# How to solve a complex equation to get the current?

In summary, the conversation discusses the derivation of an equation for current when an inductor, resistor, and capacitor are connected in series with an alternating voltage source. The equation is V=(iωL+R+1/iωC)I and it results in a complex value for current. To find the current, one can use the expressions |Z| = √(R^2 + (ωL - 1/ωC)^2) and φ = tan^-1((ωL - 1/ωC)/R). The equation for complex current is I_{complex} = (V_0 e^(iωt))/(|Z| e^(iφ)) and for real current is I_{real} =
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.

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, $$I_{real} = \frac{V_0}{|Z|} \cos(\omega t - \phi)$$

Wrichik Basu said:
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.

## 1. How do I identify the current in a complex equation?

The current can be identified by looking for the variable "I" in the equation, which represents current. It may also be explicitly stated in the equation or given as a value in the problem.

## 2. What are the steps to solve a complex equation for current?

The steps to solve a complex equation for current include: 1) Simplifying the equation by combining like terms, 2) Isolating the variable "I" on one side of the equation, 3) Using inverse operations to solve for "I", and 4) Checking your answer by plugging it back into the original equation.

## 3. How do I know if my solution for current is correct?

You can check your solution by plugging it back into the original equation and ensuring that it satisfies the equation. If the solution does not satisfy the equation, then it is incorrect.

## 4. Are there any common mistakes to avoid when solving for current in a complex equation?

One common mistake to avoid is forgetting to use inverse operations when isolating the variable "I". Another mistake is not checking your answer by plugging it back into the original equation. It is also important to be careful with signs and exponents when simplifying the equation.

## 5. Is there a specific order in which I should solve a complex equation for current?

There is no specific order in which you must solve a complex equation for current. However, it is recommended to simplify the equation first before isolating the variable "I". This can make the problem easier to solve and reduce the chances of making mistakes.

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