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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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Most popular questions from this chapter

In Fig. 30-23, a long straight wire with current ipasses (without touching) three rectangular wire loops with edge lengths L, 1.5L, and 2L. The loops are widely spaced (so as not to affect one another). Loops 1 and 3 are symmetric about the long wire. Rank the loops according to the size of the current induced in them if current iis (a) constant and (b) increasing, greatest first.

Figure shows a rod of length L = 10.0 cm that is forced to move at constant speed v = 5.0 m/s along horizontal rails. The rod, rails, and connecting strip at the right form a conducting loop. The rod has resistance 0.400惟; the rest of the loop has negligible resistance. A current i = 100 Athrough the long straight wire at distance a = 10.0 mm from the loop sets up a (non-uniform) magnetic field through the loop. (a) Find the emf. (b) Find the current induced in the loop. c) At what rate is thermal energy generated in the rod? (d) What is the magnitude of the force that must be applied to the rod to make it move at constant speed?(e) At what rate does this force do work on the rod?

A wire is bent into three circular segments, each of radius r=10 cm, as shown in Figure. Each segment is a quadrant of a circle, ab lying in the xy plane, bc lying in the yz plane, and ca lying in the zx plane. (a) If a uniform magnetic fieldB鈫�points in the positive x direction, what is the magnitude of the emf developed in the wire when B increases at the rate of 3.0 mT/s? (b) What is the direction of the current in segment bc?

Question: In Figure, two straight conducting rails form a right angle. A conducting bar in contact with the rails starts at the vertex at time t = 0and moves with a constant velocity of 5.20m/salong them. A magnetic field with B = 0.350 Tis directed out of the page. (a) Calculate the flux through the triangle formed by the rails and bar atT = 3.00S. (b) Calculate the emf around the triangle at that time. (c) If the emf is蔚=atn, where a and n are constants, what is the value of n?

The wire loop in Fig. 30-22ais subjected, in turn, to six uniform magnetic fields, each directed parallel to the axis, which is directed out of the plane of the figure. Figure 30- 22bgives the z components Bz of the fields versus time . (Plots 1 and 3 are parallel; so are plots 4 and 6. Plots 2 and 5 are parallel to the time axis.) Rank the six plots according to the emf induced in the loop, greatest clockwise emf first, greatest counter-clockwise emf last.
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