4.2: Electric Potential Energy - Potential Difference (2023)

When a free positive charge \(q\) is accelerated by an electric field, as shown in Figure \(\PageIndex{1}\), it is given kinetic energy. The process is analogous to an object accelerated by a gravitational field. It is as if the charge descends an electric hill where its electrical potential energy is converted to kinetic energy. Let's explore the work done on a charge \(q\) by the electric field in this process, so that we can develop a definition of electric potential energy.

The electrostatic or Coulomb force is conservative, which means that the work done on \(q\) is independent of the path taken. This is exactly analogous to the gravitational force in the absence of dissipative forces like 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 electrical potential energy, which has units of joules (J). The change in potential energy, \(\Delta \mathrm{PE}\), is crucial, since the work done by a conservative force is the negative of the change in potential energy; that is, \(W=-\Delta \mathrm{PE}\). For example, the work \(W\) done to accelerate a positive charge from rest is positive and results from a loss of PE, or a negative \(\Delta \mathrm{PE}\). There must be a minus sign in front of \(\Delta \mathrm{PE}\) for \(W\) to be positive. PE can be found at any point by taking one point as a reference and calculating the work required to move a charge to the other point.

Gravitational potential energy and electrical potential energy are quite analogous. Potential energy represents the work done by a conservative force and provides additional information about energy and energy transformation without the need to deal directly with the force. It is much more common, for example, to use the concept of voltage (related to electrical potential energy) than to directly address the Coulomb force.

Calculating the work directly is generally difficult, since \(W=Fd\cos \theta\) and the direction and magnitude of \(F\) can be complex for multiple charges, for irregularly shaped objects, and along arbitrary paths. . But we do know that, since \(F=qE\), the work, and therefore \(\Delta \mathrm{PE}\), is proportional to the proof load \(q\) To have a physical quantity that is independent of the test load, we defineelectric potential\(V\) (or simply potential, since electrical is understood) is the potential energy per unit charge:

\[V=\dfrac{\mathrm{PE}}{q}.\]

Since PE is proportional to \(q\), the dependency on \(q\) cancels. Therefore \(V\) does not depend on \(q\). The change in potential energy \(\Delta \mathrm{PE}\) is crucial, so we are concerned with the potential difference or potential difference \(\Delta V\) between two points, where

\[\Delta V =V_{B}-V_{A}=\dfrac{\Delta \mathrm{PE}}{q}.\]

Hepotential differencebetween points A and B, \(V_{B}-V_{A}\), is defined as the change in potential energy of a charge \(q\) moved from A to B, divided by the charge. The units of potential difference are joules per coulomb, named volt (V) after Alessandro Volta.

\[1\mathrm{V}=1\mathrm{\dfrac{J}{C}}\]

the familiar termVoltageis the common name for the potential difference. It must be taken into account that whenever a voltage is quoted, it is understood that it is the potential difference between two points. For example, each battery has two terminals, and their voltage is the potential difference between them. More fundamentally, the point you choose as zero volts is arbitrary. This is analogous to the fact that gravitational potential energy has an arbitrary zero, like sea level or perhaps the floor of a conference room.

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

\[\Delta V=\dfrac{\Delta \mathrm{PE}}{q}\: \mathrm{and}\: \Delta \mathrm{PE}=q\Delta V.\]

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

Note that the energies calculated in the example above are absolute values. The potential energy change of the battery is negative, since it loses energy. These batteries, like many electrical systems, actually move negative charge, particularly electrons. Batteries repel electrons from their negative (A) terminals through whatever circuitry is involved and attract them to their positive (B) terminals, as shown in Figure \(\PageIndex{2}\). The change in potential is \(\Delta V =V_{B}-V_{A}=+12\mathrm{V}\) and the charge \(q\) is negative, so \(\Delta \mathrm { PE}=q\Delta V\) is negative, which means that the potential energy of the battery has decreased as \(q\) has moved from A to B.

The electron volt

The energy per electron is very small in macroscopic situations like the one in the previous example: a small fraction of a joule. But on a submicroscopic scale, that energy per particle (electron, proton, or ion) can be of great importance. For example, even a small fraction of a joule can be enough for these particles to destroy organic molecules and damage living tissue. The particle can cause direct collision damage, or it can create damaging X-rays, which can also cause damage. It is useful to have a unit of energy related to submicroscopic effects. The figure \(\PageIndex{3}\) shows a situation related to the definition of said unit of energy. An electron is accelerated between two charged metal plates as it would in a television tube or an old model oscilloscope. The electron is given kinetic energy which is then converted into 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 \(\Delta PE=q\Delta V\), we can think of the joule as a coulomb-volt.

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

\[1\mathrm{ev}=(1.60\times 10^{-19}\mathrm{C})(1\mathrm{V})=(1.60\times 10^{-19}\mathrm{C}) (1\mathrm{J/C})\]

\[=1.60\times 10^{-19}J.\]

ELECTRON VOLT

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

\[1 \mathrm{eV}=(1,60\times 10^{-19} \mathrm{C})(1 \mathrm{V})=(1,60\times 10^{-19} \mathrm{C}) (1\mathrm{J/C})\]

\[=1.60\times 10^{-19} \mathrm{C}\]

An electron accelerated through a potential difference of 1 V receives 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 receive 200 eV of energy. These simple relationships between the accelerating voltage and the charges of the particles make the electron volt a simple and convenient unit of energy in such circumstances.

CONNECTIONS: POWER UNITS

The electron volt (eV) is the most common unit of energy for submicroscopic processes. This will be particularly noticeable in the chapters on modern physics. Energy is so important to so many issues that there is a tendency to define a special unit of energy for each major issue. 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 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 can break up to 6,000 of these molecules ( \(30,000 \mathrm{eV} \div 5\mathrm{eV}\) per molecule \(=6000\) molecules). Nuclear decay energies are on the order of 1 MeV (1,000,000 eV) per event and therefore can cause significant biological damage.