Calculating Self-Inductance and Capacitance in Parallel Plate Transmission Lines

[malindenmoyer]

Two very large parallel conducting plates of very large length $$l$$, and width $$w$$ are separated by a distance $$d$$. A current $$I=Jw$$ flows to the right in the lower plate and to the left in the upper plate. Each of the two currents produces a magnetic field $$\frac{B}{2}$$ between the two plates.

(a) Show that the total field between the plates is $$B=\mu_0 J$$ via Ampere's Circuital Law.

(b) Find the flux $$\phi$$ and the self inductance per unit lenght, $$\frac{L}{l}$$ for this arrangement.

(c) Find the capacitance per unit length, $$\frac{C}{l}$$.

(d) Find $$\sqrt{\frac{L}{C}}$$ in terms of $$\mu_0, \epsilon_0$$ and geometrical factors

My Attempt at Solution
Part (a) is confusing as I have not used the circuital Law for rectangular geometry. I know that Ampere's Law is given by:

$$\oint_C \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 I_{\mathrm{enc}}$$

But am confused as to how to apply it as it is not circular geometry.

In part (b) I know that:

$$\phi=BA$$

But am not sure as to what area to use, since we know B per part (a)

Solving for flux leads us one step closer to finding the self inductance which is:

$$L=\frac{N\phi}{I}$$

But again, I do not know what value to substitute in for $$N$$.

I am pretty sure I can find the capacitance per unit length per (c), and then (d) is a matter of combing (b) and (c) so that would be self explanatory. Could somebody help me get this thing started? Please keep in mind that I have a very elementary understanding of this material.

Thanks.

 

[malindenmoyer]

Anybody?