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Chapter 4: Q15E (page 1014)

(a) A long, straight solenoid has N turns, uniform cross-sectional area A, and length l. Show that the inductance of this solenoid is given by the equation L=渭0AN2l. Assume that the magnetic field is uniform inside the solenoid and zero outside. (Your answer is approximate because B is actually smaller at the ends than at the center. For this reason, your answer is actually an upper limit on the inductance.) (b) A metallic laboratory spring is typically 5cmlong and 0.15cmin diameter and has 50coils. If you connect such a spring in an electric circuit, how much self-inductance must you include for it if you model it as an ideal solenoid?

Short Answer

Expert verified

a)L=渭0AN2l

b)role="math" localid="1664194101205" L=1.11脳10-7H

Step by step solution

01

Magnetic field in a solenoid

For a long solenoid ofNturns, lengthiand currenti, the magnetic field is given byB=渭0ni=渭0NIi.

For the area of cross-section A, the magnetic flux is given by role="math" localid="1664194375207" 蠒=BA=渭0NAil.

The self-inductance is defined as L=N蠒Bi.

Clearly, after substituting the flux equation, we will getL=渭0N2Al .

02

Calculate the self-inductance

Given that I=5cmand D=.015cmandN=50.

role="math" localid="1664194642248" A=蟺D24A=蟺脳0.001524A=1.767脳10-6m2

The self-inductance is given by

L=4蟺脳10-7脳502脳1.767脳10-60.05L=1.11脳10-7H

So, the self-inductance is1.11脳10-7H.

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