Q: A blacksmith splits open a doughnut-shaped torus as pictured below:
The inner radius of the torus is 1.75 cm, and the outside radius is 2.25 cm. If the torus is made of low-carbon steel, what is its reluctance?
A: Let’s begin by viewing the torus from above prior to being split apart:
The three parameters of interest to us are the torus’ cross-sectional area ( A ), its length ( l ), and its permeability ( μ ). The cross-sectional area is that of a circle and is summed up as follows:
A = πr2
Subtracting the inner radius ( r ) value from that of the outer radius will give us the diameter ( D ) of the metallic core. Dividing this value into halves will give us the radius value that we seek:
r = ½ D
D = 2.25 cm – 1.75 cm
D = 0.50 cm
r = ( 0.5 )( 0.50 cm ) = 0.25 cm = 0.0025 m
A = ( 3.14 )( 0.0025 m )2
A = 0.000019625 = 1.96 x 10-5 m
The magnetic path length ( C ) is most accurately determined via usage of the average radius value with respect to the center of the circle:
ravg = ½ ( 1.75 cm + 2.25 cm ) = 2.0 cm = 0.020 m
l = C = 2πr
l = C = ( 2 )( 3.14 )( 0.020 m )
l = C = 0.126 m
The permeability of low-carbon steel has an estimated value of 2.0 x 10-4 Wb / At*m. We now have the information needed to solve the problem at hand:
ℛ = ( l / μA )
ℛ = [ 0.126 m / ( 2.0 x 10-4 Wb / At*m )( 1.96 x 10-5 m ) ]
ℛ = 32.1 x 106 At / Wb
Q: Mild steel has a relative permeability ( μr ) of 800. What is the reluctance of a metallic core made of mild steel, is 10 cm long, and has a cross-sectional area of 1.0 cm x 1.2 cm?
A: The relative permeability of a material compares its permeability to that of a vacuum ( μ0 ):
μr = ( μ / μ0 )
We need the actual value of permeability to solve the problem. This is obtained by multiplying the relative permeability value by ( μ0 ):
μ = ( μ0 )( μr )
μ = ( 4𝛑 x 10-7 Wb / At*m )( 800 )
μ = 1.00 x 10-3 Wb / At*m
l = 0.010 m
A = ( 0.010 m )( 0.012 m )
A = 1.2 x 10-4 m2
And finally,
ℛ = [ 0.010 m / ( 1.00 x 10-3 Wb / At*m )( 1.2 x 10-4 m2 ) ]
ℛ = 8.33 x 105 At / Wb
RENAISSANCE PHYSICS 