Problem 30 - University Physics with Modern Physics

Question

Two long, parallel wires are separated by a distance of 2.50 cm. The force per unit length that each wire exerts on the other is 4.00 \(\times\) 10\(^{-5}\) N/m, and the wires repel each other. The current in one wire is 0.600 A. (a) What is the current in the second wire? (b) Are the two currents in the same direction or in opposite directions?

Step-by-Step Solution

Answer

(a) The current in the second wire is 0.833 A. (b) The currents are in opposite directions.

Step 1: Understand the Problem

We have two parallel wires separated by a certain distance, experiencing a mutual repulsion force per unit length. We know the current in one wire and need to find the current in the second wire. Additionally, we need to determine if the currents are in the same or opposite directions.

Step 2: Recognize the Formula

The formula for the force per unit length between two parallel wires is given by:\[ F = \frac{\mu_0}{2\pi} \frac{I_1 I_2}{d} \]where \( F \) is the force per unit length, \( \mu_0 \) is the permeability of free space \((4\pi \times 10^{-7} \text{ N/A}^2)\), \( I_1 \) and \( I_2 \) are the currents in the wires, and \( d \) is the distance between the wires.

Step 3: Plug in Known Values and Solve for Current

We know:- \( F = 4.00 \times 10^{-5} \text{ N/m} \)- \( I_1 = 0.600 \text{ A} \)- \( d = 0.0250 \text{ m} \)Plug these values into the formula:\[ 4.00 \times 10^{-5} = \frac{4\pi \times 10^{-7}}{2\pi} \frac{0.600 \times I_2}{0.0250} \]Simplify and solve for \( I_2 \):\[ I_2 = \frac{4.00 \times 10^{-5} \times 2 \pi \times 0.0250}{4\pi \times 10^{-7} \times 0.600} \approx 0.833 \text{ A} \]

Step 4: Analyze the Force Direction

The problem states the wires repel each other. For parallel wires, currents in opposite directions repel, while currents in the same direction attract. Since these wires repel, the currents must be in opposite directions.

Key Concepts

Magnetic Fields

To understand the magnetic force between two parallel wires, we must first explore the concept of magnetic fields. Magnetic fields are invisible areas around a magnetic source that exert force on other magnetic materials and moving charges. When electric current flows through a wire, it generates a magnetic field around it. The direction of this field is determined by the Right-Hand Rule: if you point your thumb in the direction of the current, the curl of your fingers shows the magnetic field's direction.

This field interacts with other magnetic fields, creating forces that can either attract or repel the involved objects. Parallel wires carrying currents create magnetic fields that interact with each other. The strength and direction of these interactions are fundamental in calculating the force per unit length between the wires. Understanding this interaction is crucial because it allows us to apply mathematical laws, such as Ampere's Law, to predict and explain real-world phenomena.

Ampere's Law

Ampere's Law offers a powerful tool to relate the magnetic field around a conductor to the electric current passing through it. The law states: \[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} \]where \( \vec{B} \) is the magnetic field, \( d\vec{l} \) is a differential length element through which the field is calculated, and \( I_{enc} \) is the current enclosed by the path. Here, \( \mu_0 \) is the permeability of free space, a constant value of \( 4\pi \times 10^{-7} \text{ N/A}^2 \).

In the context of two parallel wires, Ampere's Law helps us derive the formula for the magnetic force per unit length. By recognizing that each wire creates a magnetic field, we can calculate the force experienced by the other via:\[ F = \frac{\mu_0}{2\pi} \frac{I_1 I_2}{d} \]where \( F \) is the given force per length, and \( I_1 \), \( I_2 \) are the currents in the two wires. This relationship shows the direct dependency on both the current carried and the distance between the wires. Understanding how this equation is derived from Ampere’s Law is key to solving problems related to magnetic force between wires.

Current Direction Analysis

Analyzing the direction of current is vital to determine whether two wires will attract or repel each other. The direction of the current affects the direction of the magnetic field produced around the wire. When analyzing forces between wires:

• If currents flow in the same direction, the magnetic fields will attract
• If currents flow in opposite directions, the magnetic fields will repel

Given that the problem statement mentions repulsion between the wires, this inherently tells us the currents must flow in opposite directions. This principle helps not only in academic exercises but also in practical applications, such as designing electrical circuits to minimize interference or in industrial settings where strong magnetic interactions might occur. Correctly analyzing current directions can avert possible system malfunctions and optimize circuit performance.