We have proved that the mutual inductance of coil 1 due to coil 2 are equal to the mutual inductance of coil 2 due to coil 1 (L12 =L21 =M). In a complicated situation (like this problem) that calculating the flux of one coil linking to the other coil (w21) is difficult, we can instead calculate the w21 and then calculate the mutual conductance. The figure shows a short solenoid (with length l, radius a , and n1 turns per unit length) lies on the axis of a very long solenoid (with radius b, and n2 turns per unit length). The short solenoid carries current i . We would like to determine the mutual inductance of these two coils [Since the inner solenoid is short, it has a very complicated field, and it puts a different amount of flux through each turn of the long solenoid.]. Instead we can assume that the current is flowing through the wire of the long solenoid and it is inducing on short solenoid. To find the mutual inductance , do these two steps: (a) Determine the magnetic flux density inside of a very long solenoid (, , ) using Ampere's law. (b) Now assume that a short solenoid (a,n1 ,l ) lies inside of this long solenoid. Calculate the mutual inductance of these two solenoids.

We have proved that the mutual inductance of coil 1 due to coil 2 are equal to the mutual inductance of coil 2 due to coil 1 (L12 =L21 =M). In a complicated situation (like this problem) that calculating the flux of one coil linking to the other coil (w21) is difficult, we can instead calculate the w21 and then calculate the mutual conductance. The figure shows a short solenoid (with length l, radius a , and n1 turns per unit length) lies on the axis of a very long solenoid (with radius b, and n2 turns per unit length). The short solenoid carries current i . We would like to determine the mutual inductance of these two coils [Since the inner solenoid is short, it has a very complicated field, and it puts a different amount of flux through each turn of the long solenoid.]. Instead we can assume that the current is flowing through the wire of the long solenoid and it is inducing on short solenoid. To find the mutual inductance , do these two steps: (a) Determine the magnetic flux density inside of a very long solenoid (, , ) using Ampere's law. (b) Now assume that a short solenoid (a,n1 ,l ) lies inside of this long solenoid. Calculate the mutual inductance of these two solenoids.