Coaxial waveguide (Homework 2-323)

4. Consider a cylindrical coaxial waveguide of inner radius r1 and outer radius r2. The space between conductors is vacuum.

a. Suppose you apply a sinusoidal voltage of amplitude V0 and angular frequency ω across the inner and outer conductor at one end of the waveguide; a disturbance travels down the waveguide in the lowest TEM mode (assume the waveguide is infinitely long). Find the resulting current flowing along one or the other of the conductors.

This homework is continued here

In a coaxial cable there are possible TEM modes (that is both $E_z$ and $B_z$ along cable are =0). So it remains only the transverse components of E and B that are perpendicular to each other. From equation 1.197 these E and B fields are (cylindrical coordinates $(r,ϕ,z)$)

$E ⃗(r,ϕ,z,t)=r ̂*(A*\cos⁡(kz-ωt))/r$

$B ⃗(r,ϕ,z,t)= ϕ ̂*(A*\cos⁡(kz-ωt))/(c*r)$

Since the voltage is given $V=V_0*\cos⁡ (ωt)$ at $z=0$ (one end of cable) we have

$E=-∇V$ with $∇=r ̂*∂/∂r$ in cylindrical coordinates

$V(z,t)=-∫ E_r dr=A*ln⁡ (r*\cos⁡(kz-ωt))$

and therefore $A*ln⁡(r)=V_0$

The fields are

$E=r ̂*V_0*(ln⁡(r))/r*\cos⁡ (kz-ωt)$

$B=ϕ ̂*V_0/c*(ln⁡(r))/r*\cos⁡(kz-ωt)$

The density of current in the (one of the) wires is

$μ_0 J=∇×B-μϵ*(∂E/∂t)$

In cylindrical coordinates

$∇×B ⃗=r ̂(1/r*(∂B_z)/∂ϕ-(∂B_ϕ)/∂z)+ϕ ̂((∂B_r)/∂z-(∂B_z)/∂r)+$

$+z ̂*1/r ((∂(rB_ϕ))/∂r-(∂B_r)/∂ϕ)$

$∇×B ⃗=r ̂(-(∂B_ϕ)/∂z)+z ̂*1/r ((∂(rB_ϕ))/∂r)$

$∇×B=r ̂((kV_0)/c*(ln⁡(r))/r*sin⁡(kz-ωt) )+z ̂*(1/r*V_0/c*1/r*cos⁡(…) )$

$μϵ ∂E/∂t=r ̂*(ωV_0)/c^2 *ln⁡(r)/r*\sin⁡(…)$

Therefore

$μJ=r[(kV_0)/c-(ωV_0)/c^2 ]*ln⁡(r)/r*sin⁡ (…)+z*V_0/(cr^2 )*cos⁡ (…)$ and because $ω=ck$

$μJ(z,t)=z ̂*V_0/(cr^2 )*cos⁡(kz-ωt)$

The area of a wire is $S=πr^2$ so that

$I(z,t)=S*J=(πV_0)/μc*cos⁡(kz-ωt)=πV_0*√(ϵ/μ)*\cos⁡(kz-ωt)$

At $z=0$ (one end of cable)

$I(0,t)=πV_0*\sqrt{(ϵ/μ)}*\cos⁡ (ωt)$ and $V=V_0*\cos⁡(ωt)$