100 Volts
vo,x
0 Volts
V(x)
200
150
200 Volts
100
50
-200 Volts
200 Volts
coordinates
y
z
x
(note the perpendicular intersections)
10 V
0V
y
10 V
(line of symmetry is x-axis where y=0)
0V
(x,0)
yields
10)
3

 150 V/m 
.03) .02
x(x,0)
yields
U(x)
potential energy
A
B
negative slope
(FNET to right)
unstable
equilibrium
(FNET = 0)
x
C
stable
equilibrium
(FNET = 0)
positive slope
(FNET to left)
D
U(x)
potential energy
D: FNET to right
A: stable
equilibrium
A
B
C: unstable
Equilibrium
x
C
B: FNET to left
D
U(x)
potential energy
A
B
x
C
D
V(x)
electric potential
x
A
begin
B
Radial electric vector field of a charged conducting circle
y
+
x
y
_
+
x
y
_
y
_
x
U(x,y)
potential energy
dotted lines show
constant energy
y
x
FNET to right
and forward)
U(x,y)
potential energy
(dotted lines show
constant energy:
“equipotentials”)
y
(equipotentials
closer where
steepest)
x
FNET to right
and forward)
V(x,y)
electric potential
(potential energy per unit charge)
solid lines show
electric field
+
dotted lines show
constant electric
potential
y
arrow shows electric
field direction on
positive test charge
+
x
E(x,y)
+
+
V(x,y)
+
(dotted lines sh
electric potentia
y
+
(solid lines sh
electric field)
(arrow s
on test c
x
V(x,y)
Electrical potential
energy per unit charge
+
(dotted lines show constant
electric potential while solid
lines show electric field)
y
(equipotentials
closer where
steepest)
x
V(x,y)
+
(dotted lines sh
electric potentia
y
(solid lines sh
electric field)
x
V(x,y)
solid lines show
electric field
+
dotted lines show
constant electric
potential
y
x
+
y
y
V=4 volts
A
q
B
x
V=7 volts
V=5 volts
E=?
V=7 volts
V=5 volts
E=?
d = 2 cm
y
Q must be estimated or
measured with a protractor
to calculate the legs (x and
Y components of E).
100 V/m
57 V/m
Q = 35o
82 V/m
x
75o
60o
45o
30o
15o
10o
V(x,y)
dotted lines show
constant electric
potential
solid lines show
electric field
y
arrow shows electric
field direction on
positive test charge
E(x,y)
+
x
V(x,y)
y
+
x
_
+ + + ++ + + ++ + + ++ + + ++ + + ++ + + ++ + +
V
ACROSS SECTION
L
I
ITOTAL
BATTERY
+
ID
BATTERY
BATTERY
IE
IC
IB
IA
+
+
+
BATTERY
(handle)
e
e
e
e
e
e
e
e
BATTERY
e
e
e
e
e
e
PUMP
+
e
(spinning
paddle
wheel)
R
Vsource
R
b
c
a
Vsource
d
h
g
R
f
e
R
ce
e
R
R
resistors
in series
Vsource
R
resisto
in para
R
Vsource
R
R
3V
6W
3V
6W
current can never flow
current may flow
BATTERY
the ground
+
+
(depending on the properties of the ground
BATTERY
the ground
+
BATTERY
the ground
9-VOLT
BATTERY
+ _
+ _
9-VOLT
BATTERY
+
-
Unmagnetize
being placed
S
N
N
S
S
N
N
S
S
N
Needle
compass
Draw needle
STOP
PRELAB
+ ++
+++
--- -
Uncharged conducting coin
grounded to Earth.
+
E
----
- 
The presence of positive charge creates
an electric field at the coin surface that
attracts electrons from the Earth to negat
charge the coin.
+
E
----
Removing the grounding wire leaves
the coin positively charged.
The Earth is a giant reservoir of charge,
we do not worry about the fact that it has
some miniscule amount of excess positiv
charge.

+
E
+
+++
----
The presence of positive charge creates an electric field at the
coin surface that causes macroscopic charge separation. (The
coins positive charges are forced to be far away from the positively
charged object.)

+
+
+++
----
+
E
----
++
If some fool’s hand comes into contact with the coin,
the coin’s positive charges can move even further from
the charged object by moving into the hand (and body).

+
+
+
----
+
+
+
+
+
E
----
Removal of the hand leaves a negatively charged coin.
The hand is a large reservoir of charge and we will not
worry about the miniscule amount of excess positive
charge in the hand (and body) unless a very strong electric
field had been present.

nce of positively charged object.
+
E
----

Positively charged ob
?
B.
+
V
-
1.5 V
D.
C.
+
V
-
1.5 V
1.5 V
+
V
-
1.5 V
1.5 V
V
A.
B.
4.5 V
6.0 V
4.5 V
4.5 V
b
A.
B.
6.0 V
b
A.
c
4.5 V
6.0 V
4.5 V
a
d
A.
B.
4.5 V
B.
5V
7.5 V
5V
4.5 V
b
A.
c
4.5 V
6.0 V
4.5 V
a
d
A.
B.
4.5 V
B.
5V
7.5 V
5V
4.5 V
+
Initial attraction
+
pith ball
(conductor)
repulsion after touching
solid metallic
bar with round end
very thin strip
of pure gold
metallic
enclosure
+
+
+
+
+
+
+
+
+
_
+
_ +
_
+
__
+
_ +_
+
_ +
_
+
_ +
_
+
_ +
_
+
_
+
_
+
_
BATTERY
BATTERY
+
+
A.
B.
R
Vsource
R
R
Vso
V
6.0
+
BATTERY
0
BATTERY
1.5
BATTERY
3.0
BATTERY
4.5
+
+
+
+ ++
+++
--- -
--- -
+
E
+
+++

----
R
Vsource
R
R
+
_
_
point of
intersection
+
_
_
point of
intersection
can’t happen
B
A
-
+
C
-
+
+
-
Requivalent
R2
V
R1
R2
V
1W
1W
1W
1W
2W
1W
4W
9V
RA
R2=4 W
I2=? V2=?
10 V
IBattery=?
R1= 1 W
I1=? V1=?
R2=1 W
RTotal= ?
I2=? V2=?
12 V
IBattery=?
R1= 5 W
I1=? V1=?
R2
10 V
R1
R3=4 W
I3=? V3=?
9V
IBattery=?
R1 = 1 W
I1=? V1=?
R2= 2 W
I2=? V2=?
R3=2 W
I3=? V3=?
Rtotal = 5 ohms
Ibattery = 2 amps
V1 = V2 = 8 volts
I1 = I2 = 1 amp
R4=2 W
I4=? V4=?
R1= 8 W
I1=? V1=?
R2= 8 W
I2=? V2=?
V3 = V4 = 2 volts
I3 = I4 = 1 amp
R
Vapplied
I
Vapplied
a
100 W
b
200 W
c
V
c
100 W d
a
b
e
V
f
200 W
c
100 W d
a
b
e
V
f
200 W
R
red 1
black 1
100 W
+
-
200 W
black 2
red 2
V(t)
5
t
0
t
-5
I
Vapplied
R2=1 W
I2=? V2=?
R1= 8 W
I1=? V1=?
Requivalent = ?
Ibattery = ?
V1 = ?
I1 = ?
V2 = ?
I2 = ?
C
d
a
C
R
b
c
L
I
I
I
B
rotate
I
I
B
BFar is weak
BClose is strong
Magnet
I
I
V
I
Ampere’s Law:
 
 B  ds  o I totalenclosed by
Amperian loop
L
olenoid loops enclosed in the
perian loop, each with current I.
“loop density” N/L of solenoid.
Amperian loop
hole
perian
oop
BIN
External Inductance
R
Self Inductanc
R
einduced
L
ent
B acts like battery.
wing through resistor is easily measured.
einduced
L
Oscillating voltage source causes oscillating B
Oscillating B inside inductor induces voltage e
Back EMF makes inductor seem like a resisto
External Inductance
Self Inductanc
R
R
einduced
L
einduced
ent
cillating external B causes an
d voltage einduced across the
or.
L
Oscillating voltage source
B inside inductor which in
einduced across the inducto
uctance:
R
L
I
I
I
B
brushes
S
N
I
S
N
z
a

Bo  3 [T] xˆ
{outward}
y
I
I
z
y
x
V(t)
VS
VR
VL
V(t)
VR
V?
V?
V(t)
VR
VL
VC
V(t)
VR
VC
VS
V(t)
VR
V?
V?
V(t)
VR
VC
VL
VS
V(t)
VR
VC
VL
t
VS
R [ohm]
Vsource
C [farad]
L [henry]
I(t)
Vs (t)
Vs (t)-VR(t)
-
+
Vs (t)
VR
+
VS
0
-
Vs (t)-VR(t)-VC(t)- VL(t)=0
+
-Q
-
VC
VL
+
+Q
Vs (t)-VR(t)
Vs (t)-VR(t)-VC(t)
Vs (t)-VR(t)-VC(t)
V(t)
t
[V]
[t]
[V]
[t]
[V]
[t]
Modulate Wave Transmitted by Diode to Speaker
Quantum mechanical
turn-on voltage of diode.
[V]
0
Pulses let through by the diode move speaker with
frequency of desired audio wave.
A
B
PRIMARY
SOLENOID
SECONDARY
SOLENOID
A
seconda
circuit
Lsec
B
PRIMARY
SOLENOID
SECONDARY
SOLENOID
A
seconda
circuit
Lsec/2
B
PRIMARY
SOLENOID
SECONDARY
SOLENOID
A
B
3,600 [Hz]
[Hz]
RF Modulator
carrier [Hz]
low
in
RF
out
modulated CH
1
CH
2
RF Modulator
carrier [Hz]
low
in
RF
out
modulated CH
1
antenna
modulated CH
1
ground
A
B
Iamplitude
Iamplitude
fdrive
fresonance
fdrive
fresonance
R4=4 W
Requivalent=?
I4=? V4=?
9V
IBattery=?
R1= 1 W
I1=? V1=?
R2= 2 W
I2=? V2=?
c
+
d
BATTERY
ge +Q & -Q
+
+
+
+
+
+
+
- - - - - - - - +
+
+
+
+
+
B Total charge +Q &
(-Q/2)
(+Q/2)
(-Q/2)
(+Q/2)
BATTERY
+ + + + + + + +
- - - - - - - - - - - - - - -
+ + + + + + + +
S1
S2
C
VS
R1
R2
R1 = 1x106 [
R2 = 1x105 [
C = 1x10-5 [F
Vs = 10 [V]
Thumb shows
direction of
magnetic field.
B
q
If charge q is negative,
reverse B-field direction.
Wrap fingers
in direction of
current.
B
I
I
Thumb shows
direction of
magnetic field.
voltage
“height”
1.5 [V]
+
BATTERY
0 [V]
voltage
“height”
1.5 [V]
+
BATTERY
0 [V]
a
d
voltage
“height”
voltage
b
c
1.5 [V]
BATTERY
0 [V]
1.5 [V]
0 [V]
d
a
a
b
c
d
position on ci
e
t”
voltage
c
b
1.5 [V]
+
BATTERY
a
0 [V]
e
d
a
b
c
d
e
position on circuit
a
c
b
RBULB
1.5 [V]
a
d
e
1.5 [V]
voltage
“height”
a
1.50 [V]
0 [V]
+
BATTERY
0.75 [V]
b
c
voltage
“height”
1.50 [V]
0 [V]
BATTERY
0.75 [V]
+
voltage
1.50 [V]
0.75 [V]
0 [V]
a
b
c
a
position on circuit
a
RBulb
b
RBulb
c
V
b
RBulb
c
a
V
d
e
RBulb
f
voltage
“height”
b
a
c
1.5 [V]
+
BATTERY
0 [V]
d
f
e
voltage
50 [V]
75 [V]
0 [V]
a
f
d
b
a
c
position on circuit
e
f
a
display
settings
+ positive terminal for high current meas
- negative terminal or “ground”
+ positive terminal
Voltage
VDC
A
B
? amps
mA
Amperes
mA
R
Amperes
mA
R
A
Ohms (W)
W
R
ATTERY
1.5 [V]
3V
+
d
0 [V]
BATTERY
c
c
d
a
b
3.0 [V]
+
b
1.5 [V]
3V
+
BATTE
c
BATTERY
a
red1
x-y
mode
Vamp=3 V
red2
330 W
bottom
ground
CH1
CH2
constant
voltage
• 30 V
• Ground
• 1000 V
• 2000 V
• 3000 V
ground
------- ---
-
-
grou
constant
voltage
• 30 V
• Ground
• 1000 V
• 2000 V
• 3000 V
ground
charge
separation
-
---
+
+
++
+
++
evenly
arranged
-
-
-
+
++
+
++
+
-
-
+
-
clustered
positive
-- - -+
+
+
+
++
clustered
negative
Imaginary
fshift+p
Real
Imaginary
Real
Imaginary
Im{V(t)}
Real
Re{V(t)}
V(t)=V0eiwt rotates around the complex plane in time.
Voltage
VS
VR
VL
VC
THEORY
RE
R [ohm]
C [farad]
Vsource
R
L [henry]
L
N
S
AB C
D E
x
10 [V]
8
6
4
2
0
x
0
2
4
6
8
10
10
8
6 4 2
A
0 [V]
B
x
0
2
4
6
8
10
Ampere’s Law:
 
 B  ds  o I totalenclosed by
Amperian loop
L
olenoid loops enclosed in the
perian loop, each with current I.
“loop density” N/L of solenoid.
Amperian loop
hole
perian
oop
BIN