Power Transmitted in a Coaxial Cable

[Marcus95]

Homework Statement

A coaxial transmission line consists of an inner cylindrical conductor of radius a = 1 mm and a
cylindrical outer conductor chosen to make the characteristic impedance 75 ohm. The space
between the conductors is lled with a gas which can stand a maximum eld of 105 V/m
without dielectric breakdown. Estimate the maximum mean radio-frequency power that
can be transmitted along this line into a matching load.

Homework Equations

On transmission line
$Z = \sqrt{L/C}$
For coaxial with radius a and b:
$C = \frac{2\pi \epsilon \epsilon_0}{ln(b/a)}$
$L = \frac{\mu_0 ln(b/a)}{2\pi}$

The Attempt at a Solution

The voltage between the two conductors is by Gauss's law:
$V = \int \boldsymbol{E} \cdot d\boldsymbol{r} = E_0 ln(b/a)$
hence:
$V_{max} = E_{max} ln(b/a)$
also:
$Z = \sqrt{\mu / \epsilon \epsilon_0} \frac{ln(b/a)}{2\pi}$
Using $P = VI = V^2/Z$ and asuuming $epsilon \approx 1$ gives:
$P_{max} = 0.21 GW$, but the answer is supposed to be 104 W.

What am I doing wrong? Many thanks!

 

[kuruman]

The voltage between the two conductors is by Gauss's law:$$
V = \int \boldsymbol{E} \cdot d\boldsymbol{r} = E_0 ln(b/a)$$

This equation doesn't make sense. The left side is in volts, the right side is in volts/meter. Check your derivation.

 

[Marcus95]

This equation doesn't make sense. The left side is in volts, the right side is in volts/meter. Check your derivation.

Thank you, I got it!

 

[Delta2]

This equation doesn't make sense. The left side is in volts, the right side is in volts/meter. Check your derivation.

I had my doubts about this step but for another reason. There is vector potential $A$ inside coaxial cable, so it is
$E=\nabla V+\frac{\partial A}{\partial t}$

so that
$V=\int Edr$ doesn't hold (because it is equivalent to $E=\nabla V$.
Do we make the approximation $A\approx 0$?

 

[rude man]

It's not necessary to invoke the magnetic potential.

You want to use Gauss's theorem to relate E field to potential difference between the conductors. So first you want to determine the geometry of the cable, i.e. the inner and outer radii. Is there a relation between these radii and Z₀?

Then: use Gauss's theorem relating the E field to the potential difference V and the geometry and compute the max. allowable V for the given max. E.

Don't forget they want the rms power figure (I think - wording is not clear to me). Also assume an ideal matched line.

 

[Delta2]

It's not necessary to invoke the magnetic potential. Then: use Gauss's theorem relating the E field to the potential difference V and the geometry and compute the max. allowable V for the given max. E.

Don't forget they want the rms power figure (I think - wording is not clear to me). Also assume an ideal matched line.

Can you show how you find V using Gauss's theorem(have to solve Poisson's equation?) without using that $V=\int Edr$ because that last integral becomes path dependent when there is vector potential.

 

[rude man]

Can you show how you find V using Gauss's theorem(have to solve Poisson's equation?) without using that $V=\int Edr$ because that last integral becomes path dependent when there is vector potential.

Can you show how you find V using Gauss's theorem(have to solve Poisson's equation?) without using that $V=\int Edr$ because that last integral becomes path dependent when there is vector potential.

We are dealing solely with electric, not magnetic, fields. I have never run into a vector potential such that E = ∇ x A_e the way we write H = ∇ x A (or sometimes B = ∇ x A). The former is in fact impossible if there is charge present since ∇⋅E = ρ, not zero, so no vector potential can exist. In any case, forget all vector potentials.

You can relate V to E by solving Laplace's equation for the space between the conductors but you'd still need to do step 1 below. It would in fact be good to do so as a check if you went with my suggestion which is:

(1) Find the outer radius of said space; (2) relate E(r) to V, the potential between the conductors using Gauss; (3) find where E is the largest, then you can determine the max. allowable V given the max allowable E.

To do step 1 you will need one item of data given to you.

 

[rude man]

Actually, solving Laplace's equation is the more direct route. Use the cylindrical coordinate expression ∇²V = (1/r) d/dr (r dV/dr) = 0 since there are no derivatives with respect to θ or z.

But again, your first task is to find the outer radius of the space between the conductors.