Problem 19 - Fundamentals of Physics

Question

What is the resistivity of a wire of \(1.0 \mathrm{~mm}\) diameter, \(2.0 \mathrm{~m}\) length, and \(50 \mathrm{~m} \Omega\) resistance?

Step-by-Step Solution

Answer

The resistivity is approximately \(1.96 \times 10^{-8} \, \Omega\cdot \text{m}\).

Step 1: Identify the Given Values

We are given the wire's diameter as \(1.0\, \text{mm}\), the length as \(2.0\, \text{m}\), and the resistance as \(50\,\text{m}\Omega\).

Step 2: Convert Units if Necessary

The diameter of the wire is given in millimeters. Convert it to meters: \(1.0\, \text{mm} = 0.001\, \text{m}\).

Step 3: Calculate the Cross-Sectional Area

The formula for the cross-sectional area \(A\) of a wire is \(A = \pi \left( \frac{d}{2} \right)^2\), where \(d\) is the diameter. Substitute \(d = 0.001\, \text{m}\) into the formula: \[ A = \pi \left(\frac{0.001}{2}\right)^2 = \pi \times (0.0005)^2 = \pi \times 0.00000025 \, \text{m}^2 \].

Step 4: Use the Resistivity Formula

The resistivity formula is \(R = \rho \frac{L}{A}\), where \(R\) is the resistance, \(\rho\) is the resistivity, \(L\) is the length, and \(A\) is the cross-sectional area. Rearrange for resistivity: \[ \rho = R \frac{A}{L} \].

Step 5: Substitute Values into the Formula

Substitute the known values into the formula from the previous step: \[ \rho = 50 \times 10^{-3} \Omega \times \frac{\pi \times 0.00000025\, \text{m}^2}{2\, \text{m}} \].

Step 6: Calculate the Resistivity

Perform the calculation: \[ \rho = 0.05 \times \frac{\pi \times 0.00000025}{2} = 0.05 \times 0.000000125\pi \approx 1.96 \times 10^{-8} \, \Omega\cdot \text{m} \].

Key Concepts

Cross-sectional Area of a Wire

The cross-sectional area of a wire is an important concept when dealing with problems that involve electrical properties, like resistivity. Think of a wire as a long cylinder, and the cross-sectional area is essentially the area of the circle that forms the face of this cylinder. To find the cross-sectional area, use the formula:

• A = π (d/2)²

where d is the diameter of the wire.
For example, if you have a wire with a diameter of 1.0 mm, you first convert this to meters, getting 0.001 m.
Then, substitute this value into the formula to find the area:

• A = π (0.0005)² = π × 0.00000025 m²

By understanding this process, you can easily determine the cross-sectional area for wires of any diameter, which is a crucial step in calculating other properties like resistivity.

Resistivity Formula Simplified

Resistivity is a fundamental property that dictates how strongly a material opposes the flow of electric current. Knowing how to work with the resistivity formula is essential. The main formula for resistivity is:

• R = ρ (L/A)

where R is resistance, ρ is resistivity, L is the length, and A is the cross-sectional area.
If you need to find resistivity (ρ), you rearrange this formula:

• ρ = R (A/L)

Using this formula effectively requires knowing precise values for resistance, length, and the cross-sectional area.
In cases where you know the wire's resistance, length, and diameter, you first find the cross-sectional area (as explained earlier) and then substitute all values into the formula to get the resistivity.
This process helps find out how much the wire's material restricts electrical current, which is key in designing and analyzing electrical circuits.

Importance of Unit Conversion

Unit conversion is a vital skill when working with physics problems, especially those involving resistivity. Often measurements are given in non-standard units, such as millimeters or milliohms, which need to be converted to standard units like meters or ohms.

• Always convert diameter from millimeters to meters, since resistivity calculations require the area in square meters.
• Convert resistance from milliohms to ohms, where 1 mΩ = 0.001 Ω.

These conversions ensure calculations are accurate and consistent with scientific standards.
For example, in the step-by-step solution provided, the diameter of 1.0 mm was converted to 0.001 m, and the resistance of 50 mΩ was converted to 0.05 Ω. Each conversion is essential for ensuring the final resistivity calculation is correct.
Completing accurate conversions from the beginning can save time and prevent errors, facilitating more efficient and reliable problem-solving in physics.