W12D1: RC and LR Circuits

Reading Course Notes: Sections 7.7-7.8, 7.11.3, 11.4-11.6, 11.12.2, 11.13.4-11.13.5

1

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 32-082 or 26-152

2 Outline

DC Circuits with Capacitors

First Order Linear Differential Equations

RC Circuits

LR Circuits

3

DC Circuits with Capacitors

4 Sign Conventions - Capacitor

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

!V = V " V b a

5 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)

dQ Q d Q2 dU Pabsorbed = I ΔV = = =

dt C dt 2C dt 6 RC Circuits

7 (Dis)Charging a Capacitor 1. When the direction of current flow is toward the positive plate of a capacitor, then

dQ I = + dt

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

dQ I = ! dt

8 Charging a Capacitor

What happens when we close switch S at t = 0?

9 Charging a Capacitor

Circulate clockwise

Q V IR 0 "! i = # $ $ = i C dQ I = + dt First order linear dQ `1 inhomogeneous differential = ! (Q ! C") dt RC equation

10 Energy Balance: Circuit Equation

Q ! " " IR = 0 C dQ I = + dt dQ Multiplying by I = + dt Q dQ d " 1 Q2 % ! I = I 2 R + = I 2 R + C dt dt $ 2 C ' # & (power delivered by battery) = (power dissipated through resistor) + (power absorbed by the capacitor)

11 RC Circuit Charging: Solution dQ 1 = ! (Q ! C") dt RC Solution to this equation when switch is closed at t = 0: dQ Q(t) C (1 e"t /# ) I(t) I(t) I e"t /# = ! " = + ! = 0 dt ! = RC : time constant (units: seconds)

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

http://tsgphysics.mit.edu/front/?page=demo.php&letnum=E%2010&show=0

13 Review Some Math: Exponential Decay

14 Math Review: Exponential Decay dA 1 Consider function A where: = ! A dt A decays exponentially: " A(t) = A e!t " 0

15 Exponential Behavior dA 1 Slightly modify diff. eq.: = ! ( A ! A ) dt " f A “grows” to A : f A(t) = A (1! e!t /" ) f

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

17 Concept Question: Current in RC Circuit

18 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 1. Nearly zero 2. At a maximum and decreasing 3. Nearly constant but non-zero

19 Discharging A Capacitor

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

20 Discharging a Capacitor

Circulate clockwise Q V IR 0 "! i = # = i C dQ I = ! " dt dQ Q First order linear = ! differential equation dt RC

21 RC Circuit: Discharging dQ 1 = ! Q " Q(t) = Q e!t / RC dt RC o Solution to this equation when switch is closed at t = 0 with time constant ! = RC dQ Q Q I = ! " I(t) = o e!t /# = o e!t /# dt # RC

22 Concept Questions: RC Circuit

23 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:

1. I R = IC = 0

2. I R = ! / 2R , IC = 0

3. I R = 0 , IC = ! / R

4. I R = ! / 2R , IC = ! / R 24 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: I = 0 1. C . I = ! R 2. C 3. I = ! 2R C

25 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: I = 0 1. R . 2. I = ! R R 3. I = ! 2R R

26 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 27 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?

1. It stays the same 2. Same magnitude, flips direction 3. It is cut in half, same direction 4. It is cut in half, flips direction 5. It doubles, same direction 6. It doubles, flips direction

28 LR Circuits

29 Inductors in Circuits Inductor: Circuit element with self-inductance Ideally it has zero resistance

Symbol:

30 Non-Static Fields

! ! d$ E ! d s = # B "" d t E is no longer a static field

31 Kirchhoff’s Modified 2nd Rule

! ! d& ! V = # E $ d s = + B " i "% d t i d% V B 0 ! #" i $ = i d t If all inductance is ‘localized’ in inductors then our problems go away – we just have: d I V L 0 "! i # = i d t 32 Ideal Inductor • BUT, EMF generated by an inductor is not a voltage drop across the inductor! d I ! = "L d t ! ! " E ! d s = 0 i d e a l i n d u c t o r Because resistance is 0, E must be 0! 33 Non-Ideal Inductors

Non-Ideal (Real) Inductor: Not only L but also some R

=

dI In direction of current: ! = "L " IR dt

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

35 Sign Conventions - Inductor

Moving across an inductor in the direction of current contributes dI ε = −L dt Moving across an inductor opposite the direction of current contributes dI ! = +L dt

36 LR Circuit

Circulate clockwise

dI ! " IR " L = 0 # dt

First order linear dI R # " & = ! I ! inhomogeneous dt L % R( differential equation $ '

37 RL Circuit

dI R # " & " !t /( L/ R) = ! % I ! ( ) I(t) = (1! e ) dt L $ R' R Solution to this equation when switch is closed at t = 0: ! I(t) = (1" e"t /# ) R L ! = : time constant R (units: seconds)

38 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.

39 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 40