```Superposition of Forces
r
r
F
12

qq
1 2 eˆ
r
2
4 r
0
We find the total force by
the individual forces.
Work Problem 21-18
(III) Two charges,
 Q and  4Q are a distance
apart.
These two charges are free to move
but do not because there is a third
charge nearby. What must be the
magnitude of the third charge and its
placement in order for the first two
to be in equilibrium?
21-18.
Electric Field of a Point Charge
E 
F
q0
=
kq
r
2
eˆ r
E k q
r2
k
1
4
0
Relation between F and E
If w e put a charge q1 in an electric field E ,
then the charge q1 feels a force of value
F1  q1 E
* * * T his is the really useful part.
Don’t confuse this charge q1 with the
test charge q0 or the original charges q
that produced E. The test charge q0
was used to find the electric field. This
is a real charge q1 placed in the electric
field.
Electric Field Lines for a Point Charge
Electric Field Lines for Systems of Charges
We call this
a dipole. It
is a dipole
field.
The Electric Field of a Charged Plate

E 
2 0
A Parallel-Plate Capacitor
The electric field near a conducting surface
must be perpendicular to the surface when
in equilibrium.
If w e place conductor in electric field,
the E lines m ust be  to surface. If not,
charges w ould m ove. E m ust be zero inside .
Conductor placed around a charge +Q
C onductor
21-86.
An electron moves in a circle of
radius r around a very long uniformly
charged wire in a vacuum chamber, as
shown in the figure. The charge density on
the wire is λ = 0.14 μC/m. (a) What is the
electric field at the electron (magnitude and
direction in terms of r and λ? (b) What is the
speed of the electron?
F = qE = m a
a=
qE
m
We can work all kinds of problems with
charged particles moving in electric
fields.
Electron
entering charged
parallel plates
Electric flux:
F
E
= E ×A = E A co s q
Electric flux through an
area is proportional to
the total number of field
lines crossing the area.
Flux through a closed surface:
  0 for closed surface that contains no ch arge
E
positive
negative
The net number of field lines through the
surface is proportional to the charge
enclosed, and also to the flux, giving
Gauss’s law:
E 

E dA 
Q encl
o
This can be used to find the electric field
in situations with a high degree of
symmetry.
Electric field of charged sheet
  E (2 A ) 
Q
0

A
0
w here  is charge/area on the sheet.
E2A 
E 
A

2 0
0
Electric Potential V
V 
U
q0

W
*** U nit: J/C = volt, V
q0
definition!!
Electric potential, or potential, is one of
the most useful concepts in
electromagnetism. This is a biggie!!
Electrostatic Potential Energy and
Potential Difference
The electrostatic force is
conservative – potential
energy can be defined.
Change in electric potential
energy is negative of work
done by electric force:
U b - U a = - W = - qEd
The Potentials of Charge Distributions
If the electric field is known:
ò
DV = -
rb
E ds
ra
For one point charge:
V =
q
4p e0 r
For many point charges:
V =
1
4p e0
å
i
qi
ri
The Potentials of Charge Distributions
If the electric field is known:
ò
DV = -
rb
E ds
ra
For differential charge:
dV =
dq
4p e0 r
For many point charges:
V =
1
4p e0
For a continuous charge
distribution:
V =
å
1
4p e0
i
ò
qi
ri
dq
r
Equipotential Surfaces
V =
q
4p e0 r
An equipotential is a line
or surface over which the
potential is constant.
Electric field lines are
perpendicular to
equipotentials.
The surface of a conductor
is an equipotential.
Equipotential Surfaces
Another case showing electric
field lines are perpendicular to
equipotentials.
The surface of a conductor is an
equipotential.
We can also see that
equipotentials are perpendicular
to electric fields from the
equation
òE
d
Equipotential Surfaces
Equipotential surfaces are always
perpendicular to field lines; they are
always closed surfaces (unlike field lines,
which begin and end on charges).
Electric field and
equipotentials for
electric dipole.
23-74.
Four point charges are located at the
corners of a square that is 8.0 cm on a side. The
charges, going in rotation around the square, are Q,
2Q, -3Q and 2Q, where Q = 3.1 μC. What is the total
electric potential energy stored in the system, relative
to U = 0 at infinite separation?
When a capacitor is connected to a battery, the
charge on its plates is proportional to the
voltage:
Q = CV
The quantity C is called the capacitance.
C =
Q
V
Parallel plate capacitor
V  Ed 
So
Q

0
d 
0A
Q d
A 0
and
d
V
C 

Q
V

0A
d
The capacitance
value depends only
on geometry!
Capacitors in Parallel
Capacitors in parallel
have the same voltage
across each one. The
equivalent capacitor is
one that stores the
same charge when
connected to the same
battery:
C eq = C 1 + C + C 3
(parallel)
Capacitors in Series
Capacitors in series have the same charge. In
this case, the equivalent capacitor has the
same charge across the total voltage drop.
Note that the formula is for the inverse of the
capacitance and not the capacitance itself!
V V V V  Q  Q  Q  Q
1
2
3 C
C
C
eq C
1






Q Q 1  1  1
C eq
C C
C
1
2
3
1  1  1  1
C eq C C
C
1
2
3






2
3
Effect of a Dielectric on the Electric Field of a Capacitor
E  E0
E  E 0 /  and V  V 0 / 
w here  is the dielectric
constant
A dielectric is an insulator, and is
characterized by a dielectric constant
k
.
Capacitance of a parallel-plate capacitor filled
with dielectric:
C  0 A
d
C   C0
for parallel-plate capacitor
Using the dielectric constant, we define the
permittivity:
   0
Energy in electric field
The energy U in a capacitor is














2

A

U  1CV 2  1 0
E d 
2
2
d

U  1 E 2 Ad
2 0
The volume is Ad, and the energy density u is
u  energy  2 0
 1 E2
2 0
volum e
Defibrillator
Potential energy of a
charged capacitor:
All three expressions are
equivalent!
U 
1
2
QV 
1
2
CV
2

Q
2
2C
A complete circuit is one where current can
flow all the way around. Note that the
schematic drawing doesn’t look much like the
physical circuit!
Open
circuit
Direction of Current and Electron Flow

V  IR
O hm 's law
V in this case is the em f of battery
R is the resistance of bulb

Resistors are color coded to indicate the value
of their resistance.
22  10   10%
6
Resistivity
The resistance of a wire is directly
proportional to its length and inversely
proportional to its cross-sectional area:
R= r
A
The constant ρ, the resistivity, is
characteristic of the material.
Energy and Power
P ow er
P 
U
t

dU

dt
( dQ )V
 IV
dt
The unit of electrical power is watt W (J/s).
If we use Ohm’s law with this equation, we
have
P  IV  I ( IR )  I R
2
P  IV 
V
R
V 
V
2
R
AC Voltage and Current
for a Resistor Circuit
V  V 0 sin 2  ft  V 0 sin  t
 V0 
I 

 sin  t  I 0 sin  t
R  R 
I 0  I m ax  peak current
(m axim um current)
V
Note that I and V are
in phase!!
25-36. (II) A 120-V hair dryer has
two settings: 850 W and 1250 W. (a) At
which setting do you expect the
resistance to be higher? After making a
guess, determine the resistance at (b) the
lower setting; and (c) the higher setting.
25-43. (II) How many 75-W
lightbulbs, connected to 120 V as in
Fig. 25–20, can be used without
blowing a 15-A fuse?
Resistors in Series


 V1  V 2  V 3
Current is not used up in
each resistor. Same current
I passes through each
resistor in series.


 IR eq
 IR1  IR 2  IR 3
 I ( R1  R 2  R 3 )  IR eq
R eq  R1  R 2  R 3
A parallel connection splits the current; the
voltage across each resistor is the same:
I  I1  I 2  I 3
V
R eq
1
R eq
 1
1
1 



V 



R1 R 2 R 3
R
R
R
2
3 
 1
V
V
V
 1
1
1 




R
R
R
2
3 
 1
Analyzing a Complex Circuit of Resistors
1
'
R eq
'
R eq

1

R
 R/2
1
R

2
R
R eq  R  R  R / 2
R eq  2.5 R
Kirchhoff’s Junction Rule
In +
Out -
The sum of currents meeting at a junction
must be zero. I1 – I2 – I3 = 0 or I1 = I2 + I3
Kirchhoff’s Loop Rule
The sum of
potential
differences
around any
closed circuit
loop is zero.
Our rules:
1) When going from – to + across an emf the
V is +. (+ to -, it is -).
2) When going across resistor in direction of
assumed I, the V is -. (Opposite, it is +).
Measuring the Current in a Circuit
Am m eter
We want ammeter to have very low resistance
so it will not affect circuit. Ammeters go in
series.
Measuring the Voltage in a Circuit
V oltm eter
1
R eq

1
R
1

R voltm eter
We want voltmeter to have very large resistance
so it will not affect circuit. Voltmeters go in
parallel across what is being measured.
An ohmmeter measures
resistance; it requires a
battery to provide a
current. These circuits
are much more
complicated. Rsh is a
shunt resistor to change
scales. Rser is a resistor
scale zero.
Magnetic Field Lines for a Bar Magnet
B
S
N
Imagine using a test pole N; place it at any
point and see where the force is. Just like
we do for electric fields. We actually use
small compasses to do this.
The Magnetic Force
Right-Hand Rule
FB  q v  B
Units of magnetic field: teslas
kg
1T = 1
C ×s
The Lorentz force is the sum of the electric and
magnetic forces acting on the same object:
æ
çç
çç
çç
çè
F = q E + v´ B
ö
÷
÷
÷
÷
÷
÷
÷
÷
ø
The Earth’s magnetic field is similar to that of a
bar magnet.
Note that the Earth’s
“North Pole” is really
a south magnetic
pole, as the north
ends of magnets are
attracted to it.
Operating Principle of a Mass Spectrometer
r
mv
q B
Several applications
Magnetic Force on
a Current-Carrying
Wire
F qvB
dF  I d  B
F  Id B
F  I LB
d
dq
F  IL B sin 
The Magnetic-Field Right-Hand Rule
Put thumb along
direction of current,
and fingers curl in
direction of B.
r
B ~
I
r
discuss
Magnetic Forces on a Current Loop
F 0
I
I
F = Ih B
Forces cause
a torque
F 0
Ftotal = 0
I
F  I LB
An electric motor uses the torque on a
current loop in a magnetic field to turn
magnetic energy into kinetic energy.
27-23. (II) A 6.0-MeV (kinetic energy)
proton enters a 0.20-T field, in a plane
perpendicular to the field. What is the
radius of its path? See Section 23–8.
F 2  I 2  B1
We have to look closely at fields and
forces to see how the forces occur.
Ampère’s Law
We have seen that
0I
B
2 r






 0 I  


 B ds  B  ds  B (2 r )  2 r   2 r    0 I


B
ds


I
enclosed

0
To find the field inside, we use Ampère’s law along
the path indicated in the figure.
B
ds   I enclosed
0
T he dot product of B and ds is zero everyw h ere except from cd
B
d
ds   B ds  B   0 N I
c
B   0 I N   0 nI
N  # turns, n  turns/length
B   0 nI
Induced Current Produced by a Moving Magnet
v
v
ind
We conclude that it is the change in magnetic
flux that causes induced current. F = B A
B
This is called Faraday’s Law of Induction
Induced em f 
N is num ber of turns
 N
dB
dt
Lenz’s Law
The induced current will
always be in the direction to
oppose the change that
produced it.
In d u ced em f
Û
In d u ced cu rren t
Applying Lenz’s Law to a Magnet Moving
Toward and Away From a Current Loop
Induced
current
v
v
An Electrical Generator
Falling w ater,
Current is
induced
Produces
AC power
steam
Magnetic flux
changes!
A Simple Electric Motor/Generator
Inductance
 B  LI


m agnetic flux depends on current
dB
dt
 L
dI
dt
L is called inductance (actually self ind uctance here).
The inductance L is a proportionality
constant that depends on the geometry
of the circuit
There will be a magnetic flux in Loop 1 due to current I1
flowing in Loop 1 and due to current I2 flowing in Loop 2.
 B (1)  L I  M
1 1
I
12 2
S im ilarly,
 B (2)  L I  M
2 2
I
21 1
Now it is clearer why we
call L self inductance and
M mutual inductance.
For exam ple, tw o
nearby coils
A rea A
Solenoid Self-Induction
B   0 nI
 B  N B A   0 nN IA   0 A n
 B  L I , so
L  0 An
2
I
2
O nly depends on geom etry.
General energy density
uB 
1 B
uE 
1
2
general result
2 0
2
0E
2

1B
2
u  uB  uE  
 0E 
2  0

2
30-34. (II) A 425-pF capacitor is
charged to 135 V and then quickly
connected to a 175-mH inductor.
Determine (a) the frequency of
oscillation, (b) the peak value of the
current, and (c) the maximum energy
stored in the magnetic field of the
inductor.
RL Circuit
V
0
I
Think about what will happen when the switch is
closed.
V 0  L dI  IR  0
dt
Current as a Function of Time in an RL Circuit
 L/R
I 
V0
R
1  e
 t /


V0
R
1  e
 tR / L

Oscillations in LC Circuits
Close switch.
It will discharge through
inductor, and then recharge
in opposite sense.
If no resistance, will continue
indefinitely.
The charge and current are both
sinusoidal, but with different phases.
Q  Q 0 cos( t   )
I  I 0 sin( t   )
LC Oscillations with Resistance
(LRC Circuit)
Any real (nonsuperconducting) circuit will
have resistance.
Damped Oscillations in RLC Circuits
Charge equation:
Solution:
where
and
w '= 0
w hen
R2 = 4L
C
This is a step-up
transformer – the
emf in the secondary
coil is larger than the
emf in the primary:
VP
VS

NP
NS
Lots of applications for transformers,
the bug zapper.
Power distribution
Transformers work only if the current is
changing; this is one reason why electricity is
transmitted as ac.
Single elements with AC Source
Resistors, capacitors,
and inductors have
different phase
relationships between
current and voltage
when placed in an ac
circuit.
The current through
a resistor is in phase
with the voltage.
I = I cos w t
0
Resistive element
Inductive Element
The voltage across the
inductor is determined by
I = I cos w t
0
V = L dI = - w L I sin w t
0
dt
or


0
V   L I cos   t  90 
0
V  V cos
0







0
 t  90 


Therefore, the current
through an inductor
lags the voltage by 90°.
I = I cos w t
0
t1
t2
Inductive Circuit
The voltage across the inductor is related
to the current through it:
V m ax  V   LI  X L I 0
0
0
The quantity XL is called the inductive
reactance, and has units of ohms:
X L º w L = 2p f L
For very low frequencies the inductive reactance is
small. That is because for direct currents (zero
frequency), an inductor has little or no effect. Direct
current passes right through an inductor.
Capacitive Circuit
The voltage across the
capacitor is given by




Q
 1 
0
V  I 
cos   t  90 

0  C 
C




 V cos
0





0
 t  90 


Therefore, in a capacitor,
voltage by 90°.
t1
t2
Capacitive Circuit
The voltage across the capacitor is related
to the current through it:










1

0
0
V I
cos   t  90   V cos   t  90 
0  C 
0











1  I X
V I
0
0  C 
0 C

The quantity XC is called the capacitive
reactance, and (just like the inductive
reactance) has units of ohms:
XC º
1 =
1
w C 2p f C
Effects of frequency on
capacitive reactance
Note that when the frequency increases to
large values that XC becomes very small. The
current then becomes very large.
1
XC 
Why?
C
The frequency is so high that the capacitor
doesn’t have time to fully charge. It almost
acts as a short circuit. At low frequencies, it
acts as an open circuit.
Either capacitors or inductors can be used
to make either AC or DC filters:
AC &
DC
input
LRC Series AC Circuit
Analyzing the LRC series AC circuit is
complicated, as the voltages are not in phase
– this means we cannot simply add them.
Furthermore, the reactances depend on the
frequency.
Some Applications
Diodes and Rectifiers
• A diode conducts electricity
in one direction only
• Can use diodes and
combinations of diodes to
make half- and full-wave
rectifiers
Half-wave
Rectifier
Some Applications
Full-wave
Rectifier
```