Question

A potential difference of \(12.0 \mathrm{~V}\) is applied across a wire of cross. sectional area \(4.50 \mathrm{~mm}^{2}\) and length \(1000 . \mathrm{km}\). The current passing through the wire is \(3.20 \cdot 10^{-3} \mathrm{~A}\) a) What is the resistance of the wire? b) What type of wire is this?

Short answer

Answer: The approximate resistance of the wire is 3750 Ω (ohms), and it is likely made of copper.

Step-by-step solution

Apply Ohm's Law to find resistance

Ohm's Law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance between them. Mathematically, this can be represented as: \(V = IR\) where \(V\) is the potential difference (voltage) in volts, \(I\) is the current in amperes, and \(R\) is the resistance in ohms. We need to solve for \(R\): \(R = \frac{V}{I}\)

Substitute given values and find resistance

Now we can substitute the given values into the equation from step 1: \(R = \frac{12.0 \mathrm{\,V}}{3.20 \cdot 10^{-3} \mathrm{\,A}}\) Calculate the value of \(R\): \(R \approx 3750 \,\Omega\) So, the resistance of the wire is approximately \(3750\, \Omega\).

Use the resistance formula to find resistivity

The formula for the resistance of a wire is given by: \(R = \frac{\rho L}{A}\) where \(R\) is the resistance, \(\rho\) is the resistivity, \(L\) is the length of the wire, and \(A\) is the cross-sectional area. We need to solve for \(\rho\): \(\rho = \frac{RA}{L}\)

Substitute the given values and find resistivity

Now we can substitute the given values and the calculated resistance into the equation from step 3: \(\rho = \frac{(3750 \,\Omega)(4.50 \cdot 10^{-6} \,\mathrm{m}^2)}{(1000 \cdot 10^3 \,\mathrm{m})}\) Calculate the value of \(\rho\): \(\rho \approx 1.69 \cdot 10^{-8} \,\mathrm{\Omega} \cdot \mathrm{m}\) So, the resistivity of the wire is approximately \(1.69 \times 10^{-8} \,\mathrm{\Omega} \cdot \mathrm{m}\).

Identify the type of wire

Based on the resistivity value we calculated, we can compare it to common resistivity values for different materials to determine what type of wire this is. Here are some common resistivities: - Copper: \(1.68 \times 10^{-8} \,\mathrm{\Omega} \cdot \mathrm{m}\) - Aluminum: \(2.82 \times 10^{-8} \,\mathrm{\Omega} \cdot \mathrm{m}\) - Silver: \(1.59 \times 10^{-8} \,\mathrm{\Omega} \cdot \mathrm{m}\) The resistivity of our wire, \(1.69 \times 10^{-8} \,\mathrm{\Omega} \cdot \mathrm{m}\), is closest to the resistivity of copper. Therefore, we can deduce that the wire is likely made of copper. In conclusion: a) The resistance of the wire is approximately \(3750 \,\Omega\) b) The type of wire is copper.

Key concepts

Understanding Electrical Resistance

Electrical resistance is a fundamental concept in the study of electricity. It's the measure of how much a material opposes the flow of electric current. Imagine water flowing through a pipe; if the pipe is narrow, it's harder for water to pass through. Similarly, with electricity, the narrower or longer a wire, the more resistance it has.

In the example given, after applying Ohm's Law (\(V = IR\), where V is voltage, I is current, and R is resistance), the resistance of the wire is calculated to be approximately 3750 ohms. This tells us how much the wire restricts the current flow. The higher the resistance, the lower the current for a given voltage.

Why is this important? Understanding resistance is crucial when designing circuits to ensure that they operate safely and efficiently. If resistance is too high, it can lead to less power to devices or even cause components to overheat. Therefore, engineers must carefully select materials with appropriate resistance levels for their electrical systems.

Resistivity and Its Role in Electrical Conductivity

Resistivity is a property of materials that quantifies how strongly a material opposes the flow of electric current. It's denoted by the Greek letter \rho (\rho) and is measured in ohm-meters (\(\Omega\cdot m\)).

A material with high resistivity means it's a poor conductor of electricity; conversely, low resistivity indicates good conductivity. This intrinsic property of the material doesn't change with the shape or size of the material. For instance, in our textbook problem, the resistivity helps us determine the type of metal the wire is made of. Upon calculation, the wire's resistivity is found to be approximately \(1.69 \times 10^{-8} \Omega\cdot m\), hinting that the wire is likely copper since it closely matches copper's known resistivity.

Understanding resistivity is key when selecting materials for electrical wiring and components. It helps in ensuring that the entire circuit functions as intended with minimal energy loss.

Conductivity of Materials

Conductivity is the inverse of resistivity and is a measure of a material's ability to allow the flow of electric current. It is represented by the symbol \(\sigma\) and is measured in siemens per meter (S/m).

In electrical engineering, we often talk about 'good conductors' which have high conductivity, meaning they allow current to pass through easily. These include materials like copper, silver, and gold. Conversely, materials with low conductivity such as rubber and glass are considered good insulators because they impede the flow of current.

The conductivity of a material depends on its atomic structure. Materials with more free electrons allow electricity to flow more freely. Knowing the conductivity of materials is essential for designing and implementing electrical systems, as engineers must ensure that the materials chosen will conduct electricity effectively while also meeting other design criteria such as strength, durability, and cost.