0195136047.pdf

12.4 FORCES AND TORQUES IN MAGNETIC-FIELD SYSTEMS 535

|f|=

φ^2

2

dR(x)

dx

=

1

2

(

Ni

R

) 2

dR

dx

=

i^2

2

N^2

R^2

dR

dx

=

1. 52

2

10002

0. 6285 × 10 −^8

x^2

Forx= 1 .5mm= 1. 5 × 10 −^3 m,

|f|=

1. 52

2

10002

0. 6285 × 10 −^8

( 1. 5 × 10 −^3 )^2

= 3142 .5N

The student should recognize the practical importance of force-generating capabilities

of electromechanical transducers.

EXAMPLE 12.4.3

Solenoids find application in a variety of electrically controlled valves. The magnetic structure
shown in Figure E12.4.3 is a simplified representation of a solenoid in which the flux in the air
gap activates the motion of the iron plunger.

(a) Develop a general expression for the force exerted on the iron plunger and comment on

its dependence on positionx.

(b) Determine the current through the coil ofN=100 turns to pull the plunger tox=a,

given thata=1 cm,lg=1 mm, and the spring constant isk=1 N/m. Assume the

permeability of the magnetic structure to be infinite and neglect fringing.

Spring 1)

Nonmagnetic

Movable iron bushing material

plunger

N turns

i

2 a

a

a a

x x = 0

lg

a

Fixed magnetic

core structure

(μ ∞)

(μr

Figure E12.4.3Simplified representation of a

solenoid.