III. Voltage[Physics 2702]

Dr. Bill Pezzaglia

Updated 2014Feb

III. Voltage

A. Electrostatic Energy

B. Voltage

C. Equipotentials & Electric Field

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A. Electrostatic Energy

1) Work Energy Theorem

2) Potential Energy

3) Electrostatic Potential Energy

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1. Work Energy Theorem

a) Work Definition

b) Conservative Forces

c) Work-Energy Theorem

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a) Work

• Definition: Work=Force x Displacement

• Units: Joule=Newton-Meter

• Only force parallel to path contributes:

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cosFrrFW

Hence, a force perpendicular to the path does no work!

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

• Non-conservative Forces are:• Friction (velocity dependent)• Magnetic Forces on charges• Time dependent electric fields

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EqF

gmF

c) Work-Energy Theorem

• Work=KE

• Example: mass falling distance h in a gravity field (F=mg)

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22

21

if vvmxF

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if vvmmgh

2. Potential Energy

a) Field Potential

b) Potential Energy and Work

c) Conservation of Energy

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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

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ca UUUrFW

b) Kinetic and Potential

• From the work-energy theorem: W=K

• we get change in Kinetic Energy is related to change in Potential Energy

• 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

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UK

c) Conservation of Energy

• The “total energy” is the sum of potential and kinetic energy

• If the Potential Theory is valid: K=-U

• Then it follows that total energy is conserved

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0E

3. Electrostatic Potential Energy

a) Review Gravitational Potential Energy

b) Electrostatic Potential

c) Potential of assembling charges

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a) Review: Gravitational Potential Energy

• 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:

• This would be the amount of energythat a meteor would have hitting theearth and making a big crater!

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R

MmGU

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:

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R

qQkU

Note that on previous page it was negative, while here its positive. Why?

c) Potential Energy of Assembling Charges

• 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:

• 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.

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R

kQU

2

2

1

R

QQkU 21

d) Energy of a Dipole in Electric Field

• A dipole in an electric field will have a toque on it:

• 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:

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cospEEpU

Ep

W

B. Voltage

1) Definition of Voltage

2) Sources of Voltage

3) Measuring Voltage

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1a. Definition of voltage

• Potential Energy per unit test charge:(i.e. don’t want test charge to affect field)

• 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

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q

ULimVq 0

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.

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http://www.youtube.com/v/XU8nMKkzbT8?f=videos&app=youtube_gdata&autoplay=1

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

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2. Sources of Voltage

(a)Point Charge Source

(b)Superposition of Point Charges

(c) Batteries

(d)Thermo and Piezoelectrics

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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:

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r

QkrV )(

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!

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pk

Lz

kQ

Lz

kQzV

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.

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http://www.corrosion-doctors.org/Biographies/VoltaBio.htm

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

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3. Measuring Voltage

(a)Voltmeters

(b)Oscilloscopes

(c) Piezoelectric

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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)

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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.

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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!

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C. Equipotentials & Electric Field

1) Definition of Equipotentials

2) Electric field as gradient

3) diagrams

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Notes

• Added slide on energy of dipole in a field

• Added slide on Piezoelectrics

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