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Chapter 30: Q48P (page 899)

: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

1Leq=1L2+1L2

(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

Expert verified

a)

1Leq=1L1+1L2

b)

1Leq鈭�n=1N1Ln

Step by step solution

01

Given

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

02

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

蔚=-Ldidt

03

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

Voltage across inductor is given by

蔚=-Ldidt

For parallel connection:

i=i1+i2

Differentiate with respect to t.

didt=di1dt+di2dt-蔚Leq=-蔚L1+-蔚L2

Hence,

1Leq=1L1+1L2

04

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

Equivalent inductance ofNinductors in parallel is

1Leq=1L1+1L2+1L3+...+1Ln

The generalization of above equation is

1Leq=鈭�n=1n1Ln

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In Figure, a wire forms a closed circular loop, of radius R = 2.0mand resistance4.0惟. The circle is centered on a long straight wire; at time t = 0, the current in the long straight wire is 5.0 Arightward. Thereafter, the current changes according toi=5.0A-(2.0As2)t2. (The straight wire is insulated; so there is no electrical contact between it and the wire of the loop.) What is the magnitude of the current induced in the loop at times t > 0?

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

Leq=L1+L2

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