A wire made of an unknown substance
has a resistance of 125 milliohms. The wire has a length of 1.8 meters
and a cross-sectional area of 2.35 times 10 to the negative fifth meter squared. What is the resistivity of the
substance from which the wire’s made?
Let’s begin by drawing a picture of
our wire. In the diagram, we have labeled the
length of the wire 𝐿 as 1.8 meters, the cross-sectional area of the wire 𝐴 as 2.35
times 10 to the negative fifth meter squared, the resistance of the wire 𝑅 as 125
milliohms, and we are looking for the resistivity of the wire 𝜌. In order to solve for resistivity,
we need an equation that relates the resistance, resistivity, length, and
cross-sectional area of our wire.
The equation that relates these
four variables together is 𝑅, the resistance of the wire, equals 𝜌, the
resistivity, times 𝐿, the length, divided by 𝐴, the cross-sectional area. We are solving for the
resistivity. Therefore, we must rearrange our
formula so that it solves for 𝜌. To do this, we multiply both sides
of the equation by 𝐴 over 𝐿. This will cancel out both the 𝐴
and the 𝐿 on the right side of the equation, leaving us with the relationship 𝐴𝑅
over 𝐿 is equal to 𝜌. We can now substitute in our values
for our variables.
For the cross-sectional area, we
use 2.35 times 10 to the negative fifth meter squared. For the resistance, we use 125
milliohms, and 1.8 meters for the length. We need to be careful with our
units. Right now our resistance is in
milliohms, but it needs to be converted into ohms. Recall that the prefix milli- means
10 to the negative third. 125 milliohms is the same thing as
125 times 10 to the negative third ohms. After we multiply out our numerator
and divide by our denominator, we get 1.63 times 10 to the negative six ohm meters
for our resistivity.
The length in the problem was given
to us to two significant figures. Therefore, we must report our
resistivity to two significant figures. 1.63 times 10 to the negative sixth
ohm meters rounds to 1.6 times 10 to the negative sixth ohm meters. The resistivity of the substance
from which the wire is made is 1.6 times 10 to the negative sixth ohm meters.