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).
