# RC Circuit with parallel resistor

• CrazyNeutrino
In summary, when S2 is closed, the capacitor immediately becomes isolated and current starts flowing in R1. A long time after S2 is closed, the voltage across the capacitor and R2 are the same and current stops flowing.
CrazyNeutrino

## Homework Statement

A circuit contains a capacitor of capacitance C, a power supply of emf E, two resistors of resistances R1 and R2 , and a switch S2 . Resistor R1 is in series with the power supply and R2 is in parallel with the capacitor and the power supply. S2 switches the branch that contains R2 (in parallel with the capacitor). Initially, the capacitor is CHARGED and S1 is closed while S2 is open. Switch S2 then gets closed at time t=0. Sketch the currents in resistance R1 and R2 as a function of time t.

## Homework Equations

V=IR
Kirchoffs Loop and Junction Rule
Capacitor Differential equations

## The Attempt at a Solution

I know that the solution is two exponential curves that asymptotically approach some current value from the top and from 0. I figured that the current in R1 is 0 when the switch is immediately closed while the initial current in R2 is E/R2 since the voltage across the capacitor and R2 must both equal E. My reasoning was that as t approaches infinity, the current in R1 and R2 will tend towards E/(R1+R2). However, I am not really sure why. I tried writing individual differential equations for the currents in R1, R2 and the capacitor but ended up confusing myself further. Could somebody please explain conceptually, what happens to the voltage across the capacitor and the currents in both resistors over time? I sort of understand what happens but don't particularly understand why.

If I understand the situation correctly from your description without a diagram, when you open the switch, the capacitor becomes isolated and the charge on its plates has nowhere to go. What does that imply about the voltage across its plates?

Here's a picture of the circuit. It is still in parallel with the resistor and the power supply. I don't understand why it is isolated, although, if it was I suppose the voltage across it would be the same as it was when it was initially fully charged, i.e = E.

#### Attachments

• Screen Shot 2018-04-17 at 4.35.07 PM.png
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OK, I see, thanks. I imagined a different circuit that's why I said the capacitor becomes isolated. In this case the voltage across the capacitor when S1 is closed for a long time is E. Now immediately after S2 is closed, call that t = 0, it is still E. A long time after S2 is closed, the voltage across the capacitor becomes the same as the voltage across resistor ##R_2## because the two are in parallel. Find what that voltage is and connect the dots from t = 0 to t = "a long time later" with an increasing or decreasing exponential as the case might be.

CrazyNeutrino
Fair enough I understand that part. What seems counterintuitive to me is that a current starts flowing in R1 just by introducing a new resistor. Why?

CrazyNeutrino said:
Fair enough I understand that part. What seems counterintuitive to me is that a current starts flowing in R1 just by introducing a new resistor. Why?
Because, initially at t = 0, when S2 is closed, the voltage across the capacitor, ##V_C##, is instantaneously ##V_C = E##. A long time after S2 is closed, ##V_C## is only a fraction of ##E##. Now remember that ##Q=CV_C## which means that charge has to leave the plates when the voltage is decreased. "Charge leaving the plates" means that current is flowing in the circuit, as long as ##V_C## is decreasing. Eventually, ##V_C## matches the voltage across ##R_2## at which point current stops flowing and a new steady state situation is reached.

CrazyNeutrino
Thank you so much! Couldn't visualise what was going on for the longest time but it really does make sense now.

## 1. What is an RC circuit with a parallel resistor?

An RC circuit with a parallel resistor is a type of electrical circuit that consists of a resistor (R) and a capacitor (C) connected in parallel. This means that both components are connected to the same nodes in the circuit, creating two parallel paths for the current to flow.

## 2. How does an RC circuit with a parallel resistor work?

An RC circuit with a parallel resistor works by allowing the capacitor to charge and discharge through the resistor. When a voltage is applied to the circuit, the capacitor charges up to the same voltage as the source. As the capacitor charges, the current through the resistor decreases until it reaches zero. When the capacitor discharges, the current through the resistor increases until it reaches its maximum value again.

## 3. What is the time constant of an RC circuit with a parallel resistor?

The time constant of an RC circuit with a parallel resistor is the amount of time it takes for the capacitor to charge or discharge to 63.2% of its maximum voltage. It is calculated by multiplying the resistance (R) by the capacitance (C). The time constant is also known as the "RC time constant" or "tau" (τ).

## 4. How does the value of the parallel resistor affect the behavior of an RC circuit?

The value of the parallel resistor affects the behavior of an RC circuit by changing the rate at which the capacitor charges and discharges. A larger resistor will slow down the charging and discharging process, resulting in a longer time constant. A smaller resistor will speed up the process and result in a shorter time constant.

## 5. What is the practical application of an RC circuit with a parallel resistor?

An RC circuit with a parallel resistor has many practical applications, such as in filter circuits, time delay circuits, and audio amplifiers. It is also commonly used in electronic devices to control the flow of electricity and prevent damage to components. Additionally, RC circuits are used in many electronic devices, such as computers, televisions, and radios.

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