Video Transcript
Diagrams (a) and (b) show identical sections of two conducting objects made from the
same material. The temperature of the material in diagram (b) is much higher than the temperature of
the material in diagram (a). The free electrons move in a conductor and collide with the atoms of the materials
repeatedly, making the electronsโ paths appear nearly random. When an electric field is applied across the conductor, it causes the electrons to
drift in the direction opposite to that of the electric field with a drift velocity
๐ sub 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 greater than the resistivity of
the material in diagram (b). (C) The resistivity of the material in diagram (a) is equal to the resistivity of the
material in diagram (b).
We want to determine in which diagram the resistivity is either greater or equal. Letโs first remind ourselves what resistivity is.
For an object with resistance ๐
, a cross-sectional area ๐ด, and a length ๐, the
resistivity ๐ is given by ๐ equals ๐
times ๐ด all divided by ๐. Considering the units in this expression, the SI unit of the quantity ๐ด divided by
๐ is given by meters squared over meters, which equals meters. Hence, the SI unit of resistivity is given by ohm meters.
Along with resistivity, our question talks about electrons drifting through a
conductor. The speed at which electrons drift is defined by this equation for current. Current ๐ผ equals free electron density ๐ times electron charge ๐ times the
conductorโs cross-sectional area ๐ด multiplied by the electron drift speed ๐ sub
d. As a final equation, letโs connect current ๐ผ with resistance ๐
through the
relationship known as Ohmโs law. Here, ๐ผ stands for current, ๐ for voltage, and ๐
for resistance.
Knowing all this, letโs draw some free electrons on both diagrams (a) and (b). We are told that the temperature of the material in diagram (b) is much higher than
the temperature of the material in diagram (a) and that they are made from the same
material. At a higher temperature, an ion in the conductor will tend to undergo greater
displacement from its average position than at a lower temperature. Therefore, the range of possible positions of the ions increases. We can see this from the arrows coming from the ions in diagram (b) being larger than
those in diagram (a). This means that collisions between ions and electrons are more likely. The more collisions between ions and electrons, the more the current in the conductor
is reduced and, therefore, the slower the electron drift velocity and the greater
the resistance of the conductor.
We can say that the electrons have less space to move between the ions because the
range of positions of the ions increases with increasing temperature. Notice in our equation for resistivity that if resistance increases, as it does in
this case of increased temperature, and conductor dimensions remain the same,
resistivity must increase.
Therefore, the material in diagram (b), which has a higher temperature, also has a
higher resistance and resistivity. This shows us that the correct answer is option (A). The resistivity of the material in diagram (a) is lower than the resistivity of the
material in diagram (b).
