(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A rotating magnetic dipole is built by two oscillating magnetic dipole moments, one along the y-axis and one along the x-axis. Find the vector potential at a point: (0, 0, ##z_0##) along the z-axis. Then find the magnetic field at ##z_0## . As the magnetic field is a function of time, determine the induced Electric field from Faraday's laws. The magnetic dipole moment is given as a function of time as (##m_0## is a constant)

## \vec m ## = ##m_0 [cos(\omega t) \hat x + sin(\omega t) \hat y]##

2. Relevant equations

##\vec A_{dip} (\vec r) = \frac {\mu_0} {4 \pi} \frac 1 {r^3} \vec m \times \vec r##

##\vec B = \vec \nabla \times \vec A##

Induced electric field, Faraday's law:

##\vec E = \frac 1 {4 \pi}## ## \int ## ##[({\frac {\partial \vec B} {\partial t}} {\times \mathcal {\hat r}})## ##\frac 1 {\mathcal {r^2}}]##

3. The attempt at a solution

Since I am given the magnetic dipole moment, I attempted to find the vector potential by taking the cross product of ##\vec m## with ##\vec r##. I took ##\vec r## to be: ##{<x \hat x, y \hat y, z \hat z>}## which

After taking the cross product I got:

##\vec m \times \vec r = m_0 z sin(\omega t) \hat x - m_0 z cos(\omega t) \hat y + (m_0 cos(\omega t) y - m_0 sin(\omega t) x) \hat z##

Since I'm taking the potential at the point (0, 0, ##z_0##) I think I put that into the cross product which would mean my vector potential would be equal to:

##\vec A_{dip} (\vec r) = (\frac { \mu_0 m_0 z_0} {4 \pi r^3})(sin(\omega t) \hat x - cos(\omega t) \hat y)##

I'm not entirely sure that this is right but when I try to take the cross product of ##\vec A_{dip} (\vec r)## to find the magnetic field, I came up with:

##\vec \nabla \times \vec A_{dip} =(\frac { \mu_0 m_0 z_0} {4 \pi r^3}) (2cos(\omega t) \hat x + 2sin(\omega t) \hat y)##

which has no dependence on the z coordinate. I'm wondering if I maybe approached this problem wrong? Help would be much appreciated thank you!

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# Vector Potential of a Rotating Mangetic Dipole

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