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Chapter 28: Problem 58

A neophyte magnet designer tells you that he can produce a magnetic field \(\vec{B}\) in vacuum that points everywhere in the \(x\) -direction and that increases in magnitude with increasing \(x\) . That is, \(\vec{B}=B_{0}(x / a) \hat{\imath},\) where \(B_{0}\) and \(a\) are constants with units of teslas and meters, respectively. Use Gauss's law for magnetic fields to show that this claim is impossible. (Hint: Use a Gaussian surface in the shape of a rectangular box, with edges parallel to the \(x\) ; \(y\) . and \(z\) -axes.)

Short Answer

Expert verified
The claim is impossible due to violation of Gauss's law for magnetism, which requires zero net magnetic flux through any closed surface.

Step by step solution

01

Understanding Gauss's Law for Magnetism

Gauss's law for magnetism states that the net magnetic flux through any closed surface is zero because monopoles do not exist. Mathematically, this is expressed as \( \oint \vec{B} \cdot d\vec{A} = 0 \).
02

Set Up the Gaussian Surface

To evaluate the claim, we choose a rectangular box as the Gaussian surface, aligned with its edges parallel to the \(x\), \(y\), and \(z\) axes. This surface will have faces at constant \(x\), constant \(y\), and constant \(z\).
03

Calculate the Magnetic Flux Through the Gaussian Surface

The magnetic field \( \vec{B} = B_0 \left( \frac{x}{a} \right) \hat{\imath} \) only has a component along the \(x\)-axis. Therefore, it contributes to the flux through the faces of the box perpendicular to the \(x\)-axis. The magnetic flux through a face at \(x = x_1\) is \( B_0 \frac{x_1}{a}A \), and at \(x = x_2\) is \(- B_0 \frac{x_2}{a}A \), where \(A\) is the area of the face.
04

Calculate the Net Magnetic Flux

The total magnetic flux through the entire Gaussian surface is the sum of the flux through each face. For the faces at \(x=x_1\) and \(x=x_2\), this is \(B_0 \frac{x_1}{a}A - B_0 \frac{x_2}{a}A = B_0 \frac{A}{a} (x_1 - x_2) \), which is not necessarily zero if \(x_1 eq x_2\).
05

Contradiction with Gauss's Law

According to Gauss's law for magnetism, the total net magnetic flux through any closed surface must be zero. However, our calculation shows that the net magnetic flux is \(B_0 \frac{A}{a} (x_1 - x_2)\), which is not zero unless \(x_1 = x_2\). This is a contradiction, implying that the given magnetic field configuration is impossible.

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Key Concepts

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

Magnetic Flux
Magnetic flux is a measure of the magnetic field passing through a given area. Imagine it as the number of magnetic field lines going through a surface. The amount of flux depends on:
  • The strength of the magnetic field.
  • The size of the surface it passes through.
  • The angle between the magnetic field and the normal (perpendicular) to the surface.
In mathematical terms, magnetic flux \( \Phi_B \) through a surface with area \( A \) is given by the formula: \( \Phi_B = \vec{B} \cdot \vec{A} \), where \( \vec{B} \) is the magnetic field, and \( \vec{A} = A \hat{n} \) is the area vector (area \( A \), pointing normal to the surface). If the magnetic field is uniform and perpendicular to the surface, then \( \Phi_B = B \cdot A \).
Understanding magnetic flux is crucial for analyzing how magnetic fields interact with different surfaces and is central to concepts like Gauss's Law for Magnetism.
Gaussian Surface
A Gaussian surface is an imaginary closed surface used in Gauss's Law applications. It is a powerful tool in electromagnetic theory to calculate the flux of a field. For magnetism, using an appropriate Gaussian surface helps verify that Gauss's Law for Magnetism holds, which states that the net magnetic flux through any closed surface must be zero. This is because magnetic monopoles do not exist.
When choosing a Gaussian surface, we often align its surfaces with the axes of symmetry of the problem to simplify calculations. In problems related to the rectangular box, for example, the edges may be aligned parallel to the coordinate axes.
The choice of the Gaussian surface doesn't affect the physics but makes mathematical evaluations easier or sometimes possible.
Magnetic Field
The magnetic field, often represented by \( \vec{B} \), is a vector field surrounding magnets and electric currents. It describes the magnetic influence on moving charges, magnetic dipoles, and electric currents. Magnetic fields have both direction and magnitude and can vary in space.
Understanding a magnetic field involves visualizing it with lines of force鈥攊maginary lines that illustrate the direction a north pole would move if placed within the field. The density of these lines represents the strength of the field. More field lines peeking through a surface suggest a stronger field.
Magnetic fields arise due to movements of electric charges, like in currents or changing electric fields, and are responsible for the force experienced by charged particles in movement through the field. They are fundamental in technologies like motors, generators, and various scientific instruments.
Rectangular Box
In problems involving Gauss's law, a rectangular box often serves as a simple and effective Gaussian surface. The logic behind choosing such a surface relates to ensuring that calculations remain straightforward while maximizing the symmetries of the problem.
A rectangular box as a Gaussian surface allows for easy calculation of the magnetic flux through each face. Since it has six faces, the analysis involves checking each pair of opposing faces which cancels out each other when there's no net flux. This is the hallmark of Gauss's law for magnetism.
  • Two faces for each axis: \(x\), \(y\), \(z\).
  • The choice simplifies calculations as it aligns with the problem's inherent symmetry.
  • The box's uniform surfaces make evaluating field components straightforward.
By assessing the flux through this box, contradictions in proposed magnetic field configurations can be detected with relative ease.

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Most popular questions from this chapter

A long solenoid with 60 turns of wire per centimeter carries a current of 0.15 \(\mathrm{A}\) . The wire that makes up the solenoid is wrapped around a solid core of silicon steel \(\left(K_{\mathrm{m}}=5200\right) .\) (The wire of the solenoid is jacketed with an insulator so that none of the current flows into the core.) (a) For a point inside the core, find the magnitudes of (i) the magnetic field \(\overrightarrow{\boldsymbol{B}}_{0}\) due to the solenoid current; (ii) the magnetization \(\vec{M} ;\) (iii) the total magnetic field \(\overrightarrow{\boldsymbol{B}}\) . (b) In a sketch of the solenoid and core, show the directions of the vectors \(\overrightarrow{\boldsymbol{B}}, \overrightarrow{\boldsymbol{B}}_{0}\) , and \(\overrightarrow{\boldsymbol{M}}\) inside the core.

Lamp Cord Wires. The wires in a household lamp cord are typically 3.0 mm apart center to center and carry equal currents in opposite directions. If the cord carries current to a 100 -W light bulb connected across a \(120-\mathrm{V}\) potential difference, what force per meter does each wire of the cord exert on the other? Is the force attractive or repulsive? Is this force large enough so it should be considered in the design of lamp cord? (Model the lamp cord as a very long straight wire.)

A toroidal solenoid with 500 turns is wound on a ring with a mean radins of 290 \(\mathrm{cm}\) . Find the current in the winding that is required to set up a magnetic field of 0.350 \(\mathrm{T}\) in the ring (a) if the ring is made of annealed iron \(\left(K_{m}=1400\right)\) and \((b)\) if the ring is made of sillicon steel \(\left(K_{m}=5200\right)\) .

Helmholtz Coils. Fig. 28.59 is a sectional view of two circular coils with radius \(a\) , each wound with \(N\) turns of wire carrying a current \(I,\) circulating in the same direction in both coils. The coils are separated by a distance \(a\) equal to their radii. In this configuration the coils are called Helmholtz coils; they produce a very uniform magnetic field in the region between them. (a) Derive the expression for the magnitude \(B\) of the magnetic field at a point on the axis a distance \(x\) to the right of point \(P,\) which is midway between the coils. (b) Graph \(B\) versus \(x\) for \(x=0\) to \(x=a / 2\) . Compare this graph to one for the magnetic field due to the right-hand coil alone. (c) From part (a), obtain an expression for the magnitude of the magnetic field at point \(P .\) (d) Calculate the magnitude of the magnetic field at \(P\) if \(N=300\) turns, \(I=6.00 \mathrm{A},\) and \(a=8.00 \mathrm{cm} .\) ( e) Calculate \(d B / d x\) and \(d^{2} B / d x^{2}\) at \(P(x=0)\) . Discuss how your results show that the field is very uniform in the vicinity of \(P .\)

A magnetic field of 37.2 T has been achieved at the MTT Francis Bitter National Magnetic Laboratory. Find the current needed to achieve such a field (a) 2.00 \(\mathrm{cm}\) from a long, straight wire; \((b)\) at the center of a circular coil of radius 42.0 \(\mathrm{cm}\) that has 100 turns; \((\mathrm{c})\) near the center of a solenoid with radius \(2.40 \mathrm{cm},\) length \(32.0 \mathrm{cm},\) and \(40,000\) turns.
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