Magnetic Field of Coaxial Cable Using Ampere's Law

Edited by Joe Zein Fall 2017;

The Main Idea

A coaxial cable consists of two cylindrical channels, which is how it derives its name (copper wire and copper mesh).

One of the cylinders carries the current in one direction, surround by an insulating region, and a second cylinder carries the current in the opposite direction, and they are both concentric along the same axis.

Coaxial cables are common household items used in many electronics because of their ability to avoid external interference from other fields and for their ability to carry currents for 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 with the application of Ampere's Law.

A Mathematical Model

Ampere's Law

We know that moving charges create a curly magnetic field around their trajectories. As a result, a current carrying wire creates a curly magnetic field arounds it.

Reversely, observing a curly magnetic field in space implies there should be some current producing it.

Ampere's law is a quantitative relationship that relates: 1-the sum of the magnetic field measured at infinitely many points around a path 2-the current enclosed in that path

Examples

A coaxial cable with an outer cylinder of radius R2 surround an inner cylinder of radius R1. They both carry a constant current I, however, in opposite directions. The inner cylinder carries the current to the left and the outer cylinder carries the current to the right. Find the magnitude and direction of the magnetic field when R1< r <R2, or in other words at a distance in between the two cylinders.

We simple have to apply Ampere's Law to the problem and imagine an Amperian surface with a radius of r:

The direction would be clockwise due to the right hand rule. Simply point your thumb in the direction of the current and curl your fingers in the direction of the magnetic field.

What would the magnetic field be at a distance outside of the outer cylinder?

The magnetic field would be zero. This is what is special about coaxial cables-because the current runs in opposing directions, it cancels out at regions outside of the outer cylinder. However, this is only true because the currents are EQUAL and OPPOSITE. If the outer current was greater, the magnetic field would be equal to:

in the opposite direction (counterclockwise).

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?

Coaxial cables are also used to support medical devices, which I as a Biomedical Engineer will be focused on developing and maintaining. Since the coaxial cables are effective avenues of transmission that need to avoid electromagnetic interference, they are used in ECGs, EEGs, and CT scanners.

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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