A transmission line is formed by a thin wire with radius a (along z axis) and a

Question

A transmission line is formed by a thin wire with radius a (along z axis) and a cylindrical shell with radius b as shown in figure below.

1)What is the magnetic field intensity for a <?< b?

2) Find the total flux passing through an area of length h, between the inner and outer conductor of the transmission line (assume the medium between the line and cylinder is filled with dielectric with relative permeablity of ?r).

3) Find the external inductance for unit length for this transmission line (between the inner and outer conductors).

Explanation / Answer

a)Magnetic field strength is one of two ways that the intensity of a magnetic field can be expressed. Technically, a distinction is made between magnetic field strength H, measured in amperes per meter (A/m), and magnetic flux density B, measured in Newton-meters per ampere (Nm/A), also called teslas (T).
The magnetic fields generated by currents and calculated from Ampere's Law or the Biot-Savart Law are characterized by the magnetic field B measured in Tesla. But when the generated fields pass through magnetic materials which themselves contribute internal magnetic fields, ambiguities can arise about what part of the field comes from the external currents and what comes from the material itself. It has been common practice to define another magnetic field quantity, usually called the "magnetic field strength

b)
The magnetic field can be visualized as magnetic field lines. The field strength corresponds to the density of the field lines. The total number of magnetic field lines penetrating an area is called the magnetic flux. The unit of the magnetic flux is the tesla meter squared (T · m2, also called the weber and symbolized Wb). The older units for the magnetic flux, the maxwell (equivalent to 10-8 Wb), and for magnetic flux density, the gauss (equivalent to 10-4 T), are obsolete and seldom seen today.
Flux is a q uantitative measure of the numb er of lines of a vector field that passes
perpendicularly through a surface. Figure 22.1a, shows an electric field E passing
through a portion of a surface of area A. The area of the surface is represented by a
vector A, whose magnitude is the area A of the surface, and whose direction is
perpendicular to the surface. That an area can be represented by a vector
The electric flux is defined to be
E = E * A = EA cos (22.1)
and is a quantitative measure of the number of lines of E that pass normally through
the surface area A. The number of lines represents the strength of the field. The
vector E, at the point P of figure 22.1(a), can be resolved into the components, E
the component perpendicular to the surface, and E|| the parallel component. The
perpendicular component is given by
E = E cos (22.2)
while the parallel component is given by
E|| = E sin (22.3)
The parallel component E|| lies in the surface itself and therefore does not pass
through the surface, while the perpendicular component E completely passes
through the surface at the point P. The product of the perpendicular component E
and the area A
EA = (E cos)A = EA cos = E • A = E (22.4)
is therefore a quantitative measure of the number of lines of E passing normally
through the entire surface area A. If the angle in equation 22.1 is zero, then E is
parallel to the vector A and all the lines of E pass normally through the surface

c)

External Inductance

In electromagnetism and electronics, inductance is the property of an electrical conductor by which a change in current through it induces an electromotive force in both the conductor itself[1] and in any nearby conductors by mutual inductance.Let us consider an isolated straight conductor The conductor carries a current I . Assume that the tubular element at a distance x from the center of the conductor has a field intensity Hx . Since the circle with a radius of x encloses the entire current.

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