In terms of the dc current I, how much magnetic energy is stored in the insulating medium of a 3- m-long, air –filled section of a coaxial transmission line, given that the radius of the inner conductor is 5 cm and the inner radius of the outconductor is 10 cm? Use two methods. One is to take the coaxial line an inductor and calculate the energy using . The other is integrate the energy density over the volume.

The answer is 138.5

Explanation:

STEP 1:

The inductance per unit length of a coaxial transmission line is

L′=L/ I

 =Ø/H

=μoI/2π In (b/a)

In this a is the radius of inner conductor

b is the radius of outer conductor

I is the coaxial transmission

μ is the magnetic permeability

Since the transmission of the charge exists in air, the value of the relative permeability is μr= I and permeability of free space is μo=  4π x 10-7 H/m . So the magnetic permeability will be

μ = μoμ r

μ =μ o(I) 4π x 10-7 H/m

L′= μoI/2π In (b/a)

  = (4π x 10-7 ) (2)/2π In (10/5)

  =2.77 x 10-7 H

STEP 2:

Obtain the magnetic energy stores in the magnetic field H of a volume of the coaxial transmission line containing a material with permeability μ, by using the formula given below:

Wm= 1/2 LI^2

      = 1/2 (2.77x 10^-7 I^2

      = 138.5 X 10^-9 I^2 J

      Now we will simplify the equation

Wm= 185.5I^2 nJ

So, the magnetic energy stored in insulating medium is 185.5I^2 nJ

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