Question

A copper wire has radius \(r=0.0250 \mathrm{~cm},\) is \(3.00 \mathrm{~m}\) long, has resistivity \(\rho=1.72 \cdot 10^{-8} \Omega \mathrm{m},\) and carries a current of \(0.400 \mathrm{~A}\). The wire has density of charge carriers of \(8.50 \cdot 10^{28}\) electrons \(/ \mathrm{m}^{3}\) a) What is the resistance, \(R,\) of the wire? b) What is the electric potential difference, \(\Delta V\), across the wire? c) What is the electric field, \(E\), in the wire?

Short answer

Question: Calculate and analyze the electrical properties of a 3.00 m long copper wire with a radius of 0.0250 cm carrying a current of 0.400 A. Determine (a) the resistance of the wire, (b) the electric potential difference, and (c) the electric field inside the wire.

Step-by-step solution

a) Finding the Resistance of the Wire

In order to find the Resistance of the copper wire, we can use the formula: \(R = \frac{\rho L}{A}\), where \(R\) is the resistance, \(\rho\) is the resistivity, \(L\) is the length of the wire, and \(A\) is the cross-sectional area. As the wire is cylindrical, the cross-sectional area can be calculated as: \(A = \pi r^2\) Given the radius, \(r = 0.0250 \text{ cm} = 2.50 \times 10^{-4} \text{ m}\) and length, \(L = 3.00 \text{ m}\), we can calculate the cross-sectional area and the resistance.

b) Finding the Electric Potential Difference

After finding the resistance, we can use Ohm's Law to calculate the electric potential difference, \(\Delta V\). Ohm's Law is: \(\Delta V = IR\), where \(I\) is the current and \(R\) is the resistance. Given the current, \(I = 0.400 \text{ A}\), we can calculate the electric potential difference.

c) Finding the Electric Field

We can calculate the electric field, \(E\), inside the wire using the formula: \(E = \frac{\Delta V}{L}\). Here, \(\Delta V\) is the electric potential difference we just calculated, and \(L\) is the length of the wire. Now we can find the electric field in the wire using the previously calculated values. To summarize: 1. Find the cross-sectional area of the wire: \(A = \pi r^2\) 2. Calculate the resistance of the wire: \(R = \frac{\rho L}{A}\) 3. Find the electric potential difference: \(\Delta V = IR\) 4. Calculate the electric field inside the wire: \(E = \frac{\Delta V}{L}\)

Key concepts

Ohm's Law

Ohm's Law is a fundamental concept in the field of electricity and electronics. It describes the relationship between voltage, current, and resistance in an electrical circuit. The law is usually stated as \( V = IR \), where \( V \) represents the voltage across the circuit in volts, \( I \) is the current flowing through the circuit in amperes, and \( R \) is the resistance of the circuit in ohms. This equation shows that the current in a circuit is directly proportional to the voltage and inversely proportional to the resistance.

To understand this better, think of water flowing through a pipe. The voltage is like the water pressure, the current is the amount of water flowing, and the resistance is anything that hampers the flow, like a narrower pipe. If you increase the pressure without changing the resistance, more water flows through. Similarly, with electricity, if you increase the voltage (pressure), the current (flow) increases, provided the resistance remains constant.

Moreover, Ohm's Law helps us to calculate any one of the three quantities if the other two are known. It is crucial for designing circuits and finding the right components to ensure that the devices function correctly without overheating or damage. For example, to determine the appropriate resistance for a circuit, one might rearrange the formula to \( R = \frac{V}{I} \).

Electric Potential Difference

Electric potential difference, often referred to as voltage (\( \Delta V \)), is the amount of electric potential energy per unit charge at one point in a circuit compared to another. It is the 'push' that drives an electric current around a circuit, and it's measured in volts (V). A simple analogy is a water park slide: the higher the slide (greater electric potential), the faster the water (charged particles) will move down it.

When we speak of a 9-volt battery, we are referring to the electric potential difference between its two terminals. In an electric circuit, current flows from the higher potential to the lower potential, which means it moves from a point of higher voltage to a point of lower voltage, similar to how water flows from higher to lower elevations.

Understanding electric potential difference is essential for calculating the amount of work needed to move a charge between two points in an electrical field. It directly influences the amount of current that will flow through a conductor for a given resistance. The higher the voltage, the more potential energy is available to move electrons through a circuit.

Electric Field in Conductors

The electric field within a conductor is a measure of the force that acts on a charged particle within that material. In the context of electric circuits, conductors are materials that allow the flow of electrical current, mostly metals like copper or aluminum. The electric field in a conductor, denoted as \(\text{E}\), is related to the movement of electrons, which are the charge carriers in the circuit.

The strength and direction of an electric field are directly linked to the potential difference across the conductor. This can be visualized as how much the electrons are 'pushed' through the conductor, which is determined by the voltage applied to it. The electric field is calculated by the formula \(\text{E} = \frac{\Delta V}{L}\) where \( \Delta V \) is the potential difference applied across a conductor with length \( L \). The unit for electric field is volts per meter (V/m).

It’s important to note that while the electric field causes electrons to move, it’s fairly uniform inside the conductor and strong electric fields at the surface can drive the electrons off, creating effects like corona discharge or arcing. This is why conductors used in high-voltage applications are carefully designed to manage the electric field distribution. Understanding the electric field in conductors is critical for the safe and efficient design of electrical systems.