Magnetic Field of Coaxial Cable Using Ampere's Law

Edited by Joe Zein Fall 2017; Later Edited by David Hammett Fall 2017;

The Main Idea

A coaxial cable is a type of cable used as a transmission line in many modern electronics.

Coaxial cables are so named because they are comprised of two cylindrical conductors that share a central axis (the center of the cable.) The inner cylinder carries the current in one direction, is surrounded by an insulating region, and a second cylinder outside of that region carries current in the opposite direction, and they are both concentric along the same axis of the center of the wire. An outer insulating layer often surrounds this outer region to protect the second conductor.

Coaxial cables are common household items used in many electronics because of their ability to block external interference from other electromagnetic fields and for their ability to carry currents over long distances and to be able to exclusively carry their signal in the region between the two conductors.

The magnetic field of a coaxial cable can easily be found by applying Ampere's Law!

A Mathematical Model

Ampere's Law

We know that moving charges create a curly magnetic field around their trajectories by the Biot-Savart Law.

As a result, a current-carrying wire creates a curly magnetic field around it. This magnetic field curls around the center of the wire.

We can turn this observation around and note that if there is a curly magnetic field around some region, some current must be flowing through that region. We quantify this relationship through Ampere's Law.

Ampere's Law quantifies the relationship between: 1-the sum of the magnetic field measured at infinitely many points around a path AND 2-the current enclosed in that path.

Another way to express the relationship described by Ampere's Law is that the path integral (represented by the integral with an inscribed circle in the equation below) of magnetic field along a closed path is equal to the sum of the current flowing up through that curved path times the constant mu-naught, which is the vacuum permeability, equivalent to exactly 4*pi*10^(-7).

A Computational Model

When considering calculations using Ampere's law for coaxial cables, simply realize that we may draw a boundary curve around the 2-dimensional region that we wish to investigate, and relate the path integral of the magnetic field along this curve to the current going through that region using Ampere's law.

In the case of a coaxial cable, these currents differ in direction (and may differ in magnitude), but they share a common center. Therefore, if we wish to learn about the magnetic field at certain points inside or outside of a coaxial cable, we need only draw a boundary curve around the radius from the center of these two current-carrying wires at the location at which we desire to investigate the magnetic field, and note that the magnetic field will be uniform in magnitude (but not in direction) everywhere along that surface. (The direction will always be parallel to the path and perpendicular to the radius line at that point.) Since this field is uniform, we may simply write that the magnitude of magnetic field B is equal to the product of the enclosed current and mu-naught divided by the circumference, along which the magnetic field-distance product is uniform. See the diagram and derivation below for further explanation of this.

Example and Analysis

Current flows up the inner conductor and returns down the outer conductor. The inner conductor has radius Ri; the outer conductor is a cylindrical shell with inner radius Ro,1 and outer radius Ro,2. Currents in the inner and outer conductors are uniformly distributed throughout their cross-sectional areas with densities (current/area) Ji and Jo, respectively. Ji =2000A/m2;Ri =0.3mm;Ro,1 =2.0mm;Ro,2 =2.1mm.

The goal here is to determine magnitude and direction of magnetic fields at different locations in and out the wire.

For this, we simply have to apply Ampere's Law to the problem and imagine an Amperian surface with a radius of r:

Case A: Ri< r < Ro,1.

Direction.

To evaluate direction we make use of the right hand rule. The current enclosed by a circle of radius r is the current carried by the central wire. It points out of the page. If we put our thumb in this direction, we see our fingers curling Counterclockwise. The purple arrows show the direction of B at different locations (different r).

Magnitude.

We know that:.

What would the magnitude of the magnetic field be at r = 1mm?

Case B: r > Ro,2.

Magnitude.

Show that B would be zero if and only if current is equal and opposite in direction. Also, calculate J_not, the current density in the outer copper wire.

Conclusion.

To conclude, we now know that the coax produces no magnetic field outside the wire as long as current in each wire are equal in magnitude and opposite in direction. We also know how to evaluate the field anywhere in and out the wire as long as we know the current enclosed.

Connectedness

How is this topic connected to something that you are interested in? How is it connected to your major? Is there an interesting industrial application? The main advantage of coax cables is that, as we've demonstrated, do not produce any magnetic field outside of it. This is an advantage because it allows these cables to be installed next to metals without losing power. Also, it prevents any interference coming from outer space.

Electrical engineering:. Coax cables are used in transmission of signals. Thus they have tons of applications in Its applications include feedlines connecting radio transmitters and receivers with their antennas, computer network (Internet) connections, digital audio (S/PDIF), and distributing cable television signals.

History

Oliver Heaviside designed the Coaxial Cable in 1880 in England. Over the years the cable was enhanced before its use as the first transatlantic cable in 1956. The cable was, as mentioned, was developed due to its advantages over previous cables in that it provides constant conductor spacing and resistance over electromagnetic interference, making it useful in long distance transmission lines.

References

Chabay, Sherwood. (2011). Matter and Interactions (3rd ed.). Raleigh, North Carolina: Wiley.

Coaxial cable. (n.d.). Retrieved November 27, 2016

Exam 4 Fall 2015 Question 3

Magnetic Field of Coaxial Cable Using Ampere's Law

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