Solve for Part E Learning Goal: To learn about mutual inductance from an example of a long solenoid with two windings. To illustrate the calculation of mutual inductance it is helpful to consider the specific example of two solenoids that are wound on a common cylinder. We will take the cylinder to have radius $\backslash$rho and length L$.$ Assume that the solenoid is much longer than its radius, so that its field can be determined from Amp$$re$\text{'}$s law throughout its entire length: $$B$($r$)$ d l$=\backslash$mu $\_0$ I$\_$encl. $($Figure $1)$ We will consider the field that arises from solenoid $1,$ which has n$\_1$ turns per unit length. The magnetic field due to solenoid $1$ passes $($entirely$,$ in this case$)$ through solenoid $2,$ which has n$\_2$ turns per unit length. Any change in magnetic flux from the field generated by solenoid $1$ induces an EMF in solenoid $2$ through Faraday's law of induction, $$E$($r$)$ d l$=-$d$/$d t$\backslash$Phi $\_$M$($t$).$Consider first the generation of the magnetic field by the current I$\_1($t$)$ in solenoid $1.$ Within the solenoid $($sulficiently far from its onds$),$ what is the magnitude B$1.($t$)$ of the magnetic field due to this current? Express B$\_1($t$)$ In terms of I$\_1($t$),$ variables given in the introduction, and relevant constants. Correct Wote that this fiald is independent of the radial position $($the distance from the symmetry axis$)$ for points inside the solenoid. Part B What is the flux $$h $($t$)$ generated by solenoid $1\text{'}$s magnetic field through a single turn of solenoid $27$ Express $\_1($t$)$ in terms of B$\_1($t$),$ quantities given in the Introduction, and any needed constants.Now find the electromotive force $\_2($t$)$ induced across the entirety of solenoid $2$ by the change in current in solenoid $1.$ Remember that both solenoids have length L$.$ Express your answer in terms of d I$\_2($t$)/$ d t$,$ n$\_1,$ n$\_2,$ other parameters given in the introduction, and any relevant constants. View Avaliable Hint$($s$)$ Part D This overall interaction is summarized using the symbol M$\_21$ to indicate the mutual inductance between the two windings. Based on your previous two answers. which of the following formulas do you thing is the correct one? $[\backslash$epsi $\_2($t$)=-$M$\_21$ I$\_1($t$)$; d$/$d t$\backslash$epsi $\_2($t$)=-$M$\_21$ I$\_1($t$)$; $\backslash$epsi $\_2($t$)=-$M$\_21$d$/$d i I$\_1($t$)$; I$\_1($t$)=-$M$\_21$d$/$d t$\backslash$epsi $\_2($t$)$; f$\_1($t$)=-$M$\_$m E$\_2($t$)]$ Using the formula for the mutual inductance, $\_2($t$)=-$M$\_21$d$/$d t I$\_1($t$),$ find M$\_21.$ Express the mutual inductance M$\_21$ in terms of n$\_1,$ n$\_2,$ quantities given in the introduction, and relevant physical constants.