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Magazine Chapter 20: Problem 11
A cylindrical copper cable carries a current of \(1200 \mathrm{~A}\). There is a potential difference of \(1.6 \times 10^{-2} \mathrm{~V}\) between two points on the cable that are \(0.24 \mathrm{~m}\) apart. What is the radius of the cable?
Short Answer
Expert verified
Radius is approximately \(9.8 \times 10^{-3} \mathrm{~m}\).
Step by step solution
01
Understand the Given Values
We are given the current \(I = 1200 \mathrm{~A}\), the potential difference \(V = 1.6 \times 10^{-2} \mathrm{~V}\), and the length \(L = 0.24 \mathrm{~m}\). We need to find the radius of the cylindrical cable.
02
Use Ohm鈥檚 Law
Ohm's Law relates the potential difference \(V\), current \(I\), and resistance \(R\) by the formula \(V = IR\). We can rearrange this to find the resistance: \[ R = \frac{V}{I} = \frac{1.6 \times 10^{-2}}{1200} \mathrm{~\Omega}. \]
03
Calculate the Resistance
Compute the resistance \(R\):\[ R = \frac{1.6 \times 10^{-2}}{1200} = 1.333 \times 10^{-5} \mathrm{~\Omega}. \]
04
Use the Resistance Formula for a Cylinder
The resistance of a cylindrical wire is given by \[ R = \frac{\rho L}{A}, \] where \(\rho\) is the resistivity of copper (\(1.68 \times 10^{-8} \mathrm{~\Omega \cdot m}\)) and \(A\) is the cross-sectional area of the wire.
05
Solve for the Cross-Sectional Area
Rearrange to find the area \(A\):\[ A = \frac{\rho L}{R} = \frac{1.68 \times 10^{-8} \times 0.24}{1.333 \times 10^{-5}} \mathrm{~m^2}. \]
06
Calculate the Area
Substitute and compute:\[ A = \frac{1.68 \times 10^{-8} \times 0.24}{1.333 \times 10^{-5}} = 3.024 \times 10^{-4} \mathrm{~m^2}. \]
07
Find the Radius from the Area
The area \(A\) of a circle is given by \(A = \pi r^2\). Solve for the radius \(r\):\[ r = \sqrt{\frac{A}{\pi}} = \sqrt{\frac{3.024 \times 10^{-4}}{\pi}} \mathrm{~m}. \]
08
Calculate the Radius
Compute the radius \(r\):\[ r = \sqrt{\frac{3.024 \times 10^{-4}}{\pi}} \approx 9.8 \times 10^{-3} \mathrm{~m}. \]
09
Conclusion: Final Answer
The radius of the cable is approximately \(9.8 \times 10^{-3} \mathrm{~m}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Resistance
Resistance is a property of an electrical component that opposes the flow of electric current. It is measured in Ohms (\(\Omega\)) and plays a crucial role in determining how much current flows through a conductor for a given voltage. In the context of a cylindrical copper cable, resistance is influenced by several factors.
These factors include:
These factors include:
- The material: Different materials have different inherent resistances. Copper is commonly used for cables due to its low resistivity.
- Length: The longer the cable, the higher the resistance. This is because electrons need to travel a greater distance, encountering more collisions with atoms.
- Cross-sectional Area: A larger cross-sectional area allows more paths for the current to pass through, resulting in lower resistance.
Ohm's Law
Ohm's Law is a fundamental principle in electronics and electrical engineering. It establishes a linear relationship between voltage (\(V\)), current (\(I\)), and resistance (\(R\)) through the formula:\[ V = IR \]This equation means that the voltage across a conductor is equal to the product of the current flowing through it and the resistance of the conductor.
Ohm's Law is particularly useful because:
Ohm's Law is particularly useful because:
- It provides a simple way to calculate one of the three parameters if the other two are known.
- It helps in analyzing circuits to predict how changes in voltage or resistance will affect current flow.
- It forms a foundational basis for more complex circuit analysis techniques and is essential for both theoretical and practical electrical engineering tasks.
Resistivity of Copper
Resistivity is a material-specific property that quantifies how strongly a material opposes the flow of electric current. It is often denoted by \(\rho\) and measured in Ohm-meters (\(\Omega \cdot m\)). Lower resistivity indicates better conductivity. Copper is well-known for having a very low resistivity of \(1.68 \times 10^{-8} \Omega \cdot m\), which makes it an excellent choice for electrical conductors.
Key points about resistivity:
Understanding resistivity is crucial when selecting materials for electrical wiring. It ensures that energy loss through heat is minimized while maintaining efficient power distribution.
Key points about resistivity:
- It depends on the material and its temperature鈥攎ost conductive materials increase in resistivity with higher temperatures.
- The resistivity formula for a wire takes into account the wire's length and cross-sectional area:
Understanding resistivity is crucial when selecting materials for electrical wiring. It ensures that energy loss through heat is minimized while maintaining efficient power distribution.
