diff --git a/_psets/10.md b/_psets/10.md index 5e2c544cfb9d8b9f9b663439fb1e0a01d38419e2..7ebf43cf4706fcbd57ce96723791076b88b31232 100644 --- a/_psets/10.md +++ b/_psets/10.md @@ -106,11 +106,38 @@ Using the equation for the energy in a magnetic field, describe why: {:.question} A permanent magnet is attracted to an unmagnetized ferromagnet. +The equation of interest for energy density is + +$$ +U = \frac{1}{2} (E \cdot D + B \cdot H) +$$ + +For this problem I assume the electric field is zero, so the total energy is + +$$ +U = \frac{1}{2 \mu} \int B^2 \mathrm{d} V +$$ + +An unmagnetized ferromagnet has a very high $$\mu$$, so $$U$$ is reduced by packing more field lines +into its extent. The permanent magnet's field lines are densest closest to its body, so the gradient +of the energy describes an attractive force. + ### (b) {:.question} The opposite poles of permanent magnets attract each other. +We know that $$B = \mu (H + M)$$, so + +$$ +\begin{align*} +U &= \frac{1}{2 \mu} \int \mu (H + M) \cdot H \mathrm{d} V \\ +&= \frac{1}{2} \int (H^2 + M \cdot H) \mathrm{d} V \\ +\end{align*} +$$ + +This is reduced when $$H$$ and $$M$$ are anti-aligned. + ## (13.4)