Page 209 - Engineering Electromagnetics, 8th Edition

CHAPTER 7 The Steady Magnetic Field 191

Figure 7.8 (a)Cross section of a coaxial cable carrying a uniformly
distributed current I in the inner conductor and −I in the outer conductor. The
magnetic field at any point is most easily determined by applying Amp` ere’s
circuital law about a circular path. (b) Current filaments at ρ = ρ 1 ,φ =±φ 1 ,
produces H ρ components which cancel. For the total field, H = H φ a φ .

If we choose ρ smaller than the radius of the inner conductor, the current
enclosed is
ρ 2
I encl = I
a 2
and
ρ 2
2πρH φ = I
a 2
or
Iρ
H φ = (ρ< a)
2πa 2
If the radius ρ is larger than the outer radius of the outer conductor, no current is
enclosed and
H φ = 0(ρ> c)

Finally, if the path lies within the outer conductor, we have
ρ − b
2 2
2πρH φ = I − I
2
c − b 2
2
I c − ρ 2
H φ = (b <ρ < c)
2πρ c − b 2
2
The magnetic-field-strength variation with radius is shown in Figure 7.9 for
a coaxial cable in which b = 3a, c = 4a.It should be noted that the magnetic
field intensity H is continuous at all the conductor boundaries. In other words, a
slight increase in the radius of the closed path does not result in the enclosure of a
tremendously different current. The value of H φ shows no sudden jumps.