# Circuits - Quia

Ch. 20

Electric Circuits

Electromotive force and current

In a
closed
circuit, current is driven by a “source of
emf
” which
is

either

a
battery or
a
generator
. Batteries convert
chemical potential energy into electrical energy and generators convert mec
hanical energy into electrical energy.

Emf

(symbolized with

or
V
; units are
volts
)
stands for

electromotive force

although the term force is used incorrectly

since
emf

is the source that maintains the
potential difference

(voltage) in the circuit.

The m
aximum potential
difference of a battery is the
emf

of the battery. In a
closed
circuit, the battery creates an electric field within and
parallel to the wire, directed from the positive toward the negative terminal. The electric field exerts a force
on the
free electrons causing them to move
producing
electric current
.

The diagram

below show
s

the

physical and
schematic
diagram

f
or one of the simplest circuits.

Current
(
symbolized with
I
)
is the
rate at which charge flows
.

I=

q/t

units are
C/s=A (ampere)

By convention, t
he
direction of

the flow of
current

is the same as the direction of
the
flow of
positive

charge which, in
metals, is opposite the direction of the flow of electrons.
We use
this convention
since it is
consistent with our earlier
use of a positive test charge

for defining

electric fields and potentials
.

Current is classified as either dc or ac based

on
the
motion
of the current
.

In a
d
irect current
(dc)

circuit
,

current

moves around the circuit in the same direction at all
times
. Batteries and dc generators

produce direct currents. In an
a
lternating current

(ac)
circuit, current

moves first
one direction and then the opposite, changing
direction from moment to moment. Ac

generators (power companies)

produce alternating currents.

Ohm’s law

When a potentia
l difference (voltage)

is applied across the ends of many metallic conductors, the
current is directly
proportional to the applied voltage
. This statement is known as Ohm’s law although it is not a fundamental law of
nature since it is only valid for cert
ain materials and within a certain range.

V

I

R

is the proportionality constant for the above relationship which is the resistance of the conductor.

V=IR

Rearranging the above equation for R shows that the units for resistance are
V/A =

(
ohm
)
.

Expe
riments show that
most metals have a constant resistance
over
a wide range of applied voltages

(
graph (a) below)
. Such materials are
said to be
ohmic

since they obey Ohm’s law.

Nonohmic

materials
do not obey Ohm’s law

(graph (b) below)
.

Resistanc
e and resistivity

As charge moves through
a

circuit, it encounters
resistance
, or opposition to the flow of
the
current. Resistance is the
electrical equivalent of friction. In our circuit above, the wires and the light bulb would be considered resistances
,
although usually the resistance of the wires is

neglected. The resistance of a material
is proportional to the length
and
inversely related to the
cross
-
sectional area
.

R

L/A

is the proportionality constant known as

the
resist
ivity

of the
material

and has units of ohm
-
meters (

m).

E
very
material has a characteristic resistivity that depends on its electronic structure and temperature (Table 20.1 in your
book
)
.
In

metals
,
the resistivity increases with increasing temperature
,
whereas in semiconductors the reverse is true.
Superconductors

are a class of metals and compounds whose resistance goes to virtually zero below certain
temperatures
(
critical temperature
).

Example 1
:

The five resistors shown below have the lengths
and cross
-
sectional areas indicated and are made of
material with the same resistivity. Which has the greatest resistance?

A
resistor

is a simple circuit element that provides a specified resistance. Resistors are represented by a zigzag l
ine in
circuit diagrams (a straight line represents an ideal conducting wire, or one with negligible resistance).
Resistors can
be used in circuits to control the amount of current in a conductor
.

Electrical energy and powe
r

As current moves through a circuit, electrical energy is transformed into thermal energy due to collisions with atoms in
the resistor. The amount of heat produced in joules per second is equal to the power
in

the resistor. You should
remember from prev
ious chapters the power is the rate at which work is done
, or the rate at which energy is
transferred.

Since
W=qV

and
I=q/t

and
V=IR
, by substitution it can be shown that

Of course the
unit for power is th
e
joule/second
, or
watt
.

Schematic diagrams and circuits

A
schematic diagram

is a diagram that depicts the construction of an electrical apparatus or
circuit
using symbols to
represent the different circuit elements (
emf
, resistors, capacitors, wires, s
witches, ammeters, voltmeters, etc.). The
diagram at the beginn
ing of the notes illustrates a simple

circuit

that causes a light bulb to shine. You may wonder
what would happen if you connect another bulb to the battery. It actually depends upon how you

connect the bulb,
whether you have one path (series connection) or create another path (parallel connection). We will analyze each of
these connections separately to see the
ir effect on the circuit. First, a few definitions.

An
ammeter
is an

instrumen
t used to measure current.
As shown below

left
,
a
mmeters

must be inserted into a
circuit so that the current passes directly through it (in series)
. A good ammeter is designed with a sufficiently
small resistance, so the reduction in current is negligibl
e whenever the ammeter is inserted.

A

voltmeter
is an
instrument used to measure the potential difference (voltage) between two points.
As shown
below

right
,
v
oltmeters

must be inserted in parallel in the circuit.

A good voltmeter is designed with a larg
e
resistance so the effects on voltage are negligible when it is inserted into the circuit.

Internal resistance
is

resistance in a battery or generator. In a battery,

this is due to the chemicals and in

a
generator

it

is due to
the
resi
stance in the wires and other components.
As shown below
,

interna
l resistance is
treated as

resistor connected in series with the
external
circuit
.

V

+

-

A

Resistors in series

Two or more resistors of any value placed in a circuit in such a way that the
same current passes through each of them
is called a
series
connection.

A series

connection will have a single path between two points.
A break in the circuit of
a series connection will disconnect all elements. Christmas lights often use series wiring
which is why if one bulb goes
out the entire strand goes out.
The diagram below shows a series connection with two resistors,
R

1
and

R
2
.

W
hen
resistors are connected in series, the total resistance
of the circuit
increases and the current decreases.

R
ules for series connections

1.

Cur
rent through all resistors in
series
is

the same,

because any charge that flows through the first resistor must
also flow through the second
.

I
total

= I
1

= I
2

= I
3.

The total current in the circuit is

2.

The potential difference across the entire connection (total voltage)

is equal to the sum of potential drops
(voltages) across
each resistor.

V
tota
l
=V
1
+V
2

And t
he voltage divides proportionally among the res
istances according to Ohm’s law.

3.

The equivalent resistance

is the sum of the individual
resist
ances
.

R
eq
=R
1
+R
2

Resistors in parallel

Two or more resistors o
f any value placed
in such a way that each resistor has the same potential difference is called

a
parallel
connection
.

A parallel connection will have a junction creating separate paths between two points.
A break in
one of the paths does not affect the other path.
Household circuits are generally connected so appliances, light bulbs,
etc. are co
nnected
in
parallel

so

each gets
the
same
voltage
and each can be operated independently
. The diagram
below shows a parallel connection with two resistors,
R

1
and R
2
.

When resistors are connected in parallel, the total
resistance decreases and the curre
nt coming into the junction increases, although the current through each path remains
the same.

Rules for parallel connections

1.

The potential difference (voltage) across each resistor is the same.

V
total
=V
1
=
V
2

2.

Cur
rent coming into a junction is the sum of

the currents in each path.

I
total

= I
1

+

I
2

The total current in the circuit is

The current in each path is

3.

The equivalent resistance

is the

reciprocal of the

sum of the individua
l
resist
ances

reciprocal
.

The e
quivalent resistance is always
less

than smallest resistance in group
.

Combined Series
-
Parallel Circuits

Most circuits today use both series and parallel wiring to utilize the advantages of each ty
pe.
Circuits containing
combinations of series and parallel circuits can be understood by analyzing them in steps.
When determining the
equivalent resistance for a complex circuit, you must simplify the circuit into groups of series and parallel resistor
s and
then find the equivalent resistance for each group until the circuit is reduced

to a single resistance. Work your way
backwards finding all potential drops and currents

across the individual circuits as shown in the diagram below.

Ex
ample
2:
Light

bulbs of fixed resistance 3.0

and 6.0

, a 9.0 V battery, and a switch
S

are connected as shown in
the schematic diagram
below
. The switch
S

is
initially
closed.

a.
Calculate the current in bulb
A
.

b. W
hich light

bulb is b

c.
Switch
S

is

now

opened. By checking the appropriate spaces below, indicate whether the brightness of

each light

bulb increases, decreases, or remains the same. Explain your reasoning for each light

bul
b.

i.

Bulb
A
: The brightness

increases

decreases

remains the same

Explanation:

ii.

Bulb
B
: The brightness

increases

decreases

remains the same

Explanation:

iii.

Bulb
C
: The brightness

increases

decreases

remains the same

Explanation:

Example 3:

Three resistors are arranged in a circuit as shown above. The battery has an unknown but constant emf and
negligible internal resistance.

a. Determine the equivalent resistance of the
three resistors. The current I in resistor R
3

is 0.40 ampere.

b. Determine the emf (Voltage) of the battery.

c. Determine the potential difference across resistor R
1
.

d. Determine the power dissipated in resistor R
1
.

e.
Determine the amount of charge that passes through resistor R
2

in one minute.

Kirchhoff’s rules and complex DC circuits

Electrical circuits
that
contain many paths or multiple sources of
emf
are analyzed using Kirchhoff’s rules (laws)
.
Kirch
hoff’s rules are

an application of two fundamental laws: the junction rule is based on the law of conservation of
charge and the loop rule is based on the law of conservation of energy.

Note

that Kirchhoff’s rules are
not
needed to
solve most problems on
the physics B test.

1.

The

j
unction rule

states that the

sum of the currents entering any junction must equal the sum of the currents
leaving that junction
.

To use the junction rule
, a
ssign symbols and directions to all currents

entering and leaving a jun
ction and set
them equate them. Don’t worry about direction because

if you happen to guess the wrong direction, the end
result will be negative, but the magnitude will be correct
.

2.

The

l
oop rule

states that the sum of the potential drops (voltages)
across
all elements around any closed circuit
loop must be zero
. This is
because a
ny charge that moves around a closed loop in a circuit must gain as much
energy as it loses

(change in potential energy is zero)
.

To use the
loop rule

1.

Choose a direction (clockwise

or counterclockwise)

to traverse a loop.

2.

If a resistor is traversed in
the
direction of the current, then V=
-
IR
;
i
f opposite to the current, then V=IR
.

3.

If
emf

is traversed in direction of
emf

(from

to +), then V=+
; if opposite, then V=
-
.

Example
5
:

The circuit shown above is constructed with two batteries and three resistors. The connecting wires may be
con
sidered to have negligible resistance. The current I is 2 amperes.

a. Calculate the resistance R.

b. Calculate the current in the

i
. 6
-
ohm resistor

i. 12
-
ohm resistor

c. The potential at point X is 0 volts. Calculate the electric potential at points B. C, and D in the circuit.

d. Calculate the power supplied by the 20
-
volt battery.

12

Capacitors in parallel

When capacitors are in parallel,

the potential drop

across each capacitor is the same as the battery

and the

battery
moves a charge that is the sum of the c
harges moved for each of the capacitors
.

Capacitors in series

When capacitors are in series, charge on each capacitor in is the same

and the
battery only moves an amount of
charge equal to the charge on one of the capacitors because the

charge passes by induction from one capacitor to the
next one
. Also, the terminal
voltage
of the battery
is equal to the sum of the individual
potential differences
across
each capacitor
.

Remember from Chapter 19 that to calculate
energy
stored by a capacitor

use

U
c
= ½CV
2

= ½QV

and to find
total energy

stored by a combination of capacitors add the individual values for each capacitor, or simply
do one calculation using the equivalent capacitance.

RC circuits
(circuits that contain a resi
stor(s) and a capacitor(s))

Current in the circuit and charge stored by the capacitor vary
exponentially
with time
. A
fter the switch is closed
,
graph below left,

the battery begins to charge the plates of the capacitor and continues to charge the capacit
or to the
maximum
. C
urrent is initially at a maximum but decreases as the

charge on the plates increases.
O
nce charged,
the
current is zero
.

The reverse occurs when the capacitor is discharged
, graph below right
.

q
battery
=q
1
+q
2

V
battery
=V
1
=V
2

C
eq
=C
1
+C
2

q
battery
=q
1
+q
2

V
battery
=V
1
=V
2

C
eq
=C
1
+C
2

q
battery
=q
1
=
q
2

V
battery
=V
1
+
V
2

Example 6:

A circuit contains two
resistors (10

and 20

) and two capacitors (12

F and 6

F) connected to a 6 V
battery,

as shown in the diagram above. The circuit has been connected for a long time.

a.

Calculate the total capacitance of the circuit.

b.

Calculate the current in the 1
0

resistor.

c.

Calculate the potential difference between points A and
B.

d.

Calculate the charge stored on one plate of the 6

F capacitor.

e. The wire is cut at point
P

. Will the potential difference between points
A

and
B
increase, decrease,

or

remain the same?

_____increase

_____decrease

_____remain the same

AC circuits

To analyze AC circuits, you
use the same
basic
techniques that you used to analyze DC circuits, but there is an
important distinction to note
. The direction of current across a resistor has no effect on the behavior of it (electric
energy will be converted to heat energy either way current flows), but the heating effect produced by an alternating
current with a maximum value of I
o

is

not

the s
ame as that produced by a direct current
. This is because

the magnitude
of the alternating current is continually changing, as shown
in the graphs
below. To find the average power dissipated
you must use the
rms (root mean square)

value for the

current a
nd voltage
.

Safety devices

Fuses

and
circuit breakers

are safety devices that open a circuit to prevent circuit overloads that can occur when too
many appliances are turned on at the same time or a short circuit occ
urs. A
short circuit

occurs when a path is formed
that has a very low resistance; large current results which may damage appliances or cause a fire.
A

fuse

is a short
piece of metal that melts from the heating effect of the current if the current is too
large.
A

circuit breaker

is an
automatic switch that uses a bimetallic strip to open the switch if the current exceeds some set value.
A

ground
-
fault
interrupter

(often found in electrical outlets in bathrooms or near sinks) detects small differences in
current and opens
the circuit preventing shock if tripped.