# Question Video: Comparing the Resistivity of Objects Made from Different Materials Physics • 9th Grade

Diagrams (a) and (b) show sections of equal length and equal cross-sectional area of two conducting objects made from two different materials. Both materials have the same density of free electrons and the same temperature. Free electrons move in the conductors and collide with the atoms of their materials repeatedly, making the electrons’ paths appear nearly random. When identical electric fields are applied across the two conductors, they cause the electrons in the conductors to drift in the direction opposite to that of the electric field with a drift velocity 𝑉_d . Which of the following statements correctly describes how the resistivity of the material in diagram (a) compares to the resistivity of the material in diagram (b)? [A] The resistivity of the material in diagram (a) is lower than the resistivity of the material in diagram (b). [B] The resistivity of the material in diagram (a) is equal to the resistivity of the material in diagram (b). [C] The resistivity of the material in diagram (a) is greater than the resistivity of the material in diagram (b).

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### Video Transcript

Diagrams (a) and (b) show sections of equal length and equal cross-sectional area of two conducting objects made from two different materials. Both materials have the same density of free electrons and the same temperature. Free electrons move in the conductors and collide with the atoms of their materials repeatedly, making the electrons’ paths appear nearly random. When identical electric fields are applied across the two conductors, they cause the electrons in the conductors to drift in the direction opposite to that of the electric field with a drift velocity 𝑉 subscript 𝑑. Which of the following statements correctly describes how the resistivity of the material in diagram (a) compares to the resistivity of the material in diagram (b)? (A) The resistivity of the material in diagram (a) is lower than the resistivity of the material in diagram (b). (B) The resistivity of the material in diagram (a) is equal to the resistivity of the material in diagram (b). (C) The resistivity of the material in diagram (a) is greater than the resistivity of the material in diagram (b).

Okay, so we’ve been given these two diagrams, (a) and (b). In each of the diagrams, each one of these black dots represents a single atom of that material. Let’s begin by summarizing what we’re told in the question to make some more space.

We’re told that the sections of the objects shown in each diagram have equal lengths and cross-sectional areas. We’re also told that the material is different in each case but that both materials have the same free electron density, that is, the same number of free electrons in a given volume of the material. Additionally, we know that both materials have the same temperature. Lastly, we’re told that identical electric fields applied across the two conductors cause electrons to drift in the opposite direction to the field with a drift velocity 𝑉 subscript 𝑑.

We can show this in the diagrams as follows. Let’s say that an electric field is applied that is directed to the left. Then, we know that the drift velocity is directed to the right. Now, we can notice that the black dots representing single atoms in diagram (b) are more closely spaced than those in diagram (a). This means that the material shown in diagram (b) has more atoms per unit volume than the material shown in diagram (a).

In each of the conductors, there are free electrons moving and colliding with the atoms of the materials. As a result of the electric field, the electrons are, on the whole, moving to the right with an average rightward velocity 𝑉 subscript 𝑑. However, because of all these stationary atoms in the way, these free electrons, which are drifting to the right on average, are still going to collide with these stationary atoms. Each such collision will change the direction of motion of the electron. For example, we can consider a few free electrons, represented by these blue dots, in the material shown in diagram (a).

Now, we can already see that this electron here is about to collide with this atom. Its direction will change as a result. In fact, extending the paths for each of these electrons, we find that each of them experiences a small number of collisions between one and three as it moves through the section of the material shown in the diagram.

If we now consider three similar free electrons in the material shown in diagram (b), then, extending each of the electrons paths, we can see that each electron experiences more collisions while traversing the same-sized section of material, because there are more stationary atoms in their way. That is, because the material from diagram (b) has more atoms per unit volume than the one from diagram (a), the electrons in the material from diagram (b) experience more collisions per unit volume than those from diagram (a).

Let’s recall that we also know that each material has the same temperature. This means the electrons in each material are moving with the same average speed. The only difference between the electron motion in each case then is that the electrons in the material from diagram (b) experience more collisions and so change direction more frequently. We can see from these sketches that the collisions with stationary atoms are interrupting the motion of the free electrons. That is, these collisions act to provide some resistance to the rightward drift of the electrons.

Let’s now recall that electrons are charged particles. Because these electrons have an average drift velocity directed to the right, that means that in both conductors, there is a net flow of charge from left to right. The rate of flow of electric charge with time is an electric current. So we have a current directed to the right in each of the two conductors. Now, we’ve seen that the collisions between the free electrons and the stationary atoms provide a resistance to the motion of the electrons. Because it’s the rightward motion of the electrons that’s producing the current in each conductor, then these collisions, which are causing resistance to that motion, are providing electrical resistance in each of the two conductors.

Since the free electrons in the conductor shown in diagram (b) experience more collisions for a given section of the material, we can say that these electrons experience more collisions per unit length. And that means that the conductor in diagram (b) has a greater electrical resistance per unit length. A greater resistance per unit length means a greater resistivity.

So we can say then that the material in diagram (b) has a greater resistivity than the material in diagram (a). Or, equivalently, the material in diagram (a) has a lower resistivity than the material in diagram (b). We can see that this statement matches the one given in option (A). So option (A) is the correct choice. The resistivity of the material in diagram (a) is lower than the resistivity of the material in diagram (b).