# RC Circuit with a voltage source

• RyanP
In summary, Kirchhoff's law is expressed as a differential equation. The current is determined by solving for I using an exponential form. The current amplitude and phase vary with ω.
RyanP

## Homework Statement

A voltage source E_0 cos ωt is connected in series with a resistor R and a capacitor C. Write down the differential equation expressing Kirchhoff’s law. Then guess an exponential form for the current, and take the real part of your solution to find the actual current. Determine how the amplitude and phase of the current behave for very large and very small ω, and explain the results physically.

## The Attempt at a Solution

[/B]
I got the complex current to be I(t) = [(iωE0/R)/(1/RC + iω)] * e^(iωt)

I just don't know how to convert this back to real current. I know e^(iωt) = cos(ωt)+isin(ωt), but how do I deal with the rest of the term?

RyanP said:
I know e^(iωt) = cos(ωt)+isin(ωt), but how do I deal with the rest of the term?
Plug that in and multiply out.

RyanP said:

## Homework Statement

A voltage source E_0 cos ωt is connected in series with a resistor R and a capacitor C. Write down the differential equation expressing Kirchhoff’s law. Then guess an exponential form for the current, and take the real part of your solution to find the actual current. Determine how the amplitude and phase of the current behave for very large and very small ω, and explain the results physically.

## The Attempt at a Solution

[/B]
I got the complex current to be I(t) = [(iωE0/R)/(1/RC + iω)] * e^(iωt)

I just don't know how to convert this back to real current. I know e^(iωt) = cos(ωt)+isin(ωt), but how do I deal with the rest of the term?
You have the current as the product I=G E0eiωt. G=|G|e. Find the magnitude |G| and the phase φ, then I=|G|E0ei(ωt+φ). Take the real part.

## 1. What is an RC circuit with a voltage source?

An RC circuit with a voltage source is a circuit that contains a resistor (R) and a capacitor (C) connected in series, with a voltage source (V) connected in parallel. This circuit is commonly used in electronic devices to control the flow of electric current.

## 2. How does an RC circuit with a voltage source work?

When a voltage source is connected to an RC circuit, the capacitor begins to charge, and the current starts to flow through the circuit. As the capacitor charges, the voltage across it increases until it reaches the same value as the voltage source. Once this happens, the current stops flowing, and the capacitor is fully charged.

## 3. What is the time constant in an RC circuit with a voltage source?

The time constant in an RC circuit is the time it takes for the capacitor to charge to 63.2% of its maximum charge when a voltage source is applied. It is calculated by multiplying the resistance (R) and the capacitance (C) in the circuit.

## 4. How does the time constant affect an RC circuit with a voltage source?

The time constant determines the rate at which the capacitor charges in an RC circuit. A larger time constant means that the capacitor will take longer to charge, while a smaller time constant means that the capacitor will charge more quickly.

## 5. What are the practical applications of an RC circuit with a voltage source?

RC circuits with voltage sources are commonly used in electronic devices such as filters, oscillators, and timing circuits. They are also used in power supplies to smooth out the output voltage, and in touchscreens to detect touch inputs. Additionally, they are used in various sensor circuits to convert signals into usable electrical outputs.

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