Problem 28 - Fundamentals of Physics

Question

Gives the electric potential \(V(x)\) along a copper wire carrying uniform current, from a point of higher potential \(V_{s}=12.0 \mu \mathrm{V}\) at \(x=0\) to a point of zero potential at \(x_{s}=3.00 \mathrm{~m}\). The wire has a radius of \(2.00 \mathrm{~mm}\). What is the current in the wire?

Step-by-Step Solution

Answer

The current in the wire is approximately 2.99 mA.

Step 1: Understanding the given problem

We need to find the current in the wire, which is carrying a uniform current from an initial potential of \( V_s = 12.0 \mu V \) at \( x = 0 \) to zero potential at \( x_s = 3.00 m \). We know the length and initial potential difference along the wire, with a defined wire radius of \(2.00\, \text{mm}\). Our goal is to find out the current \( I \) using Ohm's law and the wire's resistivity.

Step 2: Calculating the resistance of the wire

First, we calculate the cross-sectional area \( A \) of the wire using the formula for the area of a circle: \[ A = \pi r^2 \]. Here, the radius \( r = 2.00 \times 10^{-3} \) m. Thus, \[ A = \pi \times (2.00 \times 10^{-3})^2 \] \[ A \approx 1.257 \times 10^{-5} \text{ m}^2 \].

Step 3: Finding resistivity and resistance

Copper's resistivity \( \rho \) is approximately \( 1.68 \times 10^{-8} \text{ Ohm} \cdot \text{m} \). Using the formula for resistance, \( R = \frac{\rho L}{A} \), where \( L = 3.00 \text{ m} \), we obtain: \[ R = \frac{1.68 \times 10^{-8} \times 3.00}{1.257 \times 10^{-5}} \] \[ R \approx 0.00401 \text{ Ohms} \].

Step 4: Applying Ohm’s Law to calculate current

Ohm's Law states \( V = IR \). We rearrange it to find the current \( I \): \[ I = \frac{V}{R} \]. Substitute \( V = 12.0 \times 10^{-6} \text{ V} \) and \( R = 0.00401 \text{ Ohms} \) into the equation: \[ I = \frac{12.0 \times 10^{-6}}{0.00401} \] \[ I \approx 2.99 \times 10^{-3} \text{ Amperes} \].

Step 5: Conclusion

We have calculated the current flowing through the wire as approximately \( 2.99 \times 10^{-3} \text{ A} \), or 2.99 mA.

Key Concepts

Electrical Resistance

Electrical resistance is a fundamental concept that describes how difficult it is for electric current to flow through a material. It is measured in ohms (Ω). The resistance of a wire depends on several factors:

• Material of the wire (known as resistivity)
• Length of the wire
• Cross-sectional area of the wire

According to Ohm's Law, Resistance (\( R \)) is calculated with the formula:\[ R = \frac{\rho L}{A} \], where \( \rho \) is the resistivity of the material, \( L \) is the length of the wire, and \( A \) is its cross-sectional area.

Resistance is a crucial factor in determining how much current can flow for a given voltage.

Electric Potential

Electric potential, often referred to as voltage, is the measure of potential energy per unit charge. It determines how much work is needed to move an electric charge from one point to another. In the context of a wire, electric potential difference is what drives current through the conductor.

Electric potential is measured in volts (V). For example, a potential difference of \( 12.0 \mu V \) was used to drive current through the copper wire in our example.

Understanding electric potential helps in analyzing circuits and applying Ohm's Law, which relates the potential difference across a conductor to the current flowing through it.

Current in a Wire

The current in a wire is the flow of electric charge, typically measured in amperes (A). It represents how many charge carriers pass a point in the circuit per unit time. Current flows from a higher potential to a lower potential in a conductive material.

To find the current using Ohm’s Law, we rearrange the formula to \( I = \frac{V}{R} \), where \( I \) is the current, \( V \) is the voltage, and \( R \) is the resistance. This relationship shows how the electric potential difference and resistance affect the flow of current.

In the given problem, the current was calculated to be 2.99 mA, emphasizing how Ohm's Law can predict current for known values of voltage and resistance.

Resistivity of Copper

Copper is widely used in electrical wiring due to its low resistivity, which makes it an excellent conductor of electricity. Resistivity is the inherent property of a material that quantifies how strongly it resists the flow of electric current.

For copper, the resistivity is approximately \( 1.68 \times 10^{-8} \text{ Ohm} \cdot \text{m} \). Low resistivity means copper can carry more current for a given potential difference without as much loss of energy as heat.

Understanding the resistivity of materials, like copper, is essential when designing circuits and choosing the right materials for efficient electrical conduction.