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Chapter 20: Problem 69

Three long, straight electrical cables, running north and south, are tightly enclosed in an insulating sheath. One of the cables carries a 23.0 A current southward; the other two carry currents of \(17.5 \mathrm{~A}\) and 11.3 A northward. Use Amp猫re's law to calculate the magnitude of the magnetic field at a distance of \(10.0 \mathrm{~m}\) from the cables.

Short Answer

Expert verified
The magnetic field magnitude is approximately \(1.16 \times 10^{-7} \text{ T}\).

Step by step solution

01

Understanding the Current Directions

We are given three cables with the following current directions:1. Cable A: 23.0 A southward2. Cable B: 17.5 A northward3. Cable C: 11.3 A northwardThe net current is calculated by adding currents with their direction considered: - Southward direction positive, - Northward direction negative. Thus, the net current is: \[ I_{net} = 23.0 - 17.5 - 11.3 \text{ A} = -5.8 \text{ A} \]This indicates a net current of 5.8 A flowing northward.
02

Applying Amp猫re's Law

Amp猫re's law states that the line integral of the magnetic field \(\mathbf{B}\) around a closed loop is equal to \(\mu_0\), the permeability of free space, times the net current \(I_{net}\) enclosed by the loop:\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{net}\]For a distance \(r = 10.0 \mathrm{~m}\) from the cables, assuming the cables act like a single long wire carrying the net current, Amp猫re's law simplifies to: \[ B(2\pi r) = \mu_0 I_{net} \] where \(B\) is the magnitude of the magnetic field.
03

Solving for the Magnetic Field Magnitude

Rearrange the simplified Amp猫re's law equation to solve for the magnetic field magnitude \(B\):\[ B = \frac{\mu_0 I_{net}}{2\pi r} \]Substitute \(\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}\), \(I_{net} = 5.8 \text{ A}\), and \(r = 10.0 \mathrm{~m}\):\[ B = \frac{(4\pi \times 10^{-7}) \times 5.8}{2\pi \times 10.0} \]Simplify to find the magnitude of the magnetic field:\[ B = \frac{4 \times 10^{-7} \times 5.8}{20} = 1.16 \times 10^{-7} \text{ T} \] Thus, the magnitude of the magnetic field at a distance of 10.0 m is \(1.16 \times 10^{-7} \text{ T}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Field Calculation
When calculating the magnetic field using Amp猫re's law, it is crucial to understand that this involves both physics principles and mathematical equations. Amp猫re's law is particularly useful due to its ability to relate the magnetic field around a conductor to the electric current passing through it. If you have a net current flowing through conductors, like the cables in our problem, this current generates a magnetic field surrounding them.

To find the magnetic field, we rearrange the basic form of Amp猫re's law to incorporate the known quantities. The rearranged equation \[ B = \frac{\mu_0 I_{net}}{2\pi r} \] allows us to solve for the magnetic field, \( B \), at a distance \( r \) from the conductor. Here, \( \mu_0 \), the permeability of free space, is a constant. Once all values are substituted into this formula, we get the magnitude of the magnetic field in Tesla.

Understanding these calculations helps you directly relate electrical currents to the areas they influence through their magnetic fields.
Currents in Parallel Wires
Parallel wires carrying electrical currents impact each other due to the magnetic fields they generate. These interactions are a key concept in understanding magnetic fields. When currents flow in parallel wires in opposite directions, their magnetic fields tend to repel each other.

In our problem, we have three wires: one with current going south and two with current going north. To find the net effect, we calculate the total current by considering the directionality: subtract northward currents from southward currents. This directional approach is crucial as it impacts the net current and hence the net magnetic field created.

The formula \[ I_{net} = 23.0 - 17.5 - 11.3 \] enables us to compute the net effect of these parallel current-carrying wires, which we used in conjunction with Amp猫re鈥檚 law to determine the resultant magnetic field.
Magnetic Field Due to Current
Current through a conductor creates a magnetic field around it, a phenomenon explained through Amp猫re's Law. This magnetic field forms concentric circles around the wire with the direction determined by the right-hand rule.

In the specific exercise, the net current flows northward. This creates a magnetic field at a set distance which we calculate using known laws of electromagnetism. The direction of this field can be visualized with the thumb pointing in the direction of current and fingers indicating the circular field lines.

The source of this magnetic field is crucial in various technologies, from electric motors to data storage, highlighting the importance of understanding the practical implications of these magnetic fields due to currents.
Physics Problem Solving
Physics problem solving often requires breaking down complex concepts into manageable steps, as demonstrated with Amp猫re鈥檚 law. The goal is to interpret the problem, organize known values, and apply relevant physical laws effectively.

By calculating the net current and determining its direction, you can apply Amp猫re's law to find the magnetic field. This involves substituting known constants and solving algebraically, emphasizing attention to detail and logical decision-making.

Engaging in physics problem solving, like using these techniques to determine magnetic field strengths, strengthens critical thinking skills and boosts confidence in handling real-world physical phenomena.

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Most popular questions from this chapter

Two high-current transmission lines carry currents of \(25 \mathrm{~A}\) and 75 A in the same direction and are suspended parallel to each other \(35 \mathrm{~cm}\) apart. If the vertical posts supporting these wires divide the lines into straight \(15 \mathrm{~m}\) segments, what magnetic force does each segment exert on the other? Is this force attractive or repulsive?

An electron travels into a 0.3 T magnetic field perpendicular to its path, where it moves in a circular arc of diameter \(0.020 \mathrm{~m}\). What is the kinetic energy of the electron?

A long, straight wire carries a current \(I_{0}\) and produces a magnetic field \(B_{0}\) at a distance \(r_{0}\) from the wire's center. In terms of \(B_{0}\), determine the magnetic-field strength (a) at a distance \(r_{0} / 2\) from the wire's center and (b) at a distance \(3 r_{0}\) after the current is increased to \(6 I_{0}\).

If the magnetic field due to a long, straight, current-carrying wire has a magnitude \(B\) at a distance \(R\) from the wire's center, how far away must you be (in terms of \(R\) ) for the magnetic field to decrease to \(B / 3 ?\)

The large magnetic fields used in MRI can produce forces on electric currents within the human body. This effect has been proposed as a possible method for imaging "biocurrents" flowing in the body, such as the current that flows in individual nerves. For a magneticfield strength of \(2 \mathrm{~T}\), estimate the magnitude of the maximum force on a 1 -mm-long segment of a single cylindrical nerve with a diameter of \(1.5 \mathrm{~mm}\). Assume that the entire nerve carries a current due to an applied voltage of \(100 \mathrm{mV}\) (that of a typical action potential). The resistivity of the nerve is \(0.6 \Omega \cdot \mathrm{m}\). A. \(6 \times 10^{-7} \mathrm{~N}\) B. \(1 \times 10^{-6} \mathrm{~N}\) C. \(3 \times 10^{-4} \mathrm{~N}\) D. \(0.3 \mathrm{~N}\)
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