Why does electrical resistivity have units of $\Omega \cdot \mathrm{m}$ rather than $\Omega \cdot \mathrm{m}^3 ?$

Electrical resistivity has units of $$\Omega \cdot \mathrm{m} .$$ However, since resistivity can be described as the resistance of a unit cube, shouldn't the units therefore be $$\Omega \cdot \mathrm{m}^3$$ instead?

I ask after seeing this question to which the answer is apparently $$\left(\text{D}\right) :$$

$$\text{Resistivity can be described correctly as:} \\ \hspace{1em} \begin{array}{cl} \mathbf{A} & \text{resistance of a unit length.} \\ \mathbf{B} & \text{resistance per unit area.} \\ \mathbf{C} & \text{resistance per unit volume.} \\ \mathbf{D} & \text{resistance of a unit cube.} \end{array}$$

• "resistivity is defined as resistance of a unit cube": Absolutely not – Massimo Ortolano May 8 at 19:25
• Sorry I meant described as rather than defined as – daniel May 8 at 19:31
• @Massimo Why "absolutely not"? – my2cts May 8 at 19:46
• @my2cts Because the equality of two numerical values does not imply the equality of two corresponding quantities. Never confuse numerical values with quantities. – Massimo Ortolano May 8 at 19:55
• I like how they included both "resistance of a unit volume" and "resistance of a unit cube" as possible answers. – Nat May 8 at 20:08

The resistance $$R$$ of a body grows with bigger length $$l$$ (a longer wire has greater resistance) and shrinks with bigger cross-section area $$A$$ (a thicker wire has smaller resistance). Hence you have $$R = \rho\frac{l}{A}$$ and resistivity $$\rho$$ must have unit $$\Omega\cdot$$m.

• How does this equation link to the answer which is D (resistance of a unit cube). – daniel May 8 at 19:44
• @daniel For a cube you have $A = l^2$. – Thomas Fritsch May 8 at 19:49

Resitivity can be thought of as resistance of a unit cube, but for a unit cube, the resistance doesn't work out to $$\text{material constant} * \text{volume}$$.

Instead, restivity ($$\rho$$) is given by $$\rho = \frac {RA}{L}$$ (where $$R$$ is resistance, $$A$$ is area and $$L$$ is length of material) or to rearrange in terms of net resistance $$R = \frac {\rho L}{A}$$.

We can see that this suggests that resistance will increase with length, but decrease with area. This should make sense, because if you send the same current through a wider area, it should experience less resistance, and if you push the same current through a longer path of resistance, it should experience more net resistance over the path.

You can see from that relationship that the units will work out to resistance per unit length, not per unit volume.

• "Resitivity is defined as resistance of a unit cube": NO – Massimo Ortolano May 8 at 19:56
• @MassimoOrtolano You're right, I was just using their terminology. – JMac May 8 at 19:57
• @MassimoOrtolano YES. The unit cube must be unit in the system of unit used. Typically that means a cube 1 cm x 1 cm X 1 cm. Nonetheless, the question in question is a terrible question, and none of the answers are particularly good. – garyp May 8 at 20:25
• @garyp As several others around here, you confuse numerical values and physical quantities. It's sad to see physicists make this mistake, but I don't have enough time to fight against this wrong belief. – Massimo Ortolano May 9 at 18:39