Video Transcript
A copper wire with a resistance of 22 milliohms has a length of 6.2 meters. Find the cross-sectional area. Use 1.7 times 10 to the negative eight ohm meters for the resistivity of copper. Give your answer in scientific notation to one decimal place.
Let’s say that this is our copper wire. It has a length, we’ll call 𝐿, of 6.2 meters. And the overall resistance to the flow of charge from one end of the wire to the other is 22 milliohms; we’ll call this 𝑅. Knowing also the resistivity — we’ll refer to it as 𝜌 of the wire — we want to solve for the wire’s cross-sectional area. To help us do that, we can recall the relationship between resistance and resistivity. Resistance 𝑅 is equal to resistivity 𝜌 multiplied by the length of the material 𝐿 under consideration divided by its cross-sectional area.
Note that if we multiply both sides of this equation by 𝐴 divided by 𝑅, on the left-hand side, the resistance 𝑅 cancels out and on the right the cross-sectional area 𝐴 cancels. We find that that cross-sectional area equals the resistivity 𝜌 times the length of the material 𝐿 divided by the resistance of the material 𝑅. If we substitute into this equation our given values for resistivity, length, and resistance, we can see more clearly what the units of this expression will be.
In our numerator, we have ohms times meters times meters. Since we’re calculating an area 𝐴, we would like our final units to be in meters squared. That means that the units of ohms in our numerator must cancel with ohms in our denominator. To make that happen, let’s convert our resistance, which is currently in milliohms, into a number in units of ohms. The conversion between these units is that 1000 milliohms equals one ohm. And so to convert milliohms to ohms, we’ll take the decimal place and we’ll move it one, two, three spots to the left. This is equivalent to dividing by 1000. 22 milliohms then is equal to 0.022 ohms. Note that now we will cancel out units of ohms from our numerator and denominator.
When we calculate 𝐴 and round it to one decimal place, in scientific notation, it’s 4.8 times 10 to the negative six meters squared. This is the cross-sectional area of our copper wire.
