The bulk resistivity of copper is 1.6 uOhms-cm. What is the resistance between opposite faces of a cube of copper 1 cm on a side? What if it is 10 cm on a side?
The correct answer and explanation is :
Correct Answer:
The resistance $R$ of a material is given by the formula:
$$
R = \rho \cdot \frac{L}{A}
$$
Where:
- $R$ = resistance (Ohms)
- $\rho$ = resistivity (Ohm·cm)
- $L$ = length of the material current flows through (cm)
- $A$ = cross-sectional area perpendicular to the current (cm²)
Case 1: Cube of Copper – 1 cm on a side
- $\rho = 1.6 \, \mu\Omega\cdot\text{cm} = 1.6 \times 10^{-6} \, \Omega\cdot\text{cm}$
- $L = 1 \, \text{cm}$
- $A = 1 \, \text{cm}^2$
$$
R = (1.6 \times 10^{-6}) \cdot \frac{1}{1} = 1.6 \times 10^{-6} \, \Omega
$$
Case 2: Cube of Copper – 10 cm on a side
- $\rho = 1.6 \times 10^{-6} \, \Omega\cdot\text{cm}$
- $L = 10 \, \text{cm}$
- $A = 10^2 = 100 \, \text{cm}^2$
$$
R = (1.6 \times 10^{-6}) \cdot \frac{10}{100} = 1.6 \times 10^{-7} \, \Omega
$$
Explanation (Approx. 300 Words):
The resistance of a material depends on three factors: its resistivity ($\rho$), the length through which current flows ($L$), and the cross-sectional area ($A$) of the material. Resistivity is a property intrinsic to the material—in this case, copper—which indicates how strongly the material resists the flow of electric current. For copper, this resistivity is extremely low, which makes it an excellent conductor.
In this problem, we are given the resistivity of copper as $1.6 \, \mu\Omega\cdot\text{cm}$ or $1.6 \times 10^{-6} \, \Omega\cdot\text{cm}$, and we must compute the resistance across opposite faces of cubes with two different sizes: 1 cm and 10 cm on a side.
For the 1 cm cube, the current flows through a length of 1 cm, and the cross-sectional area is also 1 cm². Plugging these into the formula, we get a resistance of $1.6 \times 10^{-6} \, \Omega$, which is extremely small, indicating minimal opposition to current.
For the 10 cm cube, the length increases to 10 cm, but the area increases to 100 cm². This increase in area more than compensates for the increase in length, resulting in a lower resistance of $1.6 \times 10^{-7} \, \Omega$. This demonstrates a key principle: while increasing the length increases resistance, increasing the cross-sectional area decreases it more rapidly, which is why thicker wires have less resistance.