Unit of impedance

Question

^{simulate this circuit – Schematic created using CircuitLab}

For the circuit above (for steady state I guess, if it is important) impedance of the capacitor is calculated using $$Z_c=\frac{1}{j\cdot\omega\cdot C}$$ So,$$Z_c =-j2000\;\Omega$$My question is why unit of \$Z_c\$ is not\$\;\frac{\Omega}{rad}\$ because \$\omega\$ has unit of \$\frac{rad}{second}\$ and C has unit of Farad which is equivalent \$\frac{second}{\Omega}\$ and I assumed \$j\$ is dimensionless.

I will be glad if you help me. Also I am not sure whether or not my question is reasonable.

Answer 1

Radians are a ratio (arc length over radius), so a dimensionless quantity.

The units of \$\omega\$ are 1/s. That's also true of the Laplace transform's complex frequency variable s.

Answer 2

If you are familiar with RC circuits, you know that RC has dimensions of time. So C has units of sec/ohm and 1/C has units of ohm/sec. When you multiply this by 1/frequency, you get (ohm/sec) * sec = ohms