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W12D1: RC and LR Circuits Reading Course Notes: Sections , , , , Class 18

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**Announcements Math Review Week 12 Tuesday 9pm-11 pm in 26-152**

PS 9 due Week 13 Tuesday April 30 at 9 pm in boxes outside or

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**Outline DC Circuits with Capacitors**

First Order Linear Differential Equations RC Circuits LR Circuits Class 15

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**DC Circuits with Capacitors**

Week 04, Day 2 DC Circuits with Capacitors 4 Class 09 4

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**Sign Conventions - Capacitor**

Moving across a capacitor from the negatively to positively charged plate increases the electric potential Class 14

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Power - Capacitor Moving across a capacitor from the positive to negative plate decreases your potential. If current flows in that direction the capacitor absorbs power (stores charge) Class 14

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RC Circuits Class 15

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**(Dis)Charging a Capacitor**

When the direction of current flow is toward the positive plate of a capacitor, then 2. When the direction of current flow is away from the positive plate of a capacitor, then Class 15

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**Charging a Capacitor What happens when we close switch S at t = 0?**

Class 15

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**Charging a Capacitor Circulate clockwise**

First order linear inhomogeneous differential equation Class 15

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**Energy Balance: Circuit Equation**

Multiplying by (power delivered by battery) = (power dissipated through resistor) + (power absorbed by the capacitor) Class 14

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**RC Circuit Charging: Solution**

Solution to this equation when switch is closed at t = 0: (units: seconds) Class 15

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**Demonstration RC Time Constant Displayed with a Lightbulb (E10)**

Class 27

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**Review Some Math: Exponential Decay**

Class 23

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**Math Review: Exponential Decay**

Consider function A where: A decays exponentially: Class 15

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**Exponential Behavior Slightly modify diff. eq.: A “grows” to Af:**

Class 23

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**Homework: Solve Differential Equation for Charging and Discharging RC Circuits**

Class 15

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**Concept Question: Current in RC Circuit**

Class 15

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**Concept Question: RC Circuit**

An uncharged capacitor is connected to a battery, resistor and switch. The switch is initially open but at t = 0 it is closed. A very long time after the switch is closed, the current in the circuit is Nearly zero At a maximum and decreasing Nearly constant but non-zero Class 15

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**Concept Q. Answer: RC Circuit**

Answer: 1. After a long time the current is 0 Eventually the capacitor gets “completely charged” – the voltage increase provided by the battery is equal to the voltage drop across the capacitor. The voltage drop across the resistor at this point is 0 – no current is flowing. Class 15

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**Discharging A Capacitor**

At t = 0 charge on capacitor is Q0. What happens when we close switch S at t = 0? Class 15

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**Discharging a Capacitor**

Circulate clockwise First order linear differential equation Class 15

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**RC Circuit: Discharging**

Solution to this equation when switch is closed at t = 0 with time constant Class 15

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**Concept Questions: RC Circuit**

Class 15

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**Concept Question: RC Circuit**

Consider the circuit at right, with an initially uncharged capacitor and two identical resistors. At the instant the switch is closed: Class 15

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**Concept Question Answer: RC Circuit**

Initially there is no charge on the capacitor and hence no voltage drop across it – it looks like a short. Thus all current will flow through it rather than through the bottom resistor. So the circuit looks like: Class 15

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**Concept Q.: Current Thru Capacitor**

In the circuit at right the switch is closed at t = 0. At t = ∞ (long after) the current through the capacitor will be: . 20

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**Con. Q. Ans.: Current Thru Capacitor**

Week 10, Day 1 Con. Q. Ans.: Current Thru Capacitor Answer 1. After a long time the capacitor becomes “fully charged.” No more current flows into it. Class 23 28

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**Concept Q.: Current Thru Resistor**

In the circuit at right the switch is closed at t = 0. At t = ∞ (long after) the current through the lower resistor will be: . 20

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**Concept Q. Ans.: Current Thru Resistor**

Week 10, Day 1 Concept Q. Ans.: Current Thru Resistor Answer 3. Since the capacitor is “fullly charged” we can remove it from the circuit, and all that is left is the battery and two resistors. So the current is Class 23 30

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**Group Problem: RC Circuit**

For the circuit shown in the figure the currents through the two bottom branches as a function of time (switch closes at t = 0, opens at t = T>>RC). State the values of current (i) just after switch is closed at t = 0+ (ii) Just before switch is opened at t = T-, (iii) Just after switch is opened at t = T+ Class 15

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**Concept Q.: Open Switch in RC Circuit**

Week 11, Day 2 Concept Q.: Open Switch in RC Circuit Now, after the switch has been closed for a very long time, it is opened. What happens to the current through the lower resistor? It stays the same Same magnitude, flips direction It is cut in half, same direction It is cut in half, flips direction It doubles, same direction It doubles, flips direction 20 Class 25 32

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**Con. Q. Ans.: Open Switch in RC Circuit**

Answer: 1. It stays the same The capacitor has been charged to a potential of so when it is responsible for pushing current through the lower resistor it pushes a current of , in the same direction as before (its positive terminal is also on the left) Class 15

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LR Circuits Class 23

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**Inductors in Circuits Inductor: Circuit element with self-inductance**

Ideally it has zero resistance Symbol: Class 23

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**E is no longer a static field**

Non-Static Fields E is no longer a static field Class 23

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**Kirchhoff’s Modified 2nd Rule**

If all inductance is ‘localized’ in inductors then our problems go away – we just have: Class 23

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Ideal Inductor BUT, EMF generated by an inductor is not a voltage drop across the inductor! Because resistance is 0, E must be 0! Class 23

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Non-Ideal Inductors Non-Ideal (Real) Inductor: Not only L but also some R = In direction of current: Class 25

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**Circuits: Applying Modified Kirchhoff’s (Really Just Faraday’s Law)**

Class 23

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**Sign Conventions - Inductor**

Moving across an inductor in the direction of current contributes Moving across an inductor opposite the direction of current contributes Class 14

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**LR Circuit Circulate clockwise**

First order linear inhomogeneous differential equation Class 23

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**RL Circuit (units: seconds)**

Solution to this equation when switch is closed at t = 0: (units: seconds) Class 23

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RL Circuit t=0+: Current is trying to change. Inductor works as hard as it needs to to stop it t=∞: Current is steady. Inductor does nothing. Class 23

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**Group Problem: LR Circuit**

For the circuit shown in the figure the currents through the two bottom branches as a function of time (switch closes at t = 0, opens at t = T>>L/R). State the values of current (i) just after switch is closed at t = 0+ (ii) Just before switch is opened at t = T-, (iii) Just after switch is opened at t = T+ Class 15

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