For parallel plane transmission line with two conducting plates of conductivity σ, thickness = t, and separation of 'd' and dielectric parameters ε, μ and length 'b' following is given : For b >> d (A) \(L=\dfrac{\mu b}{t}\) Choose the most appropriate answer from the options given below :
(B) \(\text{Capacitance}=\dfrac{\varepsilon b}{d}\)
(C) \(G=\dfrac{\sigma b}{s}\)
(D) \(Z_{0}=\sqrt{\dfrac{\mu}{\varepsilon}}\left(\dfrac{d}{b}\right)\)
(B), (C), (D) Only
Statement (A) has the wrong dimension in its denominator; the other three are the standard parallel-plate results — option 4.
The geometry. Two plates of width b separated by d, with \(b\gg d\) so that fringing may be ignored and the field treated as uniform between them.
(B) — capacitance per unit length. The ordinary parallel-plate formula with area per unit length equal to b:
\(C=\dfrac{\varepsilon\times\text{area}}{\text{separation}}=\dfrac{\varepsilon b}{d}\)
(C) — shunt conductance per unit length. Leakage through the dielectric follows the same geometry, with conductivity in place of permittivity:
\(G=\dfrac{\sigma b}{d}\)
(the symbol printed as s is the plate separation). The pattern is worth noting: G and C share the same geometric factor \(b/d\), so their ratio is always \(G/C=\sigma/\varepsilon\) whatever the shape of the line.
(D) — characteristic impedance. For a lossless line,
\(Z_{0}=\sqrt{\dfrac{L}{C}}=\sqrt{\dfrac{\mu d/b}{\varepsilon b/d}}=\sqrt{\dfrac{\mu}{\varepsilon}}\left(\dfrac{d}{b}\right)\)
This is the intrinsic impedance of the dielectric scaled by the geometry — a wide, closely spaced line has a low impedance, exactly as one expects.
(A) is the false statement, and the derivation of (D) shows why. The inductance per unit length of a parallel-plate line is
\(L=\dfrac{\mu d}{b}\)
— the separation over the width, since the magnetic flux threads the region between the plates. Statement (A) writes \(\mu b/t\), using the plate thickness t in the denominator and inverting the ratio. The thickness governs the plates' own series resistance through the skin effect, not the inductance of the field between them.
A check that catches (A) at once : substituting \(L=\mu b/t\) into \(Z_{0}=\sqrt{L/C}\) would give an impedance depending on the plate thickness, which is plainly wrong — making the plates thicker cannot change a line's characteristic impedance. Statement (D), which the same option set accepts, is inconsistent with (A) and consistent with the correct \(L=\mu d/b\).
Hence, the correct statements are (B), (C) and (D).
