Question

Inductors in series. Two inductors L₁ and L₂ are connected in series and are separated by a large distance so that the magnetic field of one cannot affect the other.(a)Show that the equivalent inductance is given by

${\mathbf{L}}_{\mathbf{eq}}\mathbf{=}{\mathbf{L}}_{\mathbf{1}}\mathbf{+}{\mathbf{L}}_{\mathbf{2}}$

(Hint: Review the derivations for resistors in series and capacitors in series. Which is similar here?) (b) What is the generalization of (a) for N inductors in series?

Short answer

a)$L_{eq}=L_1+L_2$

b)$L_{eq}=\sum _{n=1}^n{L}_n$

Step-by-step solution

Given

i) $L_1$- Inductance of inductor 1.

ii) $L_2$- Inductance of inductor 2.

iii) $L_{eq}$- Equivalent inductance of series connection.

iv) Hint: Review the derivation for resistors in series and capacitors in series.

Understanding the concept

Net voltage across the series connection of inductors is the sum of voltages across each inductor.

Formulae:

Net voltage for series connection of inductors:

$$\begin{gathered} V=V_1+V_2 \\ {V}_1-voltageacross{L}_1 \\ {V}_2-voltageacross{L}_2 \end{gathered}$$

Show that the equivalent inductance is given by Leq=L1+L2

For series connection, the current is the same for each inductor.

$$\begin{gathered} V=V_1+V_2 \\ {L}_{eq}I=L_{1}I+L_{2}I \end{gathered}$$

Hence,

$Leq=L_1+L_2$

(b) Find the generalization of (a) for   conductors in series

Equivalent inductance fornumbers of inductors in series is

$L_{eq}=L_1+L_2+L_3+\cdots +L_n$

Generalization oftheabove formula is

$L_{eq}=\sum _{n=1}^n{L}_n$