Question #42564

A toroidal magnetic core of volume $V = 10^{-3} \text{ m}^3$ is subjected to an alternating magnetic field $H(t)$ oscillating between $-H_0$ and $H_0$, where $H_0 = 100 \text{ A/m}$, at a frequency $f = 50 \text{ Hz}$. The non-symmetric hysteresis loop of the material is modeled by the branches: $B_{\pm}(H) = \mu H \pm \alpha (H_0^2 - H^2) \quad \text{for } H \in [-H_0, H_0]$ where $\mu = 6 \times 10^{-2} \text{ T}\cdot\text{m/A}$ and $\alpha = 3 \times 10^{-4} \text{ T}\cdot\text{m}^2\text{/A}^2$. The core is thermally insulated. By the First Law of Thermodynamics, the net work done by the magnetic field per cycle is entirely dissipated as heat. The total energy dissipated per cycle per unit volume includes both hysteresis loss and eddy current loss. The eddy current loss per cycle per unit volume is given by $W_{\text{eddy}} = k f B_{\text{max}}^2$, where $k = 0.5 \text{ J}\cdot\text{m}^{-3}\text{Hz}^{-1}\text{T}^{-2}$ and $B_{\text{max}}$ is the maximum magnetic field achieved in the cycle. If the temperature of the core increases by $1.0 \text{ K}$ in $10 \text{ s}$, what is the heat capacity $C$ of the core?