The circuit operates in sinusoidal regime.
\$V_e\$ operates at 50 Hz frequency.
The eff. voltages (rms) of \$V_e\$ and \$V_a\$ are, respectively, \$100\$ V and \$50\$ V.
simulate this circuit – Schematic created using CircuitLab
I tried to solve it with 2 methods:
First method - phasors diagrams
\$I=\frac{V_A}{100}=\frac{1}{2}\$ A
So \$|V_C|=\sqrt{V_e^2-V_A^2}=\sqrt{100^2-50^2}=\sqrt{7500}=50\sqrt{3}\$
Now: \$|V_C|=|\frac{1}{jwC} I|=\frac{1}{wC}|I|=\frac{1}{wC}\frac{1}{2}\implies C=\frac{1}{2\omega|V_C|}=\frac{1}{2\cdot50^2 \sqrt3}\approx115.47 \mu F\$
Second method
\$V_C=V_e-V_A=50\$ V
\$I_R=\frac{V_A}{100}=\frac{1}{2}\$ A
\$V_C=I\frac{1}{j\omega C}\implies C=\frac{I}{V_C\cdot j\omega}=\frac{\frac{1}{2}}{j 50\cdot 50}=0.0002j\$ F
Why is in this case \$C\$ a complex number?
However, in both cases, the result appears to be wrong, since the solution is \$18.38 \mu F\$.
Could anybody explain-me where I am wrong with the two ways, please?