The resistivity of a wire is given by R = pl/A. Where p is the resistivity of the material, L is the length of the wire and A is the cross sectional area of the wire. a) If the diameter of the wire is .35mm, calculate the cross sectional area of the wire.
The Correct Answer and Explanation is :
To calculate the cross-sectional area ( A ) of a wire given its diameter, we can use the formula for the area of a circle:
[
A = \pi r^2
]
where ( r ) is the radius of the wire. The radius is half the diameter. Given that the diameter of the wire is ( 0.35 \, \text{mm} ), we can first convert this to meters to be consistent with SI units:
[
\text{Diameter} = 0.35 \, \text{mm} = 0.35 \times 10^{-3} \, \text{m} = 3.5 \times 10^{-4} \, \text{m}
]
Now, the radius ( r ) can be calculated as:
[
r = \frac{\text{Diameter}}{2} = \frac{3.5 \times 10^{-4} \, \text{m}}{2} = 1.75 \times 10^{-4} \, \text{m}
]
Next, we can substitute this radius into the area formula:
[
A = \pi (1.75 \times 10^{-4} \, \text{m})^2
]
Calculating this gives:
[
A \approx \pi (3.0625 \times 10^{-8}) \approx 9.621 \times 10^{-8} \, \text{m}^2
]
Final Answer
The cross-sectional area ( A ) of the wire is approximately ( 9.62 \times 10^{-8} \, \text{m}^2 ).
Explanation
The resistivity formula ( R = \frac{pL}{A} ) shows that the resistance ( R ) of a wire depends on its material (represented by resistivity ( p )), its length ( L ), and its cross-sectional area ( A ). In this scenario, we focused on calculating the cross-sectional area, which is a critical factor in determining the wire’s electrical resistance.
By knowing the diameter of the wire, we first converted it into a radius, as the formula for the area of a circle requires the radius. The diameter was provided in millimeters, so it was essential to convert this into meters to ensure the area is expressed in square meters, aligning with SI unit conventions.
The final area calculation provides insight into the wire’s ability to conduct electricity: a larger cross-sectional area typically leads to lower resistance, making the wire more efficient for transmitting electrical current. This principle is crucial in electrical engineering and physics, as it allows for the design and selection of wires for various applications, balancing factors like current capacity and heating effects.