# data[].attributes.attachment_link.readable_text
Si introducimos una carga q' en el seno de un campo eléctrico, la carga sufrirá la acción de una fuerza eléctrica y como consecuencia de esto, adquirirá cierta energía potencial eléctrica (también conocida como energía potencial electrostática). Si lo vemos desde una perspectiva más simple, podemos pensar que el campo eléctrico crea un área de influencia donde cada uno de sus puntos tienen la propiedad de poder conferir una energía potencial a cualquier carga que se sitúe en su interior.

A partir de este razonamiento, se establece una nueva magnitud escalar propia de los campos eléctricos denominada potencial eléctrico y que representa la energía potencial electrostática que adquiere una unidad de carga positiva si la situamos en dicho punto.

El potencial eléctrico en un punto del espacio de un campo eléctrico es la energía potencial eléctrica que adquiere una unidad de carga positiva situada en dicho punto.

donde:

V es el potencial eléctrico en un punto del campo eléctrico. Su unidad en el S.I. es el julio por culombio (J/C) que en honor a Alesandro Volta recibe el nombre de Voltio.

Ep es la energía potencial eléctrica que adquiere una carga testigo positiva q' al situarla en ese punto.

El hecho de que todas las magnitudes sean escalares, permite que el estudio del campo eléctrico sea más sencillo. De esta forma, si conocemos el valor del potencial eléctrico V en un punto, podemos determinar que la energía potencial eléctrica de una carga q situada en él es:

Potencial eléctrico creado por una carga puntual

Tal y como estudiamos en el apartado de intensidad del campo eléctrico, una única carga q es capaz de crear un campo eléctrico a su alrededor. Si en dicho campo introducimos una carga testigo q' entonces, atendiendo a la definición de energía potencial eléctrica de dos cargas puntuales:

El potencial eléctrico del campo eléctrico creado por una carga puntual q se obtiene por medio de la siguiente expresión:

donde:

V es el potencial eléctrico en un punto.

K es la constante de la ley de Coulomb.

q es la carga puntual que crea el campo eléctrico.

r es la distancia entre la carga y el punto donde medimos el potencial.

Si observas detenidamente la expresión puedes darte cuenta de que:

Si la carga q es positiva, la energia potencial es positiva y el potencial eléctrico V es positivo.

Si la carga q es negativa, la energía el potencial es negativa y el potencial eléctrico V es negativo.

Si no existe carga, la energía potencial y el potencial eléctrico es nulo.

El potencial eléctrico no depende de la carga testigo q' que introducimos para medirlo.

Potencial eléctrico creado por varias cargas puntuales

Si el campo eléctrico es creado por varias cargas puntuales, el potencial eléctrico en un punto sigue el principio de superposición:

El potencial eléctrico originado por n cargas puntuales en un punto de un campo eléctrico es la suma escalar de los potenciales eléctricos en dicho punto creados por cada una de las cargas por separado.

o lo que es lo mismo:

Diferencia de Potencial Eléctrico

Si dos puntos de un campo eléctrico poseen distinto potencial eléctrico, entre ambos puntos existe lo que se denomina una diferencia de potencial o tensión, ΔV. Este valor se encuentra íntimamente relacionado con el trabajo eléctrico. Por definición, el trabajo que debe realizar un campo eléctrico para trasladar una carga q desde un punto A a otro B dentro del campo se obtiene por medio de la siguiente expresión:

Si aplicamos la definición de potencial eléctrico, obtenemos que:

La diferencia de potencial eléctrico entre dos puntos A y B de un campo eléctrico es el opuesto del trabajo realizado por el campo eléctrico para trasladar una unidad de carga positiva desde el punto A al B.

Potencial Eléctrico y el Movimiento de las Cargas

Si analizamos detenidamente la expresión de la diferencia de potencial:

Tal y como vimos en el apartado del trabajo eléctrico, el trabajo realizado por una fuerza eléctrica para desplazar una carga q desde un punto A a otro B, sin presencia de fuerzas externas, es un valor positivo. Si estudiamos que ocurre si la carga q es positiva o negativa, obtenemos que:

Por tanto:

Las cargas positivas se mueven desde zonas de mayor potencial eléctrico a zonas de menor potencial eléctrico.

Las cargas negativas se mueven desde zonas de menor potencial eléctrico a zonas de mayor potencial eléctrico.

Teniendo en cuenta que tal y como estudiamos en el apartado de intensidad del campo eléctrico, las cargas positivas se mueven en el sentido de dicha intensidad entonces, la intensidad de campo eléctrico se dirige siempre desde zonas de mayor potencial a zonas de menor potencial.

La intensidad de campo eléctrico apunta siempre hacia potenciales decrecientes.

# data[].attributes.attachment_link.title
Introducción al Concepto de Potencial Eléctrico

# data[].attributes.attachment_link.description
Si introducimos una carga q' en el seno de un campo eléctrico, la carga sufrirá la acción de una fuerza eléctrica y como consecuencia de esto, adquirirá cierta energía potencial eléctrica (también conocida como energía potencial electrostática).

# data[].attributes.attachment_link.readable_text
OpenStaxCollege

Learning Objectives

Define electric potential and electric potential energy.

Describe the relationship between potential difference and electrical potential energy.

Explain electron volt and its usage in submicroscopic process.

Determine electric potential energy given potential difference and amount of charge.

When a free positive charge is accelerated by an electric field, such as shown in [link], it is given kinetic energy. The process is analogous to an object being accelerated by a gravitational field. It is as if the charge is going down an electrical hill where its electric potential energy is converted to kinetic energy. Let us explore the work done on a charge by the electric field in this process, so that we may develop a definition of electric potential energy.

The electrostatic or Coulomb force is conservative, which means that the work done on is independent of the path taken. This is exactly analogous to the gravitational force in the absence of dissipative forces such as friction. When a force is conservative, it is possible to define a potential energy associated with the force, and it is usually easier to deal with the potential energy (because it depends only on position) than to calculate the work directly.

We use the letters PE to denote electric potential energy, which has units of joules (J). The change in potential energy, , is crucial, since the work done by a conservative force is the negative of the change in potential energy; that is, . For example, work done to accelerate a positive charge from rest is positive and results from a loss in PE, or a negative . There must be a minus sign in front of to make positive. PE can be found at any point by taking one point as a reference and calculating the work needed to move a charge to the other point.

Gravitational potential energy and electric potential energy are quite analogous. Potential energy accounts for work done by a conservative force and gives added insight regarding energy and energy transformation without the necessity of dealing with the force directly. It is much more common, for example, to use the concept of voltage (related to electric potential energy) than to deal with the Coulomb force directly.

Calculating the work directly is generally difficult, since and the direction and magnitude of can be complex for multiple charges, for odd-shaped objects, and along arbitrary paths. But we do know that, since , the work, and hence , is proportional to the test charge To have a physical quantity that is independent of test charge, we define electric potential (or simply potential, since electric is understood) to be the potential energy per unit charge:

This is the electric potential energy per unit charge.

Since PE is proportional to , the dependence on cancels. Thus does not depend on . The change in potential energy is crucial, and so we are concerned with the difference in potential or potential difference between two points, where

The potential difference between points A and B, , is thus defined to be the change in potential energy of a charge moved from A to B, divided by the charge. Units of potential difference are joules per coulomb, given the name volt (V) after Alessandro Volta.

The familiar term voltage is the common name for potential difference. Keep in mind that whenever a voltage is quoted, it is understood to be the potential difference between two points. For example, every battery has two terminals, and its voltage is the potential difference between them. More fundamentally, the point you choose to be zero volts is arbitrary. This is analogous to the fact that gravitational potential energy has an arbitrary zero, such as sea level or perhaps a lecture hall floor.

In summary, the relationship between potential difference (or voltage) and electrical potential energy is given by

Potential Difference and Electrical Potential Energy

The relationship between potential difference (or voltage) and electrical potential energy is given by

The second equation is equivalent to the first.

Voltage is not the same as energy. Voltage is the energy per unit charge. Thus a motorcycle battery and a car battery can both have the same voltage (more precisely, the same potential difference between battery terminals), yet one stores much more energy than the other since . The car battery can move more charge than the motorcycle battery, although both are 12 V batteries.

Suppose you have a 12.0 V motorcycle battery that can move 5000 C of charge, and a 12.0 V car battery that can move 60,000 C of charge. How much energy does each deliver? (Assume that the numerical value of each charge is accurate to three significant figures.)
Strategy
To say we have a 12.0 V battery means that its terminals have a 12.0 V potential difference. When such a battery moves charge, it puts the charge through a potential difference of 12.0 V, and the charge is given a change in potential energy equal to .

So to find the energy output, we multiply the charge moved by the potential difference.
Solution
For the motorcycle battery, and . The total energy delivered by the motorcycle battery is

Similarly, for the car battery, and

Discussion
While voltage and energy are related, they are not the same thing. The voltages of the batteries are identical, but the energy supplied by each is quite different. Note also that as a battery is discharged, some of its energy is used internally and its terminal voltage drops, such as when headlights dim because of a low car battery. The energy supplied by the battery is still calculated as in this example, but not all of the energy is available for external use.

Note that the energies calculated in the previous example are absolute values. The change in potential energy for the battery is negative, since it loses energy. These batteries, like many electrical systems, actually move negative charge-electrons in particular. The batteries repel electrons from their negative terminals (A) through whatever circuitry is involved and attract them to their positive terminals (B) as shown in [link]. The change in potential is and the charge is negative, so that is negative, meaning the potential energy of the battery has decreased when has moved from A to B.

The energy per electron is very small in macroscopic situations like that in the previous example-a tiny fraction of a joule. But on a submicroscopic scale, such energy per particle (electron, proton, or ion) can be of great importance. For example, even a tiny fraction of a joule can be great enough for these particles to destroy organic molecules and harm living tissue. The particle may do its damage by direct collision, or it may create harmful x rays, which can also inflict damage. It is useful to have an energy unit related to submicroscopic effects. [link] shows a situation related to the definition of such an energy unit. An electron is accelerated between two charged metal plates as it might be in an old-model television tube or oscilloscope. The electron is given kinetic energy that is later converted to another form-light in the television tube, for example. (Note that downhill for the electron is uphill for a positive charge.) Since energy is related to voltage by we can think of the joule as a coulomb-volt.

On the submicroscopic scale, it is more convenient to define an energy unit called the electron volt (eV), which is the energy given to a fundamental charge accelerated through a potential difference of 1 V. In equation form,

On the submicroscopic scale, it is more convenient to define an energy unit called the electron volt (eV), which is the energy given to a fundamental charge accelerated through a potential difference of 1 V. In equation form,

An electron accelerated through a potential difference of 1 V is given an energy of 1 eV. It follows that an electron accelerated through 50 V is given 50 eV. A potential difference of 100,000 V (100 kV) will give an electron an energy of 100,000 eV (100 keV), and so on. Similarly, an ion with a double positive charge accelerated through 100 V will be given 200 eV of energy. These simple relationships between accelerating voltage and particle charges make the electron volt a simple and convenient energy unit in such circumstances.

Connections: Energy Units

The electron volt (eV) is the most common energy unit for submicroscopic processes. This will be particularly noticeable in the chapters on modern physics. Energy is so important to so many subjects that there is a tendency to define a special energy unit for each major topic. There are, for example, calories for food energy, kilowatt-hours for electrical energy, and therms for natural gas energy.

The electron volt is commonly employed in submicroscopic processes-chemical valence energies and molecular and nuclear binding energies are among the quantities often expressed in electron volts. For example, about 5 eV of energy is required to break up certain organic molecules. If a proton is accelerated from rest through a potential difference of 30 kV, it is given an energy of 30 keV (30,000 eV) and it can break up as many as 6000 of these molecules ( ). Nuclear decay energies are on the order of 1 MeV (1,000,000 eV) per event and can, thus, produce significant biological damage.

The total energy of a system is conserved if there is no net addition (or subtraction) of work or heat transfer. For conservative forces, such as the electrostatic force, conservation of energy states that mechanical energy is a constant.

Mechanical energy is the sum of the kinetic energy and potential energy of a system; that is, . A loss of PE of a charged particle becomes an increase in its KE. Here PE is the electric potential energy. Conservation of energy is stated in equation form as

or

where i and f stand for initial and final conditions. As we have found many times before, considering energy can give us insights and facilitate problem solving.

Calculate the final speed of a free electron accelerated from rest through a potential difference of 100 V. (Assume that this numerical value is accurate to three significant figures.)
Strategy
We have a system with only conservative forces. Assuming the electron is accelerated in a vacuum, and neglecting the gravitational force (we will check on this assumption later), all of the electrical potential energy is converted into kinetic energy. We can identify the initial and final forms of energy to be
Solution
Conservation of energy states that

Entering the forms identified above, we obtain

We solve this for :

Entering values for gives

Discussion
Note that both the charge and the initial voltage are negative, as in [link]. From the discussions in Electric Charge and Electric Field, we know that electrostatic forces on small particles are generally very large compared with the gravitational force. The large final speed confirms that the gravitational force is indeed negligible here. The large speed also indicates how easy it is to accelerate electrons with small voltages because of their very small mass. Voltages much higher than the 100 V in this problem are typically used in electron guns. Those higher voltages produce electron speeds so great that relativistic effects must be taken into account. That is why a low voltage is considered (accurately) in this example.

Find the ratio of speeds of an electron and a negative hydrogen ion (one having an extra electron) accelerated through the same voltage, assuming non-relativistic final speeds. Take the mass of the hydrogen ion to be

A bare helium nucleus has two positive charges and a mass of (a) Calculate its kinetic energy in joules at 2.00% of the speed of light. (b) What is this in electron volts? (c) What voltage would be needed to obtain this energy?

Integrated Concepts

Integrated Concepts
The temperature near the center of the Sun is thought to be 15 million degrees Celsius . Through what voltage must a singly charged ion be accelerated to have the same energy as the average kinetic energy of ions at this temperature?

Integrated Concepts

(a)

Integrated Concepts
A lightning bolt strikes a tree, moving 20.0 C of charge through a potential difference of . (a) What energy was dissipated? (b) What mass of water could be raised from to the boiling point and then boiled by this energy? (c) Discuss the damage that could be caused to the tree by the expansion of the boiling steam.

(a)

(b)

Integrated Concepts
A battery-operated car utilizes a 12.0 V system. Find the charge the batteries must be able to move in order to accelerate the 750 kg car from rest to 25.0 m/s, make it climb a high hill, and then cause it to travel at a constant 25.0 m/s by exerting a force for an hour.

Integrated Concepts
Fusion probability is greatly enhanced when appropriate nuclei are brought close together, but mutual Coulomb repulsion must be overcome. This can be done using the kinetic energy of high-temperature gas ions or by accelerating the nuclei toward one another. (a) Calculate the potential energy of two singly charged nuclei separated by by finding the voltage of one at that distance and multiplying by the charge of the other. (b) At what temperature will atoms of a gas have an average kinetic energy equal to this needed electrical potential energy?

Unreasonable Results

(a)

Construct Your Own Problem

Glossary

electric potential

potential energy per unit charge

potential difference (or voltage)

change in potential energy of a charge moved from one point to another, divided by the charge; units of potential difference are joules per coulomb, known as volt

electron volt

the energy given to a fundamental charge accelerated through a potential difference of one volt

mechanical energy

sum of the kinetic energy and potential energy of a system; this sum is a constant

# data[].attributes.attachment_link.title
Electric Potential Energy: Potential Difference

# data[].attributes.attachment_link.description
OpenStaxCollege Define electric potential and electric potential energy. Describe the relationship between potential difference and electrical potential energy. Explain electron volt and its usage in submicroscopic process. Determine electric potential energy given potential difference and amount of charge. When a free positive charge is accelerated by an electric field, such as shown in [link], it is given kinetic energy.