Question

:Inductors in parallel. Two inductors L₁ and L₂ are connected in parallel and 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

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

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

Short answer

a)

$\frac{1}{L_{eq}}=\frac{1}{L_1}+\frac{1}{L_2}$

b)

$\frac{1}{L_{eq}}\sum _{n=1}^N\frac{1}{L_n}$

Step-by-step solution

Given

Hint: Review the derivation for resistors in parallel and capacitors in parallel.

Understanding the concept

Net current through the parallel connection of inductors is the sum of currents through each inductor, and the voltage across each inductor remains the same.

Formulae:

Voltage across the inductor is given by

$\epsilon =-L\frac{di}{dt}$

(a) Show that the equivalent inductance is given by 1Leq=1L1+1L2

Voltage across inductor is given by

$\epsilon =-L\frac{di}{dt}$

For parallel connection:

$i=i_1+i_2$

Differentiate with respect to t.

$$\begin{gathered} \frac{di}{dt}=\frac{d{i}_1}{dt}+\frac{d{i}_2}{dt} \\ -\frac{\epsilon }{L_{eq}}=\frac{-\epsilon }{L_1}+\frac{-\epsilon }{L_2} \end{gathered}$$

Hence,

$\frac{1}{L_{eq}}=\frac{1}{L_1}+\frac{1}{L_2}$

(b) Find out the generalization of (a) for N inductors in parallel?

Equivalent inductance ofNinductors in parallel is

$\frac{1}{L_{eq}}=\frac{1}{L_1}+\frac{1}{L_2}+\frac{1}{L_3}+...+\frac{1}{L_n}$

The generalization of above equation is

$\frac{1}{L_{eq}}=\sum _{n=1}^n\frac{1}{L_n}$