Work associated with polarization and magnetization

[dRic2]

Homework Statement

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Relevant Equations

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Hi, this is not an exercise. In some lecture notes the authors states that from elementary EM I should familiar with the fact that the work associated with polarization and magnetization of a material is given by $\delta W = E dP$ and $\delta W = BdM$. I have to admit that I am super rusty about EM so I was wondering if my reasoning to get here is correct.

- polarization: from Lorentz force without B field and identifying with $d \mathbf l$ the displacement from the "equilibrium position" (P = 0) of the bound charges, I get $\mathbf F \cdot d \mathbf l = (q_{bound} + q_{ext}) \mathbf E \cdot d \mathbf l$. Since I'm considering a neutral material $q_{ext} = 0$, I can write $\delta W = \mathbf E \cdot ( q_{bound} d \mathbf l )$. Here I naively identify $q_{bound} d \mathbf l = d \mathbf P$ and conclude $\delta W = \mathbf E \cdot d \mathbf P$

- magnetization: I recall that for a magnetic dipole $\mathbf m$, $U = - \mathbf m \cdot \mathbf B$. Since the filed is supposed to be constant I just take the differential and it follows immediately that $\delta W = \mathbf B \cdot d \mathbf m$

 

[hutchphd]

Notice this then leads to the respective energy density E⋅D/2 and B⋅H/2 as it must. Jackson does this all rather well I think.

 

[dRic2]

Yes, didn't have a copy available. I'm going to check tomorrow. Thanks