Referring to the coaxial cable as shown in Fig. 5.32, determine the inductance per unit length of the cable from the magnetic energy stored in the cable.
In the region a < ρ < b, inserting Eq. (5-106b) into Eq. (5-126), the magnetic energy is
BB=2πρμoIaaϕ (a < ρ < b) (5-106b)
Wm=2μ∫νH2dv=2μ1∫νB2dv [J] (5-126)
Wm1=2μo1∫ρ=aρ=b∫ϕ=0ϕ=2π[2πρμoI]2ρdρdϕ=4πμoI2lnab (5-128a)
In the region 0 ≤ ρ ≤ a , inserting Eq. (5-106c) into Eq. (5-126), the magnetic energy is
BB=2πa2μoρIaaϕ (0 ≤ ρ ≤ a) (5-106c)
Wm2=2μo1∫ρ=0ρ=a∫ϕ=0ϕ=2π[2πa2μoρI]2ρdρdϕ=16πμoI2 (5-128b)
The inductance per unit length of the cable is obtained by inserting Eq. (5- 128) into Eq. (5-127):
L=I22Wm [J] (5-127)
L=I22(Wm1+Wm2)=2πμolnab+8πμo [H/m] (5-129)
We have the same result as Eq. (5-110).
L=IΛ1+Λ2=2πμolnab+8πμo≡Lex+Lin [H/m] (5-110)