Question #95564

A long cylindrical paramagnetic core is placed in an environment at absolute zero temperature ($T_0 = 0\text{ K}$). The core is subjected to a superposed external magnetic field intensity $H(t) = H_0 + H_0 \sin(\omega t)$, where $H_0$ is a positive constant in $\text{A m}^{-1}$. Assume the material is highly paramagnetic such that its susceptibility obeys Curie's Law $\chi = C/T \gg 1$ (where $C$ is the Curie constant in $\text{K}$), allowing the approximation for the magnetic field inside the core to be $B(t) \approx \mu_0 \chi H(t)$. The time-varying component of the magnetic field induces eddy currents, resulting in a time-averaged power dissipation per unit volume given by $P_{in} = \alpha B_{ac, rms}^2$, where $B_{ac, rms}$ is the root-mean-square value of the AC component of the magnetic field inside the core, and $\alpha$ is a constant in $\text{W m}^{-3} \text{T}^{-2}$. The core dissipates heat to its surroundings at a rate per unit volume given by $P_{out} = \beta T$, where $\beta$ is a constant in $\text{W m}^{-3} \text{K}^{-1}$ and $T$ is the steady-state absolute temperature of the core. Assuming uniform temperature distribution, what is the total root-mean-square magnetic field $B_{total, rms}$ inside the core in the steady state?