Show that the Fermi momentum $p_{F}$ of a gas of $N$ non-interacting electrons in volume $V$ is

$$p_{F}=\left(3 \pi^{2} \hbar^{3} \frac{N}{V}\right)^{1 / 3}$$

Consider the electrons to be effectively massless, so that an electron of momentum $p$ has (relativistic) energy $c p$. Show that the mean energy per electron at zero temperature is $3 c p_{F} / 4$.

When a constant external magnetic field of strength $B$ is applied to the electron gas, each electron gets an energy contribution $\pm \mu B$ depending on whether its spin is parallel or antiparallel to the field. Here $\mu$ is the magnitude of the magnetic moment of an electron. Calculate the total magnetic moment of the electron gas at zero temperature, assuming $\mu B$ is much less than $c p_{F}$.