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Electric Potential Energy and Potential Difference
Electric potential energy results from forces between charges; potential difference is the energy needed to move a charge from point A to B.
Learning Objective

Calculate the potential energy between the charges
Key Points
 Electric potential energy is a type of potential energy that results from Coulomb forces. The potential energy (UE) between charges q and Q can be calculated as a function of distance between the charges (r):
$U_E(r)=\frac {qQ}{4 \pi \epsilon_0 r}$ .  The formula for potential energy can be modified for potential between many charges, so long as the interactions of each charge with every other charge in the system are considered. For example, potential between three charges can be solved using the following formula:
$U_E=\frac {1}{4 \pi \epsilon_0 r} (\frac {Q_1Q_2}{r{12}}+\frac {Q_2Q_3}{r{23}}+\frac {Q_1Q_3}{r{13}})$ .  Potential difference, or voltage, is the difference in electric potential energy between two points. It is denoted by ∆V and has units of volts, or joules per Coulomb.
Terms

coulomb
In the International System of Units, the derived unit of electric charge; the amount of electric charge carried by a current of 1 ampere flowing for 1 second. Symbol: C

potential energy
The energy an object has because of its position (in a gravitational or electric field) or its condition (as a stretched or compressed spring, as a chemical reactant, or by having rest mass)
Full Text
Electric potential energy is a type of potential energy that results from Coulomb forces. It is measured in joules and depends on the positioning of charged particles relative to one another, as well as the magnitude of their respective charges.
The potential energy (U_{E}) between charges q and Q can be calculated as a function of distance between the charges (r):
If there are three or more charges, the above formula can be modified so that the potential energies between all charges are summed. Consider, for example, the case involving charges Q_{1}, Q_{2} and Q_{3}:
In this example, r_{12} represents the distance between Q_{1} and Q_{2}, r_{23} represents the distance between Q_{2} and Q_{3}, and r_{13} represents the distance between Q_{1} and Q_{3}. The above formula can be modified for any number of charges.
Potential Difference
Potential difference , or voltage, is the difference in electric potential energy between two points. It is denoted by ∆V and has units of volts, or joules per Coulomb.
Voltage denotes the work per unit charge that must be done against a static electric field to move a charge from one point to another. It may represent a source of energy, or lost, stored or used energy. Voltage also is defined such that negative charges are pulled towards higher voltages, while positive charges move towards lower voltages. Thus, current in wires flows from higher to lower voltages.
Potential difference is independent of path taken from one point to another, and may be measured by any of a number of instruments . These include the voltmeter, the potentiometer, and the oscilloscope. It is most typically measured in circuits, and in such situations can be calculated using Ohm's Law, which will be covered in a later atom.
Potential difference in a static field
When a charge q moves from point A to point B, the potential difference is independent of path taken.
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Key Term Reference
 Law
 Appears in these related concepts: TwoComponent Forces, Physics and Other Fields, and Models, Theories, and Laws
 atom
 Appears in these related concepts: Overview of Atomic Structure, Description of the Hydrogen Atom, and Stable Isotopes
 circuit
 Appears in these related concepts: Combinations of Capacitors: Series and Parallel, Microwaves, and Maxwell's Equations
 current
 Appears in these related concepts: Reporting LongTerm Liabilities, The Battery, and Magnetic Force Between Two Parallel Conductors
 electric field
 Appears in these related concepts: Maxwell's Predictions and Hertz' Confirmation, Gauss's Law, and Ampere's Law: Magnetic Field Due to a Long Straight Wire
 electric potential
 Appears in these related concepts: Relation Between Electric Potential and Field, Parametric Surfaces and Surface Integrals, and Potentials and Charged Conductors
 energy
 Appears in these related concepts: Surface Tension, Energy Transportation, and Introduction to Work and Energy
 force
 Appears in these related concepts: Force of Muscle Contraction, Force, and First Condition
 magnitude
 Appears in these related concepts: Multiplying Vectors by a Scalar, Roundoff Error, and Components of a Vector
 ohm
 Appears in these related concepts: Resonance in RLC Circuits, Poiseuille's Equation and Viscosity, and Current and Voltage Measurements in Circuits
 potential
 Appears in these related concepts: What is Potential Energy?, Conservative and Nonconservative Forces, and Linear Expansion
 potential difference
 Appears in these related concepts: Uniform Electric Field, Energy Conservation, and The ElectronVolt
 potentiometer
 Appears in this related concept: Null Measurements
 relative
 Appears in these related concepts: Relative Deprivation Approach, Addition of Velocities, and Relative Velocity
 static
 Appears in these related concepts: Friction: Static, Time and Motion, and Alternative Views
 voltage
 Appears in these related concepts: The Nernst Equation, Electric Potential Due to a Point Charge, and Principles of Electricity
 work
 Appears in these related concepts: Heat and Work, Free Energy and Work, and The First Law of Thermodynamics
Sources
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Source: Boundless. “Electric Potential Energy and Potential Difference.” Boundless Physics. Boundless, 26 May. 2016. Retrieved 26 Sep. 2016 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/electricpotentialandelectricfield18/overview139/electricpotentialenergyandpotentialdifference4988078/