AC Voltage applied to coaxial cable

[insynC]

Homework Statement

A coaxial cable consists of two long, concentric, perfect conductors. The inner conductor is a right circular cylinder of radius a, while the outer conductor is a thin, hollow annular cylinder of inner radius b.

Suppose a half-infinite length of such a coaxial cable stretches between z = 0 and z = +∞. A signal generator is attached between the two conductors at z = 0, and a periodic voltage V(t) = V(0)sin(ωt) is applied. What is the current I(z,t) that flows along the surface of the inner conductor at some fixed z > 0?

The Attempt at a Solution

This question was posed for revision of a previous course. But in that course little if any AC circuit work was done. The precursor question to this was to find the capacitance and inductance per unit length of an infinite coaxial cable and these were fine, but I'm completely lost with this.

Any suggestions on how to get started?

Thanks

 

[Jhero]

for your post! I am a scientist who specializes in electrical engineering and I am happy to help you with this problem.

To start, we need to understand the basic principles of an AC circuit. In an AC circuit, the current and voltage vary with time in a periodic manner. The voltage is typically described by a sine or cosine function, as shown in the problem statement. The current in the circuit is related to the voltage by Ohm's Law, which states that the current is equal to the voltage divided by the resistance.

In this problem, we are dealing with a coaxial cable, which consists of two conductors separated by an insulating material. The current will flow through the inner conductor and return through the outer conductor. Since the outer conductor is a perfect conductor, we can assume that there is no resistance in that part of the circuit. Therefore, the current flowing through the inner conductor will be equal to the voltage divided by the resistance of the inner conductor.

To find the resistance of the inner conductor, we need to use the formula for resistance, which is equal to the resistivity of the material times the length divided by the cross-sectional area. In this case, the length of the inner conductor is infinite, so we can ignore that term. The cross-sectional area of the inner conductor is given by the formula for the area of a circle, πr^2, where r is the radius of the inner conductor. Therefore, the resistance of the inner conductor is equal to the resistivity of the material times πa^2.

Now, to find the current at a fixed z > 0, we need to use the voltage and resistance we just calculated in Ohm's Law, I = V/R. Since the voltage is given by V(t) = V(0)sin(ωt), the current will also vary with time in a sinusoidal manner. Therefore, the current at a fixed z > 0 can be described by the equation I(z,t) = V(0)sin(ωt)/[πa^2ρ], where ρ is the resistivity of the material.

I hope this helps you get started on solving this problem. Let me know if you have any further questions. Good luck!