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 rho and length L. Assume that the solenoid is much longer than its radius, so that its field can be determined from Ampere's law throughout its entire length: B(r) middot dl = mu_0 I_encl. 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) middot dl = -d/dt Phi_M (t). Consider first the generation of the magnetic field by the current I_1 (t) in solenoid 1. Within the solenoid (sufficiently far from its ends), 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 constant