Which of the following formulas correctly relates the resistivity 𝜌 of a substance to the resistance of an object with a length 𝑙 that is made of the substance if the object has a cross-sectional area 𝐴 and a resistance 𝑅? (A) 𝜌 is equal to 𝑅 times 𝑙 divided by 𝐴. (B) 𝑅 is equal to 𝜌 times 𝑙 divided by 𝐴. (C) 𝑅 is equal to 𝜌 times 𝐴 divided by 𝑙. (D) 𝑅 is equal to 𝜌 times 𝐴 times 𝑙.
Each of these four equations purports to be a mathematically correct way of relating the resistivity of a substance 𝜌 to the resistance of an object made with that substance 𝑅. Even if we don’t recall offhand the relationship that connects these quantities, we’ll still be able to work toward our answer by recognizing that for any of these equations to be true, the units on the left-hand side must equal those on the right-hand side.
Concerning units, let’s recall that in the SI system, the units of electrical resistance are ohms. So for example, if we had a wire of this length and we were to pass current through that wire, there would be some amount of electrical resistance for this wire and we record that in ohms. But then imagine this. Say that we have a wire of the same material and the same cross section as the one before, but just that now its length is shorter. If we once again pass current through this smaller section of wire, we will get a resistance, but it will be smaller than the resistance along the length of this wire. This tells us that resistance is a quantity that doesn’t just depend on the material an object is made of, but also on that object’s dimensions.
In contrast to this is the resistivity 𝜌 of a material. For two objects made of the same material, like these two wires, the resistivity of the one object is the same as the resistivity of the other. Unlike resistance, resistivity does not depend on the dimensions of an object, only on the material that it’s made of. We can see both the similarity of resistivity to resistance and the difference between these two by looking at the units of resistivity. Both resistivity and resistance involve units of ohms. But in the resistivity, it’s this extra, we could call it dimension of length, that makes it so that the resistivity of this wire is the same as this one even though the two wires have different lengths.
Knowing the units of these two quantities will help us choose which of our answer options is correct. Note that if we stay within the SI system, then the base units of length will be meters, while the units of area will be meters squared. Knowing all this, let’s now study the units on either side of these equations, starting with answer option (A). For this equation to be correct, it must be the case that the units of resistivity, ohms times meters, equal the units of resistance, ohms, times the unit of length, meters, divided by the unit of area, meters squared. We see though that on the right-hand side, one factor of meters cancels from numerator and denominator, leaving us with the overall units of ohms divided by meters. This equality does not hold then. So we won’t choose answer option (A).
Moving on to answer option (B), here on the left we have resistance, which has units of ohms. And this is claimed to be equal to the units of resistivity, ohms times meters, times the units of length, meters, all divided by meters squared, units of area. In this case, we see that the two factors of meters in our numerator cancel out with the two factors of meters in our denominator. The simplified units on the right-hand side then are ohms. This tells us that for answer option (B), the units on either side of the equation agree with one another.
To confirm that this is our correct answer, let’s move on to options (C) and (D). In answer choice (C), on the left-hand side we have units of ohms, those of resistance, and these are said to be equal to ohms times meters multiplied by meters squared all divided by meters. On the right-hand side of this expression, one factor of meters cancels out. That leaves us with ohms times meters squared, which we know is not equal to ohms. The units in answer choice (C) do not agree with one another.
Lastly, we’ll look at answer option (D). Here, resistance with units of ohms is said to be equal to resistivity with units of ohms times meters multiplied by area with units of meters squared multiplied by length with units of meters. All together, this right-hand side has units of ohms times meters to the fourth. This is certainly different from units of ohms. We eliminate answer choice (D) as well. This confirms our answer choice as option (B). This is the only formula that correctly relates the resistivity of a substance to the resistance of an object with a length 𝑙 that is made of that substance if the object has a cross-sectional area 𝐴 and resistance 𝑅.