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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 Objectives

Calculate the potential energy between the charges

Define the electric potential energy

Define the electric potential difference
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

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)

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
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.
Key Term Reference
 Law
 Appears in this related concepts: TwoComponent Forces, Physics and Other Fields, and Models, Theories, and Laws
 atom
 Appears in this related concepts: Early Ideas about Atoms, Stable Isotopes, and John Dalton and Atomic Theory
 circuit
 Appears in this related concepts: Combinations of Capacitors: Series and Parallel, Microwaves, and Maxwell's Equations
 current
 Appears in this related concepts: Reporting LongTerm Liabilities, Magnetic Force Between Two Parallel Conductors, and The Junction Rule
 electric field
 Appears in this related concepts: Gauss's Law, Uniform Electric Field, and Maxwell's Predictions and Hertz' Confirmation
 electric potential
 Appears in this related concepts: Relation Between Electric Potential and Field, Parametric Surfaces and Surface Integrals, and Electric Potential Due to a Point Charge
 energy
 Appears in this related concepts: Energy Transportation, Surface Tension, and Introduction to Work and Energy
 force
 Appears in this related concepts: Work Done by a Variable Force, Driven Oscillations and Resonance, and Glancing Collisions
 magnitude
 Appears in this related concepts: Roundoff Error, Multiplying Vectors by a Scalar, and Components of a Vector
 ohm
 Appears in this related concepts: Resonance in RLC Circuits, Poiseuille's Equation and Viscosity, and Current and Voltage Measurements in Circuits
 potential
 Appears in this related concepts: What is Potential Energy?, Conservative and Nonconservative Forces, and Linear Expansion
 potential difference
 Appears in this related concepts: EMF and Terminal Voltage, Energy Conservation, and The ElectronVolt
 potentiometer
 Appears in this related concept: Null Measurements
 relative
 Appears in this related concepts: Relative Deprivation Approach, Relative Velocity, and Addition of Velocities
 static
 Appears in this related concepts: Translational Equilibrium, Time and Motion, and Alternative Views
 voltage
 Appears in this related concepts: Conductors, The Millikan OilDrop Experiment, and Principles of Electricity
 work
 Appears in this related concepts: Conservation of Energy in Rotational Motion, Heat and Work, and Free Energy and Work
Sources
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Cite This Source
Source: Boundless. “Electric Potential Energy and Potential Difference.” Boundless Physics. Boundless, 14 Nov. 2014. Retrieved 23 Apr. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/electricpotentialandelectricfield18/overview139/electricpotentialenergyandpotentialdifference4988078/