In a long bar having the shape of rectangular parallelepiped with sides $a$, $b$, and $c$ ($a \gg b \gg c$), made from the semiconductor InSb flows a current $I$ parallel to the edge $a$. The bar is in externam magnetic field $B$ which is parallel to the edge $c$. The magnetic field produced by the current $I$ can be neglected. The currect carriers are electrons. The average velocity of electrons in a semiconductor in the presence of an electric field only is $v= \mu E$, where $\mu$ is called mobility. Of the magnetic field is also present, the electric field is no longer parallel to the current. This phenomenon is knows as the Hall effect.

A1 Determine what the magnitude and the direction of the electric field in the bar is, to yield the current described above.

A2 Calculate the difference of the electric potential between the opposite points on the surface of the bar in the direction of the edge $b$.

A3 Find the analytical expression for the DC component of the electric potential difference if the current and the magnetic field are alternating (AC); $I=I_0 \sin \omega t$ and $B= B_0 \sin (\omega t + \delta)$

A4 Design and explain the electric circuit wich would make possible, be exploting the result A3, to measure the power consumption of an electric apparatus connected with the AC network.

The electron mobility in $\rm InSb$ is $7.8~\text{m}^2\text{T}/\text{V}\text{s}$, the electron concentration in $\rm InSb$ is $2.5\cdot10^{22}~\text{m}^{-3}$, $I=1.0~\text{A}$, $B=0.10~\text{T}$, $b=1.0~\text{cm}$, $c=1.0~\text{mm}$, $e_0=1.6 \cdot 10^{-19}~\text{C}$