# 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.

#### 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

• 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

• 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)

#### Figures

1. ##### 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.

2. ##### Electric Potential Difference

A brief overview of electric potential difference and electric potential energy for beginning physics students.

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 (UE) between charges q and Q can be calculated as a function of distance between the charges (r):

$U_E(r)=\frac {qQ}{4 \pi \epsilon_0 r}$

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 Q1, Q2 and Q3:

$U_E=\frac {1}{4 \pi \epsilon_0 r} (\frac {Q_1Q_2}{r{12}}+\frac {Q_2Q_3}{r{23}}+\frac {Q_1Q_3}{r{13}})$

In this example, r12 represents the distance between Q1 and Q2, r23 represents the distance between Q2 and Q3, and r13 represents the distance between Q1 and Q3. The above formula can be modified for any number of charges.

### Potential Difference

Potential difference Figure 2, 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 (Figure 1). 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 Glossary

atom
The smallest possible amount of matter which still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons.
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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
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current
The time rate of flow of electric charge.
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electric field
A region of space around a charged particle, or between two voltages; it exerts a force on charged objects in its vicinity.
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electric potential
The potential energy per unit charge at a point in a static electric field; voltage.
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energy
A quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent.
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force
A physical quantity that denotes ability to push, pull, twist or accelerate a body which is measured in a unit dimensioned in mass × distance/time² (ML/T²): SI: newton (N); CGS: dyne (dyn)
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Force
A force is any influence that causes an object to undergo a certain change, either concerning its movement, direction or geometrical construction.
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Law
A concise description, usually in the form of a mathematical equation, used to describe a pattern in nature
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magnitude
A number assigned to a vector indicating its length.
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ohm
in the International System of Units, the derived unit of electrical resistance; the electrical resistance of a device across which a potential difference of one volt causes a current of one ampere; symbol: Ω
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particle
A very small piece of matter, a fragment; especially, the smallest possible part of something.
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potential
A curve describing the situation where the difference in the potential energies of an object in two different positions depends only on those positions.
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potential difference
The difference in potential energy between two points in an electric field; the difference in charge between two points in an electrical circuit; voltage.
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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)
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potentiometer
an instrument that measures a voltage by opposing it with a precise fraction of a known voltage, and without drawing current from the unknown source.
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relative
Expressed in relation to another item, rather than in complete form.
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static
Fixed in place; having no motion.
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voltage
The amount of electrostatic potential between two points in space.
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work
A measure of energy expended in moving an object; most commonly, force times displacement. No work is done if the object does not move.