Download voltage_2014feb_2702

III. Voltage
[Physics 2702]
Dr. Bill Pezzaglia
Updated 2014Feb
2
III. Voltage
A. Electrostatic Energy
B. Voltage
C. Equipotentials & Electric Field
3
A. Electrostatic Energy
1) Work Energy Theorem
2) Potential Energy
3) Electrostatic Potential Energy
4
1. Work Energy Theorem
a) Work Definition
b) Conservative Forces
c) Work-Energy Theorem
a) Work
• Definition: Work=Force x Displacement
• Units: Joule=Newton-Meter
• Only force parallel to path contributes:
 
W  F  r  Fr cos 
Hence, a force perpendicular to the path does no work!
5
b) Conservative Forces
• For a “conservative force” the work is independent of the path, it
only depends upon the endpoints.
• Conservative Forces are
• Gravity
• Electrostatics


F  mg


F  qE
• Non-conservative Forces are:
• Friction (velocity dependent)
• Magnetic Forces on charges
• Time dependent electric fields
6
7
c) Work-Energy Theorem
• Work=KE

Fx  m v f  vi
1
2
2
• Example: mass falling distance h in a
gravity field (F=mg)

mgh  m v f  vi
1
2
2
2

2

8
2. Potential Energy
a) Field Potential
b) Potential Energy and Work
c) Conservation of Energy
a) Field Potential
• For a conservative force, the total work done
by the field on the test particle over a path can
be equated to the difference of the potential
energy of endpoints
 
W  F  r  U  U a  U c
9
b) Kinetic and Potential
10
• From the work-energy theorem: W=K
• we get change in Kinetic Energy is related to change in
Potential Energy
K  U
• For example, if a ball drops in a gravity field a distance
“h”, the potential energy decreases by U=mgh, which
gives the ball kinetic energy
c) Conservation of Energy
• The “total energy” is the sum of potential and
kinetic energy
E U  K
• If the Potential Theory is valid: K=-U
• Then it follows that total energy is conserved
E  0
11
12
3. Electrostatic Potential Energy
a) Review Gravitational Potential Energy
b) Electrostatic Potential
c) Potential of assembling charges
a) Review: Gravitational Potential Energy
13
• Near surface of earth, where gravitational field is constant g=9.8
m/s2, then the change of potential energy of lifting a mass “m” up
a distance “h” is just: U=mgh
• For large distances, gravity follows the inverse square law. A
body “m” falling from infinity to the surface of the earth (mass
“M”) will have a change of potential energy of:
Mm
U  G
R
• This would be the amount of energy
that a meteor would have hitting the
earth and making a big crater!
b) Electrostatic Potential Energy
• Electric fields also follow the inverse square law.
Hence a small test charge “q” pushed from infinity
onto a massive ball of charge “Q” of radius “R” will
have a change of potential energy of:
qQ
U   k
R
Note that on previous page it was negative, while here its positive. Why?
14
c) Potential Energy of Assembling Charges
15
• If you assemble two charges (such as a dipole) from charges
which started out at opposite ends of the universe, the energy it
would take is:
Q1Q2
U  k
R
• The total energy “stored” by putting total charge “Q” on the ball
of radius “R” is a slightly different problem, because initially there
is very little field you have to fight, but as you add charge the
Electric field increases and it takes that much more work to add
the next piece.
1 kQ2
U  
2 R
16
d) Energy of a Dipole in Electric Field
• A dipole in an electric field will
have a toque on it:
 
  p E

• The work done to twist the
dipole a small amount of angle
would be torque times angular
displacement:
• The energy of a dipole in an
electric field can be easily
expressed as:


W    
 
U   p  E   pE cos 
17
B. Voltage
1) Definition of Voltage
2) Sources of Voltage
3) Measuring Voltage
1a. Definition of voltage
• Potential Energy per unit test charge:
(i.e. don’t want test charge to affect field)
18
U 
V  Lim 
q 0
q
• Units: Volt=Joule/Coulomb
• Voltage is the “pressure” that makes charges
move (current flow).
• Even if there is no test charge to experience it,
voltage exists
1b. Cathode Ray Tube
•
A CRT (Cathode Ray Tube) is a
vaccuum tube with a large voltage
across the electrodes. Electrons are
emitted by the Cathode and
accelerate towards the anode.
•
Kinetic energy the electrons gain is
hence: U=eV
•
1 eV = 1 electron volt is the energy of
one electron accelerated through one
volt = 1.6x10-19 Joules.
http://www.youtube.com/v/XU8nMKkzbT8?f=videos&app=youtube_gdata&autoplay=1
19
1c. Particle Accelerators
SLAC (Stanford
Linear Accelerator
Center) accelerates
electrons to 50 GeV
of energy
Note: the E=mc2
rest-mass energy of
a proton is only 938
MeV
20
21
2. Sources of Voltage
(a) Point Charge Source
(b) Superposition of Point Charges
(c) Batteries
(d) Thermo and Piezoelectrics
2a. Charge as Source of Voltage
• Define the voltage at infinity to be zero
• Voltage a distance “r” from the center of a
spherical charge Q is:
Q
V (r )  k
r
22
2.b Voltage of a Dipole
• Basically you use “superposition” of
voltages of two monopoles.
• Voltage of dipole along its z-axis drops
off like the square of the distance!
kQ
kQ
p
V ( z) 


k
2
1
1
z  2 L  z  2 L  z
23
2c. Batteries are a source of voltage
• Volta (1745-1827) “The
Newton of electricity”
•1800 develops first battery
(approximately 30 volts)
•By adding batteries together
in series, one can make as
big as voltage as you want.
http://www.corrosion-doctors.org/Biographies/VoltaBio.htm
24
2d. Piezoelectrics etc
Some devices that are useful as detectors
• Thermoelectrics: some materials will
create a voltage across them due to a
temperature difference
• Pyroelectrics: heating some materials
will create a voltage across them
• Piezoelectrics: 1880 Pierre Curie
demonstrates effect that some crystals
generate a voltage when deformed
25
26
3. Measuring Voltage
(a) Voltmeters
(b) Oscilloscopes
(c) Piezoelectric
3a. Voltmeters
• The classic voltmeter does not actually measure
voltage directly
• Instead, it is really an ammeter (galvanometer)
measuring current through a “shunt resistor” “R”
• The product of current times “R” gives the
voltage (indirectly)
27
3b. Oscilloscope
• Oscilloscopes are used to measure voltage (especially of
AC signals). They are essentially a CRT tube with
deflection plates.
• The amount of deflection of the beam is proportional to
the voltage across the plates.
28
29
3c. Converse Piezoelectric Effect
• 1881 Gabriel Lippmann predicts converse
should be true, changing voltage across a crystal
would cause it to deform.
• Used to make “piezo speaker” (e.g. in your cell
phone as can be made very small and thin!)
• Or, can be used as a device to measure voltage!
30
C. Equipotentials & Electric Field
1) Definition of Equipotentials
2) Electric field as gradient
3) diagrams
31
Notes
•
•
Added slide on energy of dipole in a field
Added slide on Piezoelectrics