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Chapter 28: Problem 35

A closed curve encircles several conductors. The line integral \(\phi \overrightarrow{\boldsymbol{B}} \cdot d \vec{l}\) around this curve is \(3.83 \times 10^{-4} \mathrm{T} \cdot \mathrm{m} .\) (a) What is the net current in the conductors? (b) If you were to integrate around the curve in the opposite direction, what would be the value of the line integral? Explain.

Short Answer

Expert verified
(a) 303.4 A; (b) Line integral is \(-3.83 \times 10^{-4} \mathrm{T} \cdot \mathrm{m}\).

Step by step solution

01

Use Amp猫re's Law

According to Amp猫re's Law, the line integral of the magnetic field \( \overrightarrow{\boldsymbol{B}} \) around a closed path is equal to \( \mu_0 \) times the net current \( I \) enclosed by the path. The formula for this is: \( \oint \overrightarrow{\boldsymbol{B}} \cdot d \vec{l} = \mu_0 I \).Given that \( \oint \overrightarrow{\boldsymbol{B}} \cdot d \vec{l} = 3.83 \times 10^{-4} \mathrm{T} \cdot \mathrm{m} \), we can use this equation to find the net current, \( I \).
02

Solve for Net Current

We know that:\[ \oint \overrightarrow{\boldsymbol{B}} \cdot d \vec{l} = \mu_0 I \]Solving for \( I \), we have:\[ I = \frac{\oint \overrightarrow{\boldsymbol{B}} \cdot d \vec{l}}{\mu_0} \]Assuming \( \mu_0 = 4\pi \times 10^{-7} \text{ T} \cdot \text{m/A} \), substitute \( 3.83 \times 10^{-4} \text{ T} \cdot \text{m} \) for the line integral:\[ I = \frac{3.83 \times 10^{-4}}{4\pi \times 10^{-7}} \approx 303.4 \text{ A} \].
03

Consider the Opposite Direction Integral

If the integration is performed in the opposite direction, this simply inverts the sign of the line integral due to the reversal of path direction. Thus, the value becomes \(-3.83 \times 10^{-4} \mathrm{T} \cdot \mathrm{m} \). Amp猫re's Law itself remains unchanged as it is a scalar relation dependent on the enclosed current and measures direction change with a negative sign.

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Key Concepts

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

Magnetic Field
A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. It is represented by the symbol \( \overrightarrow{\boldsymbol{B}} \), and it's a vector field.
  • Think of the magnetic field lines like invisible threads pulling magnetic objects towards or away from a magnet.
  • These lines give us a visual cue about the strength and direction of the magnetic influence.
In the context of this problem, we are particularly interested in how the magnetic field interacts with electrical currents. When a current flows through conductors, it generates a magnetic field around them. This relationship is fundamental in electromagnetism and ties directly into Amp猫re's Law. Understanding the interaction between currents and magnetic fields is essential in fields like electrical engineering and physics.The strength of a magnetic field can be expressed in teslas \( \text{T} \), which measure the force on moving charges within the field. It's critical to understand how these fields contribute to practical applications like electric motors, MRI machines, and more.
Line Integral
In mathematics, a line integral is a type of integral where a function is integrated along a curve. In physics, particularly electromagnetism, it describes how a field behaves over a path or boundary. For the magnetic field \( \overrightarrow{\boldsymbol{B}} \), we use a line integral to understand its cumulative effect across a closed loop.
  • The line integral is computed using \( \oint \overrightarrow{\boldsymbol{B}} \cdot d \vec{l} \), representing the integral of the magnetic field over the path \( d \vec{l} \).
  • This integral helps in calculating the total magnetic effect along a given path around current-carrying conductors.
When we perform the line integral of a magnetic field around a closed curve, according to Amp猫re's Law, it gives us the net current encircled by that path. If the direction of integration changes, the integral鈥檚 sign is reversed, implying that the line integral is direction-sensitive. Thus, flipping the integration path reverses the final sign of the calculated line integral, but the fundamental relationship described by Amp猫re's Law remains unaffected.
Net Current
Net current indicates the total current flowing through conductors encircled by a chosen closed path. Amp猫re's Law provides the concept that relates the magnetic field around a path to the current enclosed by it.
  • According to Amp猫re's Law, \( \oint \overrightarrow{\boldsymbol{B}} \cdot d \vec{l} = \mu_0 I \), where \( \mu_0 \) is the permeability of free space, a constant. By solving this equation for \( I \), we find the net current.
  • In our exercise, the given line integral allowed us to compute the enclosed current as approximately 303.4 A.
Permeability of free space \( \mu_0 \) is critical to understanding the relationship between magnetic fields and currents. Its value helps us convert the magnetic effects observed around conductors into quantifiable current values. This principle is crucial in designing and understanding electrical circuits, power systems, and many related technologies. Understanding the net current鈥檚 calculation helps engineers and scientists design more efficient magnetic and electromagnetic systems.

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Most popular questions from this chapter

A pair of long, rigid metal rods, each of length \(L,\) lie parallel to each other on a perfectly smooth table. Their ends are connected by identical, very light conducting springs of force constant \(k\) (Fig. 28.55 ) and negligible unstretched length. If a current \(I\) runs through this circuit, the springs will stretch. At what separation will the rods remain at rest? Assume that \(k\) is large enough so that the separation of the mods will be much less than \(L\) .

A conductor is made in the form of a hollow cylinder with inner and outer radii a and \(b\) , respectively. It carries a current I uniformly distributed over its cross section. Derive expressions for the magnitude of the magnetic field in the regions (a) \(r < a\) ; (b) \(a< r b\).

A wooden ring whose mean diameter is 14.0 \(\mathrm{cm}\) is wound with a closely spaced toroidal winding of 600 turns. Compute the magnitude of the magnetic field at the center of the cross section of the windings when the current in the windings is 0.650 \(\mathrm{A}\) .

Fields within the Atom. In the Bohr model of the hydrogen atom, the electron moves in a circular orbit of radius \(5.3 \times 10^{-11} \mathrm{m}\) with a speed of \(2.2 \times 10^{6} \mathrm{m} / \mathrm{s}\) . If we are viewing the atom in such a way that the electron's orbit is in the plane of the paper with the electron moving clockwise, find the magnitude and direction of the electric and magnetic fields that the electron produces at the location of the nucleus (treated as a point).

A long, straight wire carries a current of 2.50 A. An electron is traveling in the vicinity of the wire. At the instant when the electron is 4.50 \(\mathrm{cm}\) from the wire and traveling with a speed of \(6.00 \times 10^{4} \mathrm{m} / \mathrm{s}\) directly toward the wire, what are the magnitude and direction (relative to the direction of the current) of the force that the magnetic field of the current exerts on the electron?
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