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]

The image above shows the setup of the problem: a magnetic dipole located at the origin, with its moment \( m_0 \) directed along the positive \( z \)-axis, and a constant magnetic field \( H_0 \) pointing downward along the negative \( z \)-axis. The spherical surface of radius \( r = \left( \frac{m_0}{2\pi H_0} \right)^{1/3} \) surrounds the dipole.

This symmetry helps explain why the net magnetic flux through the spherical surface is zero.

Would you like further details on any specific step? Here are some related questions:

1. What is the mathematical derivation of the magnetic field due to a dipole?
2. How does the symmetry of the dipole and external field lead to zero net flux?
3. Can we calculate the exact flux contributions at different points on the sphere?
4. What role does the choice of sphere radius \( r = \left( \frac{m_0}{2 \pi H_0} \right)^{1/3} \) play?
5. How can we visualize the individual flux contributions at different points on the surface?

**Tip:** Understanding the symmetry of fields is key when evaluating flux integrals over closed surfaces, especially for dipole fields.

A small magnetic dipole located at the origin with moment m⃗ = m_0 â_z is in a region with a constant magnetic field H⃗ = -H_0 â_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.

Explore the solution to a magnetic dipole problem where zero net flux passes through a spherical surface. The problem involves key magnetostatic concepts, including magnetic flux and dipole fields. Ideal for undergraduate physics students.

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