Solving Spinning Iron Wire Angular Velocity Change

AI Thread Summary
The discussion centers on calculating the change in angular velocity of a vertically suspended iron wire subjected to a strong magnetic field. It highlights the relevance of the Poynting vector and angular momentum in the context of electromagnetic fields. The initial approach involves using Faraday's law to relate the magnetic field to an induced electric field, but concerns arise about the simplicity of this method. The conversation suggests a more straightforward calculation using the properties of iron, specifically the relationship between angular momentum and spin. Ultimately, the discussion emphasizes the need to consider iron's unique characteristics in solving the problem effectively.
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Homework Statement



Iron atoms (atomic mass 56) contain two free electron spins that can align with an external magnetic field. An iron wire 3 cm long and 1 mm in diameter is suspended vertically and is free to rotate about its axis. A strong magnetic field parallel to the wire's axis is applied. How large is the resulting change in its angular velocity.

Homework Equations



\oint \vec{E} \cdot \vec{dl} = - \frac{d}{dt} \int \vec{B} \cdot \vec{da}

Poynting vector: \vec{S} = \mu_0 \vec{E} \times \vec{B}.

Bohr's magneton might come in handy: \mu_b = \frac{e \hbar}{2 m_e}

The Attempt at a Solution



My first thought was to find the Poynting vector because the angular momentum contained in the fields is proportional to \vec{r} \times \vec{S}. Assuming the wire is in the z direction, we can write that the applied magnetic field is \vec{B} = B \hat{z}. This would create a magnetic flux through the x-y plane, and hence create an electric field in the \hat{\phi} direction from the Faraday law. But that means that \vec{S} \approx \hat{\phi} \times \hat{z} = \hat{r} and therefore there would be no angular momentum change in the fields.

I feel this is too simplistic and possibly wrong, especially since we didn't use any of iron's properties.
 
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So as always I overcomplicate stuff. It is easy to just calculate L = N hbar/2, where N is the number of spin, given the density of iron. L = I w, where I is the moment of inertia. Super simple. Ugh.
 
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