Problem 58 - University Physics with Modern Physics

Question

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.)

Step-by-Step Solution

Answer

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

Step 1: 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 \( \oint \vec{B} \cdot d\vec{A} = 0 \).

Step 2: 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 \(z\).

Step 3: 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.

Step 4: 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 \(x_1 eq x_2\).

Step 5: 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.

Key Concepts

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—imaginary 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.