DC CircuitsDC Circuits

The circuitThe circuitResistance in combinationsResistance in combinations

Kirchhoff’s RulesKirchhoff’s RulesRC transient circuitsRC transient circuits

dq

dWε

Work done by a battery on charge

Here AB εε

Real Battery and Single Loop circuits… What’s the current ?Real Battery and Single Loop circuits… What’s the current ?

Conservation of energy: Kirchoff’s first Law: Sum of voltages in a closed loop is zero.

2)( Rr

R

RiP

Rri

0iRir

ViRirV

2

2

R

aa

Real Circuit with ammeter and voltmeter

Equivalent ResistanceEquivalent ResistanceResistors in SeriesResistors in Series

Series requirementsSeries requirements– Conservation of energyConservation of energy– Potential differences addPotential differences add– Current is constantCurrent is constant

n21 VVVV ...Apply Ohm’s Law to each resistorApply Ohm’s Law to each resistor

n21eq IRIRIRIR ...

n21eq RRRR ...

Resistors in parallelResistors in parallel

Parallel requirementsParallel requirements– Charge conservationCharge conservation– Currents must addCurrents must add– Potential difference is Potential difference is

same across each resistorsame across each resistor

321 iiii

Apply Ohm’s Law to each resistorApply Ohm’s Law to each resistor

n21eq RV

RV

RV

RV

...n21eq R1

R1

R1

R1

...

Example 1

What is current through battery?

What is current through i2 ?

Kirchhoff’s RulesKirchhoff’s Rules1 The algebraic sum of the currents The algebraic sum of the currents

entering a junction is zero.entering a junction is zero. ((ConservationConservation of Charge of Charge))

2 The algebraic sum of the changes in The algebraic sum of the changes in electric potential difference around electric potential difference around any closed circuit loop is zero.any closed circuit loop is zero. ((Conservation of EnergyConservation of Energy))

Signs for Rule 2Signs for Rule 2The direction of travel when traversing The direction of travel when traversing the loop is from a to b.the loop is from a to b.

Problem 2

Find the currents in each of the three legs of the circuit,

321 i,i,i

Three unknowns, need three equations. Also since batteries are in there cannot reduce the resistances since none in parallel or series

Example: Applying Kirchhoff’s RulesExample: Applying Kirchhoff’s Rules

Apply Kirchhoff’s first rule to Apply Kirchhoff’s first rule to the three wire junction at the three wire junction at the bottom of the diagramthe bottom of the diagram

0III 312

Apply Kirchhoff’s second Apply Kirchhoff’s second rule to the closed path in red, rule to the closed path in red, traversing it clockwisetraversing it clockwise

0I05I03V05 21 ...

Apply Kirchhoff’s second rule Apply Kirchhoff’s second rule to the closed path in green, to the closed path in green, traversing it clockwisetraversing it clockwiseNote the sign changes for Note the sign changes for some of the elementssome of the elements

0I07V010I03V05 31 ....

Another, example: applying Kirchhoff’s RulesAnother, example: applying Kirchhoff’s Rules

Solve the equations Solve the equations simultaneously for the values simultaneously for the values if I. If I is negative the if I. If I is negative the current is in the opposite current is in the opposite directiondirection

0III 312

0I05I03V05 21 ...

0I07I03V05 31 ...

A7740I

A9150I

A1410I

3

2

1

.

.

.

RC Circuits and Time dependenceRC Circuits and Time dependenceTime dependence

Recall Lab 7! Resistor slows down the charging

of the capacitor

Time dependent behavior Time dependent behavior (transient) 2 cases: switch at (transient) 2 cases: switch at

““a” or at “b”a” or at “b”

a)a) ChargingChargingb)b) dischargingdischarging

In position “a” Charging the In position “a” Charging the CapacitorCapacitor

Use Kirchhoff’s Loop ruleUse Kirchhoff’s Loop rule

RCt

c

RCt

R

e1V

e1Cq

RRC

q

dt

dq

0IRq

0VV

(t)

(t)

(t)(t)

C-

- c

Or the voltage across capacitor is …..

What’s VR across resistor?Find the current and multiply by R

,or ....

get and derivative take

(t)

(t)

(t)(t)

(t)

RCt

R

RCt

RCt

e

Rdt

dqV

eR

I

dt

dqI

e1Cq

RCt

0c

RCt

0

RCt

0

R

eVV

eCVeqq

RC

q

dt

dq

0VV

(t)

(t)

(t)(t)

c

Discharging Position b:

1. Charging the Capacitor1. Charging the Capacitor

Note:Note: RCis called the time constantis called the time constantWhat are the units?What are the units?