A certain metal has a resistivity of $1.68 \times 10 ^ { - 8 } \Omega \cdot \mathrm { m }$. You have a long spool of wire made from this metal. If this wire has a diameter of $0.15 \mathrm {~mm}$, how long should you cut a segment so its resistance will be $15 \Omega$ ?

Question

Quizplus · Accepted Answer

The resistance $R$ of a wire can be calculated using the formula $R = \rho \frac{L}{A}$, where $\rho$ is the resistivity of the material, $L$ is the length of the wire, and $A$ is the cross-sectional area of the wire. The area $A$ of a wire with a circular cross-section is given by $A = \pi r^2$, where $r$ is the radius of the wire. Given the diameter is $0.15 \, \text{mm}$, the radius $r$ is $0.075 \, \text{mm}$ or $0.075 \times 10^{-3} \, \text{m}$. Substituting the given values and solving for $L$ yields a length of approximately $16 \, \text{m}$ to achieve a resistance of $15 \, \Omega$.

A Certain Metal Has a Resistivity Of 1.68×10−8Ω⋅m1.68 \times 10 ^ { - 8 } \Omega \cdot \mathrm { m }1.68×10−8Ω⋅m

A Certain Metal Has a Resistivity Of $1.68 \times 10 ^ { - 8 } \Omega \cdot \mathrm { m }$