let’s look now at the next part of our question.
Which of the following statements correctly states how the average time taken for a free electron to cross from one side of the section to the other side in diagram (a) compares to the average time taken for a free electron to cross from one side of the section to the other in diagram (b)? (A) The time taken for the section in diagram (a) is greater. (B) The time taken for the section in diagram (b) is greater. (C) The time taken is the same for both sections.
As we’ve seen, the magnified section in diagram (a) is larger than the magnified section in diagram (b). We can tell this by the fact that each of these large red dots represents a positively charged ion, while each of the smaller blue dots represents a free electron. We can see that in each row in the section in diagram (a), there are four ions, while each row in the section in diagram (b) has just two. Physically then, the side-to-side distance across the section in diagram (a) is larger than that in diagram (b).
This means that when a free electron crosses these two sections, either the section in diagram (a) or the section in diagram (b), since the free electrons will have the same average speed as the two circuits are identical, an electron will tend to take longer to cross the section in diagram (a) because that section is longer. That is, the free electron has to cover more distance to complete this section than it does to cross the section in diagram (b). On average then, the time taken for the free electron to cross the section in diagram (a) is greater. We choose answer option (A).
Let’s look now at the last part of our question.
Which of the following statements correctly states how the resistance of the section of conducting wire in diagram (a) compares to the resistance of the section of conducting wire in diagram (b)? (A) The resistance of the section in diagram (a) is greater. (B) The resistance of the section in diagram (b) is greater. (C) The resistance is the same for both sections.
Unlike resistivity, the resistance of a section of wire does depend on that section’s length. For example, if we had two wires made of the same material and each having the same cross-sectional area, while the resistivities of these two wires will be the same, their resistances are not. The longer wire, the one requiring electrons to move through a greater distance, has the greater resistance. The same thing holds true here for our two sections in our diagrams (a) and (b). Because the magnified section in diagram (a) is larger — that is, it consists of four ions left to right rather than the two in the section in diagram (b) — the overall electrical resistance of this longer section will be greater.
For our answer then, we choose option (A). The resistance of the section in diagram (a) is greater.
