Chapter 28:

DC and RC Circuits

Kirchhoff’s Rules

Limitations of DC Transmission

• Large currents in wires produce heat and energy losses by

P = I2R.

• Large expensive conductors would be needed or else very

high voltage drops (and efficiency losses) would result.

• High loads of direct current could rarely be transmitted for

distances greater than one mile without introducing

excessive voltage drops.

• Direct current can not easily be changed to higher or lower

voltages. Separate electrical lines are needed to distribute

power to appliances that used different voltages, for

example, lighting and electric motors.

Advantages of AC Transmission

• Alternating Current can be transformed to ‘step’ the

voltage up or down with transformers.

• Power is transmitted at great distances at HIGH voltages

and LOW currents and then stepped down to low voltages

for use in homes (240V) and industry (440V).

• Convert AC to DC with a rectifier in appliances.

AC is more efficient for Transmission

& Distribution of electrical power than

DC over long distances! But what if the

power generation is LOCAL?

In The Future…. Long Distance AC Power Transmission may

not be needed!!! New DC technologies and

Solar Power will make AC obsolete!

War of Currents 1880’s

Thomas Edison, American

inventor and businessman,

pushed for the development

of a DC power network.

George Westinghouse,

American entrepreneur

and engineer, backed

financially the

development of a

practical AC power

network.

Nikola Tesla,

Serbian inventor,

physicist, and

electro-mechanical

engineer, was

instrumental in

developing AC

networks.

The first electric chair, which was used to execute William Kemmler in 1890

Edison wired NYC with DC. He carried out a campaign to discourage the use of

AC, including spreading information on fatal AC accidents, killing animals, and

lobbying against the use of AC in state legislatures. Edison opposed capital

punishment, but his desire to disparage the system of alternating current led to the

invention of the electric chair. Harold P. Brown, who was at this time being secretly

paid by Edison, constructed the first electric chair for the state of New York in

order to promote the idea that alternating current was deadlier than DC.

Edison's Publicity Campaign

The first electric chair,

which was used to execute

William Kemmler in 1890

Nebraska: Only state that requires it.

15-second-long jolt of 2,450 volts of

electricity (~ 8 Amps)

GE & Edison: We bring good things to light.

More than a 1000 killed since 1890!

Any practical distribution system will use voltage levels quite sufficient for a dangerous amount of current to flow, whether it uses alternating or direct current. Ultimately, the advantages of AC power transmission outweighed this theoretical risk, and it was eventually adopted as the standard worldwide after Nikola Tesla designed the first AC hydroelectric power plant at Niagara Falls, New York which started producing electrical power in 1895.

Is AC Deadlier than DC?

They are BOTH Deadly!

Is AC Deadlier than DC?

• Low frequency (50 - 60 Hz) AC currents can be more dangerous than similar levels of DC current since the alternating fluctuations can cause the heart to lose coordination, inducing ventricular fibrillation, which then rapidly leads to death.

• High voltage DC power can be more dangerous than AC, however, since it tends to cause muscles to lock in position, stopping the victim from releasing the energized conductor once grasped.

• Frequency MATTERS.

Electric Shock

•Electric Shock occurs when current is

produced in the body, which is caused by an

impressed voltage.

•Voltage is the CAUSE

•Current does the DAMAGE

What causes electric Shock in the human

body, Voltage or Current?

Current (A) Effect

0.001 Can be felt

0.005 Painful

0.010 Causes involuntary muscle

spasms

0.015 Causes loss of muscle control

0.070 If through heart, serious!

If current lasts for 1 s - FATAL!

Dry Skin Body Resistance: 500,000

Wet Skin Body Resistance: 1000

Electric Shock

Question

What current would you draw if you

were unfortunate to short-circuit a

120 V line with dry hands?

Wet hands?

DRY: I = V/R = 120 V/500,000 = .00024 (live!)

WET: I = V/R = 120 V/1000 = .12 (dead!)

Dry Skin Body Resistance: 500,000

Wet Skin Body Resistance: 1000

Ground Wire

• Electrical equipment manufacturers use electrical cords that have a third wire, called a ground

• This safety ground normally carries no current and is both grounded and connected to the appliance

Ground-Fault Interrupters

(GFI)

• Special power outlets

• Used in hazardous areas

• Designed to protect people from electrical

shock

• Senses currents (< 5 mA) leaking to ground

• Quickly shuts off the current when above

this level

Electric Shock Therapy

ELECTRO CONVULSIVE

THERAPY

An electric shock is applied to produce a convulsive

seizure. The shock is typically between 140 - 170 volts and

lasts between 0.5 and 1 seconds. No explanation of how it

works.

Used in the treatment of:

1.Chronic endogenous depression

2.Bipolar disorder.

3.Acute mania.

4.Certain types of schizophrenia

In the U.S. 33,000 - 50,000 people receive ECT each year

• The current is the same in each device.

• The equivalent resistance of the circuit is the sum

of the individual resistances.

Series Circuits

• The voltage of each device is the full

voltage of the EMF source (the battery)

• The total current is divided between

each path:

Parallel Circuits

1 2

1 1 1

totalR R R

1 2totalR R R

Find Everything

1 2 215.0V, 10.0 , 20.0 , 30.0 ,V R R R

YOU DO IT.

For the circuit shown, find the equivalent resistance of the

circuit, the total current drawn by the battery, and the

current through and the potential difference across each

resistor. Place your results in a table for ease of reading.

Voltage in a single-resistor circuit

with an Ideal 1.5 Volt battery

• The emf e is the voltage labeled on the battery. It is greater than the terminal voltage. DV=IR

• The actual potential difference DV between the terminals of the battery depends on the current in the circuit and the internal resistance of the battery: DV = e – Ir

• If the internal resistance of the battery is zero, the terminal voltage equals the emf : DV = e

• R is the load resistance: DV = IR

Electromotive Force: EMF

e = DV+ Ir

Power

• The total power output of the battery is

• This power is delivered to the external resistor (I 2 R) and to the internal resistor (I2 r) is maximum when R = r!! (see nice proof in text)

I V Ie D

2 2I R I r

Single Circuit: Multiple Batts What is the direction of current?

Book: Chooses Wrong Current Directions to show you that the rule still

works, you get negative currents. But try to pick the correct direction!

The 12 V WINS!

When polarities of the batteries are opposed, one gets CHARGED.

I

Book

Kirchhoff’s Rules When resistors are connected so that the circuits

formed cannot be reduced to a single equivalent

resistor you can use two rules, called Kirchhoff’s

rules, to solve the problem by generating systems

of equations to find the unknowns!!

Gustav Kirchhoff • 1824 – 1887

• German physicist

• Worked with Robert

Bunsen

• They

– Invented the

spectroscope and

founded the science of

spectroscopy

– Discovered the elements

cesium and rubidium

– Invented astronomical

spectroscopy

Kirchhoff’s Rules for Spectra: 1859

Bunsen

German physicist who developed the spectroscope and the science of

emission spectroscopy with Bunsen.

Kirkoff

* Rule 1 : A hot and opaque solid, liquid or highly compressed gas emits a continuous spectrum.

* Rule 2 : A hot, transparent gas produces an emission spectrum with bright lines.

* Rule 3 : If a continuous spectrum passes through a gas at a lower temperature, the transparent

cooler gas generates dark absorption lines.

Kirchhoff’s Junction Rule

• Junction Rule

– The sum of the currents at any junction must

equal zero

• Currents directed into the junction are entered

into the -equation as +I and those leaving as -I

• A statement of Conservation of Charge

– Mathematically, 0

junction

I

More about the Junction Rule

• I1 - I2 - I3 = 0

• Required by

Conservation of

Charge

• Diagram (b) shows a

mechanical analog

What are the

magnitude and the

direction of the

current in the fifth

wire?

A. 15 A into the junction

B. 15 A out of the junction

C. 1 A into the junction

D. 1 A out of the junction

E. Not enough data to determine

A. 15 A into the junction

B. 15 A out of the junction

C. 1 A into the junction

D. 1 A out of the junction

E. Not enough data to determine

What are the

magnitude and the

direction of the

current in the fifth

wire?

Kirchhoff’s Loop Rule

• Loop Rule

– The sum of the potential differences across all

elements around any closed circuit loop must

be zero

• A statement of Conservation of Energy

• Mathematically,

closedloop

0VD

Using the Loop Rule • Traveling around the loop

from a to b

• In (a), the resistor is

traversed in the direction

of the current, the

potential across the

resistor is – IR

• In (b), the resistor is

traversed in the direction

opposite of the current, the

potential across the

resistor is is + IR

Loop Rule

• In (c), the source of emf is traversed in the direction of the emf (from – to +), and the change in the electric potential is +ε

• In (d), the source of emf is traversed in the direction opposite of the emf (from + to -), and the change in the electric potential is –ε

which means you are CHARGING the battery!

I

I

Discharging

Charging

Junction Equations

from Kirchhoff’s Rules

• Use the junction rule as often as needed, so

long as each time you write an equation,

you include in it a current that has not been

used in a previous junction rule equation

– In general, the number of times the junction

rule can be used is one fewer than the number

of junction points in the circuit

0junction

I

Loop Equations from

Kirchhoff’s Rules

• The loop rule can be used as often as

needed so long as a new circuit element

(resistor or battery) or a new current appears

in each new equation

• You need as many independent equations as

you have unknowns

closedloop

0VD

Kirchhoff’s Rules Equations

• In order to solve a particular circuit problem,

the number of independent equations you need

to obtain from the two rules equals the number

of unknown currents

• Any capacitor acts as an open branch in a

circuit

– The current in the branch containing the capacitor is

zero under steady-state conditions

• Junction Rule:

• Loop Rule:

Kirchhoff’s Rules

•First use junction rule and assign values to the current - guess the directions!

•Then use the loop rule CONSISTENTLY (e.g. Clockwise) on each loop.

•Any capacitor acts as an open branch in a circuit once under steady state conditions

0junction

I

closedloop

0VD

Multiple Loop Circuit

Find the Currents! Book: Chooses Wrong Current Directions to show you that the rule still

works, you get negative currents. But try to pick the correct direction!

I1 I2

I3 I3

I1

Kirchhoff Problem

Taking R = 1.00 kM and ε = 250V, determine

the direction and magnitude of the current in

the horizontal wire between a and e.

Many-Loop – Many Equations! You can solve systems of equations with simple algebra or

by using linear algebra (Cramer’s Rule) but I can’t teach

you that! I’ve put a document on the website if you want to

teach yourself. BUT YOU DON’T NEED TO! Algebra

will work and I won’t put a circuit with more than two

loops on the exam! Also, you calculator can solve systems

of equations!

RC Circuits: Charging The capacitor continues to charge until it reaches its maximum

charge (Q = Cε). Once the capacitor is fully charged, the current

in the circuit is zero and the The potential difference across the

capacitor matches that supplied by the battery.

RC Circuit: Charging

• The charge on the capacitor varies

with time

q(t) = Ce(1 – e-t/RC) = Q(1 – e-t/RC)

t is the time constant

• t = RC

• The current can be found

I( ) t RCεt e

R

• The time constant t has units of time represents the time required

for the charge to increase from zero to 63.2% of its maximum

• The energy stored in the charged capacitor is ½ Qe = ½ Ce2

Discharging a Capacitor in an RC

Circuit • When a charged capacitor is

placed in the circuit, it can be discharged – q = Qe-t/RC

• The charge decreases exponentiallyAt t = t = RC,

the charge decreases to 0.368 Qmax

– In other words, in one time constant, the capacitor loses 63.2% of its initial charge

• The current is:

I t RCdq Qt e

dt RC

I( ) t RCεt e

R

q(t) = Q(1 – e-t/RC)

Active Figure: RC Circuit Charging

An RC Problem

The switch in the figure has been in position a for a long time. It is

changed to position b at t = 0s. What are the charge on the capacitor

and the current I through the resistor

a)immediately after the switch is closed?

b)What is the time constant t ?

c)at t = 50 s?

d)at t = 200 s?

What is the current in the

resistors?

What is the emf and internal

resistance of the battery?