Chapter 30: Q47P (page 899)

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
$L_{eq} = L_{1} + L_{2}$
(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

Expert verified

a) $L_{e q} = L_{1} + L_{2}$

b) $L_{e q} = \sum_{n = 1}^{n} L_{n}$

Step by step solution

Given

i) $L_{1}$ - Inductance of inductor 1.

ii) $L_{2}$ - Inductance of inductor 2.

iii) $L_{e q}$ - 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:

$V = V_{1} + V_{2} V_{1} - v o l t a g e a c r o s s L_{1} V_{2} - v o l t a g e a c r o s s L_{2}$

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

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

$V = V_{1} + V_{2} L_{e q} I = L_{1} I + L_{2} I$

Hence,

$L e q = L_{1} + L_{2}$

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

Equivalent inductance fornumbers of inductors in series is

$L_{e q} = L_{1} + L_{2} + L_{3} + \dots + L_{n}$

Generalization oftheabove formula is

$L_{e q} = \sum_{n = 1}^{n} L_{n}$

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Short Answer

Step by step solution

Given

Understanding the concept

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

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

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