Problem 22 - University Physics with Modern Physics

Question

Two long. parallel transmission lines, 40.0 \(\mathrm{cm}\) apart, carry \(25.0-\mathrm{A}\) and 75.0 - A currents. Find all locations where the net magnetic field of the two wires is zero if these currents are in (a) the same direction and (b) the opposite direction.

Step-by-Step Solution

Answer

(a) 0.10 m from Wire 1; (b) 0.20 m from Wire 1.

Step 1: Identify the Formula for Magnetic Field Due to a Long Straight Current-Carrying Wire

The magnetic field created by a long straight wire carrying current can be calculated using Ampère's Law. The formula is given by:\[ B = \frac{\mu_0 I}{2 \pi r} \]where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space \( (4\pi \times 10^{-7} \;\text{T m/A}) \), \( I \) is the current in the wire, and \( r \) is the distance from the wire.

Step 2: Calculate the Magnetic Field Expressions for Each Wire

Let \( I_1 = 25.0 \;\text{A} \) and \( I_2 = 75.0 \;\text{A} \). Assume Wire 1 is at \( x = 0 \) and Wire 2 is at \( x = 0.40 \;\text{m} \). The magnetic field at a point \( x \) due to each wire is:For Wire 1: \[ B_1(x) = \frac{\mu_0 I_1}{2 \pi x} \]For Wire 2: \[ B_2(x) = \frac{\mu_0 I_2}{2 \pi (0.40\, \text{m} - x)} \].

Step 3: Set Up Equation for Net Magnetic Field Being Zero In the Same Direction

For the currents in the \textbf{same direction}, the net magnetic field is zero when the fields due to each wire cancel each other out. This results in:\[ B_1(x) = B_2(x) \].Using the expressions from Step 2:\[ \frac{\mu_0 I_1}{2 \pi x} = \frac{\mu_0 I_2}{2 \pi (0.40 - x)} \]\[ \frac{25}{x} = \frac{75}{0.40 - x} \]

Step 4: Solve the Equation for Currents in Same Direction

Cross-multiply and solve the equation:\[ 25(0.40 - x) = 75x \]\[ 10 - 25x = 75x \]\[ 10 = 100x \]\[ x = 0.10 \;\text{m} \]Thus, when the currents are in the same direction, the net magnetic field is zero at \( x = 0.10 \;\text{m} \) from Wire 1.

Step 5: Set Up Equation for Currents In Opposite Directions

For the currents in \textbf{opposite directions}, the fields add up and cancel each other out over different distances. Set up the equation:\[ B_1(x) + B_2(x) = 0 \].Use the same expressions, recognizing that the fields now have opposite senses.

Step 6: Solve the Equation for Currents in Opposite Directions

In the case of opposite directions:\[ \frac{25}{x} = \frac{75}{x-0.40} \]Rearrange and solve: \[ 25(x - 0.40) = 75x \]\[ 25x - 10 = 75x \]\[ 10 = 50x \]\[ x = 0.20 \;\text{m} \]Thus, when the currents are in opposite directions, the net magnetic field is zero at \( x = 0.20 \;\text{m} \) from Wire 1.

Key Concepts

Ampère's Law

Ampère's Law is a fundamental concept in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through the loop. It is expressed mathematically as:\[\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}\]Where:

• \( \mathbf{B} \) is the magnetic field,
• \( d\mathbf{l} \) is a differential length element of the closed path,
• \( \mu_0 \) is the permeability of free space,
• \( I_{enc} \) is the current enclosed by the path.

This law is incredibly useful in deriving the magnetic field around current-carrying conductors, such as wires and loops.
For example, the magnetic field around a long straight wire can be derived using this law, resulting in a circular magnetic field around the wire. The formula for this scenario is:\[ B = \frac{\mu_0 I}{2 \pi r} \]This effectively shows how the magnetic field strength diminishes as you move further away from the wire. Understanding Ampère's Law is crucial for analyzing the magnetic fields in practical engineering tasks such as designing electrical circuits or transmission lines.

Current-Carrying Wire

A wire carrying an electric current generates a magnetic field around it. This phenomenon is a fundamental principle in electromagnetism. When current flows through a wire, it creates circular magnetic field lines around the wire.
The direction of the magnetic field can be determined using the right-hand rule:

• If the thumb of your right hand points in the direction of the current, your fingers curl in the direction of the magnetic field.

For a long straight wire, the magnetic field decreases as the distance from the wire increases, according to the formula mentioned earlier:\[ B = \frac{\mu_0 I}{2 \pi r} \]Here, \( B \) represents the magnetic field, \( I \) is the current through the wire, and \( r \) is the distance from the wire.
In real-world applications, such as in transmission lines, understanding the behavior of magnetic fields around current-carrying wires is essential for efficiently managing electromagnetic interactions and reducing energy losses.

Transmission Lines

Transmission lines are systems used to transport electrical energy from one place to another. They are an integral part of the power grid and involve a series of conductors, typically wires, that carry electrical currents over long distances.
The design of transmission lines involves understanding the current and the resulting magnetic fields around the wires.

• This is crucial for ensuring the efficient and safe operation of the power grid.

The magnetic fields created by the currents in the wires can interact with each other, an effect that can be analyzed using Ampère’s Law. For example, in a setup with parallel transmission lines, the interaction of their magnetic fields can result in phenomena such as cancelling each other out or reinforcing each other under different conditions. In practical terms, knowing where the magnetic fields cancel out is important to minimize interference with nearby lines and optimize the layout of electrical networks. Transmission engineers often need to calculate the precise locations where net magnetic fields become zero, as shown in typical problems involving current-carrying wires running in parallel with each other.