In this problem, we will figure out how conductivity transforms between frames of reference.

A laboratory is filled with a material whose conductivity is $.$ An electric field of magnitude $E$ points in the $x$ direction. This causes a current to flow normal to a square loop with sides of length $l.$

$($a$)$ In an interval of time $t,$ how much charge $q$ flows through the loop?

$2$

This laboratory is now placed on a large train that is moving in the $x$ direction with very large velocity $v.$ An interval of time $t$ passes on the train. We observe all of this as the train goes past us$.$

$($b$)$ During the "train time" interval $t,$ what time interval $t\text{'}$ passes according to us$?$ What area $A^{\text{'}}$ do we observe? How much charge $q^{\text{'}}$ passes through the loop in this interval?

$($c$)$ Using the results of part $($b$),$ compute the current density $J^{\text{'}}$ we see.

$($d$)$ Using the result of part $($c$)$ plus the rule for transforming electric fields, compute the conductivity ${}^{\text{'}}$ that we measure.

The lab is now taken off the train and placed on an elevator which climbs upward, in the $y$ direction, with the same very large velocity $v.$ An interval of time $t$ passes on the elevator. We observe the lab go past us while standing next to the elevator doors.

$($e$)$ During the "elevator time" interval $t,$ what time interval $t^{\text{'}}$ passes according to us$?$ What area $A^{\text{'}}$ do we observe? How much charge $q^{\text{'}}$ passes through the loop in this interval?

$($f$)$ Using the results of part $($e$),$ compute the current density $J^{\text{'}}$ we see.

$($g$)$ Using the result of part $($f$)$ plus the rule for transforming electric fields, compute the conductivity ${}^{\text{'}}$ that we measure. Does it agree with your result for part $($d$)?$ Should it$?$