Math Problem Statement

Problem 4A: A small magnetic dipole located at the origin with moment m ⃗=m_0 a ̂_z is in a region with a constant magnetic field H ⃗=-H_0 a ̂_z, where H_0>0. Show that there is zero net flux passing through any portion of the sphere at r=(m_0/(2πH_0 ))^(1/3 ) Be sure to show a picture of the problem. [Hint: This is not a Gauss’s law problem]

Solution

The diagram illustrates a magnetic dipole located at the origin with its moment m0m_0 pointing along the zz-axis. The surrounding sphere, with radius r=(m02πH0)1/3r = \left( \frac{m_0}{2\pi H_0} \right)^{1/3}, is influenced by both the dipole’s magnetic field and the external constant magnetic field H0H_0, which points in the negative zz-direction.

The flux contributions from the dipole and external field cancel out, resulting in zero net flux through the sphere.

Would you like further clarification or details on any part of the problem?

Here are 5 related questions:

  1. How does the symmetry of the dipole field contribute to zero net flux?
  2. What is the general form of the magnetic dipole field for any orientation?
  3. Can you explain why this problem is not related to Gauss's law?
  4. How would the flux change if the external field were not aligned with the dipole?
  5. What is the role of the sphere's radius in this problem?

Tip: In problems involving magnetic dipoles, always check the symmetry of the magnetic field to simplify flux calculations.

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Math Problem Analysis

Mathematical Concepts

Electromagnetic Theory
Magnetic Dipoles
Flux Calculation

Formulas

Magnetic field of a dipole: B(r) = (μ₀/4π) * [(3(m⃗·r⃗)r⃗ - m⃗)/r⁵] for r > 0
External field: H⃗ = -H₀â_z
Sphere radius: r = (m₀/(2πH₀))^(1/3)

Theorems

Superposition Principle
Magnetic Dipole Field Symmetry

Suitable Grade Level

Undergraduate level (Physics)

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