Question

Two long cylindrical coils with uniform windings of the same length and nearly the same radius have inductances L₁ and L₂. The coils are coaxially inserted into each other and connected to a current source as shown in Fig. 110. The directions of the current in the circuit and in the turns are shown by arrows.

Determine the inductance L of such a composite coil.

Answer 1

Let N₁ be the number of turns of the coil of inductance L₁, and N₂ be the number of turns of the coil of inductance L₂. It should be noted that the required composite coil of inductance L can be treated as a coil with N = N₁ +N₂ turns. If the relation between the inductance and the number of turns is known, L can be expressed in terms of L₁ and L₂. For a given geometrical configuration of the coil, such a relation must actually exist because inductance is determined only by geometrical configuration and the number of turns of the coil (we speak of long cylindrical coils with uniform winding). Let us derive this relation.

From the superposition principle for a magnetic field, it follows that the magnetic field produced by a current I in a coil of a given size is proportional to the number of turns in it. Indeed, the doubling of the number of turns in the coil can be treated as a replacement of each turn by two new closely located turns. These two turns will produce twice as strong a field as that produced by a single turn since the fields produced by two turns are added. Therefore, the field in a coil with twice as many turns is twice as strong. Thus, B ∝ N (B is the magnetic induction, and the current is fixed). It should be noted that the magnetic flux embraced by the turns of the coil is

Thus, we obtain L = kN² for a given geometry. Further, we take into account that N₁ = √(L₁/k), N₂ = √(L₂/k), and hence L = k (N₁ + N₂)². Consequently,