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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

AnnouncementsMath 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

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Outline

DC Circuits with Capacitors

First Order Linear Differential Equations

RC Circuits

LR Circuits

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DC Circuits with Capacitors

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Sign Conventions - Capacitor

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

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

dt

dU

C

Q

dt

d

C

Q

dt

dQVIP

2

2

absorbed

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RC Circuits

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(Dis)Charging a Capacitor1. 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

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Charging a Capacitor

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

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Charging a Capacitor

Circulate clockwise

First order linear inhomogeneous differential equation

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Energy Balance: Circuit Equation

Multiplying by

(power delivered by battery) = (power dissipated through resistor)

+ (power absorbed by the capacitor)

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RC Circuit Charging: Solution

Solution to this equation when switch is closed at t = 0:

(units: seconds)

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DemonstrationRC Time ConstantDisplayed with a

Lightbulb (E10)

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

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Review Some Math:Exponential Decay

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Math Review: Exponential Decay

Consider function A where:

A decays exponentially:

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Exponential Behavior

Slightly modify diff. eq.:

A “grows” to Af:

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Homework: Solve Differential Equation for Charging and Discharging RC Circuits

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Concept Question:Current in RC Circuit

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Concept Question: RC CircuitAn 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 zero2. At a maximum and

decreasing3. Nearly constant but non-zero

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Concept Q. Answer: RC Circuit

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.

Answer: 1. After a long time the current is 0

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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?

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Discharging a Capacitor

Circulate clockwise

First order linear differential equation

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RC Circuit: Discharging

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

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Concept Questions:RC Circuit

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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:

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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:

Answer: 3.

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1. .

2.

3.

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:

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Con. Q. Ans.: Current Thru Capacitor

After a long time the capacitor becomes “fully charged.” No more current flows into it.

Answer 1.

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1. .

2.

3.

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:

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Concept Q. Ans.: Current Thru Resistor

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

.

Answer 3.

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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+

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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 same2. Same magnitude, flips direction3. It is cut in half, same direction4. It is cut in half, flips direction5. It doubles, same direction6. It doubles, flips direction

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

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)

Answer: 1. It stays the same

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LR Circuits

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Inductors in Circuits

Inductor: Circuit element with self-inductance Ideally it has zero resistance

Symbol:

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Non-Static Fields

E is no longer a static field

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Kirchhoff’s Modified 2nd Rule

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

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• BUT, EMF generated by an inductor is not a voltage drop across the inductor!

Ideal Inductor

Because resistance is 0, E must be 0!

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Non-Ideal Inductors

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

In direction of current:

=

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

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Sign Conventions - Inductor

Moving across an inductor in the direction of current contributes

dILdt

Moving across an inductor opposite the direction of current contributes

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LR Circuit

Circulate clockwise

First order linear inhomogeneous differential equation

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RL Circuit

Solution to this equation when switch is closed at t = 0:

(units: seconds)

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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.

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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+