1. **State the problem:** We need to find the resistivity $\rho$ of a wire given its length $L = 500$ m, cross-sectional area $A = 2.6$ mm$^2$, and resistance $R = 5 \Omega$. 2. **Formula used:** The resistance of a wire is related to resistivity by the formula: $$ R = \rho \frac{L}{A} $$ where $R$ is resistance, $\rho$ is resistivity, $L$ is length, and $A$ is cross-sectional area. 3. **Rearrange the formula to solve for resistivity:** $$ \rho = R \frac{A}{L} $$ 4. **Convert units:** Cross-sectional area $A = 2.6$ mm$^2 = 2.6 \times 10^{-6}$ m$^2$ (since $1$ mm$^2 = 10^{-6}$ m$^2$). 5. **Substitute values:** $$ \rho = 5 \times \frac{2.6 \times 10^{-6}}{500} $$ 6. **Calculate:** $$ \rho = 5 \times 5.2 \times 10^{-9} = 2.6 \times 10^{-8} \text{ ohm meters} $$ 7. **Convert to micro-ohm meters ($\mu\Omega m$):** Since $1 \Omega m = 10^{6} \mu\Omega m$, $$ \rho = 2.6 \times 10^{-8} \times 10^{6} = 0.026 \mu\Omega m $$ **Final answer:** The resistivity of the wire is $0.026 \mu\Omega m$.