How to solve a complex equation to get the current?

[Adesh]

I was reading The Feynman Lectures on physics http://www.feynmanlectures.caltech.edu/I_23.html chapter 23, section 4. In it he derives the equation for current when inductor, resistor and capacitor is connected in series with an alternating voltage source, he derives this equation:-
V=(iωL+R+1/iωC)I
It's a complex equation, so if we are given Voltage, Inductance , Resistance and Capacitance the value we will get is a complex one, so how can we find current from this equation? How to use this equation ?

Thank you.

 

[Wrichik Basu]

Actually,
$$|Z| = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}$$ and the phase is $$\phi = \tan^{-1}\left(\dfrac{\omega L - \frac{1}{\omega C}}{R}\right)$$ Your equation reduces to $$I_{complex} = \frac{V_0 e^{i \omega t}}{ |Z| e^{i \phi}}$$ for complex current. Take the real part, $$I_{real} = \frac{V_0}{|Z|} \cos(\omega t - \phi)$$

 

[Adesh]

Actually,
$$|Z| = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}$$ and the phase is $$\phi = \tan^{-1}\left(\dfrac{\omega L - \frac{1}{\omega C}}{R}\right)$$ Your equation reduces to $$I_{complex} = \frac{V_0 e^{i \omega t}}{ |Z| e^{i \phi}}$$ for complex current. Take the real part, $$I_{real} = \frac{V_0}{|Z|} \cos(\omega t - \phi)$$

You have helped to a great extent. Thank you so much.

 

[Wrichik Basu]

As an addendum, the expression $\omega L$ is known as inductive reactance denoted by $X_L$, and $1/(\omega C)$ is known as capacitive reactance denoted by $X_C$. I believe you already know these, but I am posting this for future visitors.