What Is The Change In The Electron's Kinetic Energy

Learning Objectives

By the end of this section, you lot will be able to:

• Ascertain electrical potential and electric potential energy.
• Describe the human relationship between potential divergence and electric potential energy.
• Explain electron volt and its usage in submicroscopic process.
• Determine electric potential energy given potential difference and corporeality of accuse.

Figure one. A charge accelerated past an electrical field is analogous to a mass going down a colina. In both cases potential energy is converted to another form. Work is done past a forcefulness, but since this force is conservative, we can write Due west= –ΔPE.

When a free positive charge q is accelerated past an electric field, such as shown in Figure 1, it is given kinetic energy. The process is analogous to an object being accelerated past a gravitational field. Information technology is as if the charge is going down an electric hill where its electric potential energy is converted to kinetic energy. Permit us explore the work done on a accuse q by the electric field in this procedure, so that we may develop a definition of electric potential energy.

The electrostatic or Coulomb force is conservative, which means that the work washed on q is independent of the path taken. This is exactly analogous to the gravitational forcefulness in the absence of dissipative forces such every bit friction. When a force is bourgeois, it is possible to ascertain a potential free energy associated with the force, and it is usually easier to deal with the potential energy (because it depends merely on position) than to summate the work directly.

We utilize the messages PE to denote electric potential energy, which has units of joules (J). The alter in potential energy, ΔPE, is crucial, since the work washed by a conservative force is the negative of the change in potential energy; that is, Westward = –ΔPE. For example, work West done to accelerate a positive charge from rest is positive and results from a loss in PE, or a negative ΔPE. There must be a minus sign in front of ΔPE to brand W positive. PE can be found at whatever point by taking one point as a reference and calculating the piece of work needed to move a accuse to the other signal.

Potential Energy

W = –ΔPE. For example, work W washed to accelerate a positive charge from rest is positive and results from a loss in PE, or a negative ΔPE. In that location must be a minus sign in front end of ΔPE to brand W positive. PE can be found at any point by taking i point as a reference and calculating the work needed to motility a charge to the other betoken.

Gravitational potential energy and electrical potential energy are quite analogous. Potential free energy accounts for work done past a conservative force and gives added insight regarding energy and energy transformation without the necessity of dealing with the forcefulness straight. It is much more mutual, for example, to utilise the concept of voltage (related to electric potential energy) than to bargain with the Coulomb force straight.

Calculating the work directly is generally difficult, since W =Fd cosθ and the management and magnitude of F tin be complex for multiple charges, for odd-shaped objects, and along arbitrary paths. But nosotros exercise know that, since F =qE, the work, and hence ΔPE, is proportional to the test charge q. To have a physical quantity that is independent of test accuse, nosotros define electric potential V (or simply potential, since electric is understood) to be the potential energy per unit accuse [latex]V=\frac{\text{PE}}{q}\\[/latex].

Electric Potential

This is the electrical potential energy per unit charge.

[latex]\displaystyle{V}=\frac{\text{PE}}{q}\\[/latex]

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

[latex]\displaystyle\Delta{Five}=V_{\text{B}}-V_{\text{A}}=\frac{\Delta{\text{PE}}}{q}\\[/latex]

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

[latex]i\text{V}=1\frac{\text{J}}{\text{C}}\\[/latex]

Potential Difference

The potential difference between points A and B, V _B –V _A, is defined to be the change in potential free energy of a accuse q moved from A to B, divided past the charge. Units of potential difference are joules per coulomb, given the proper name volt (V) after Alessandro Volta.

[latex]\displaystyle{one}\text{V}=1\frac{\text{J}}{\text{C}}\\[/latex]

The familiar term voltage is the mutual name for potential difference. Keep in mind that whenever a voltage is quoted, it is understood to be the potential difference between 2 points. For example, every battery has 2 terminals, and its voltage is the potential deviation between them. More fundamentally, the point you cull to exist 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 divergence (or voltage) and electrical potential energy is given by [latex]\Delta{Five}=\frac{\Delta\text{PE}}{q}\\[/latex] and ΔPE =qΔ5.

Potential Difference and Electric Potential Energy

The relationship between potential deviation (or voltage) and electrical potential energy is given past

[latex]\Delta{V}=\frac{\Delta\text{PE}}{q}\\[/latex] and ΔPE =qΔV

The 2nd equation is equivalent to the commencement.

Voltage is non the same equally energy. Voltage is the energy per unit charge. Thus a motorcycle bombardment and a car battery can both take the same voltage (more than precisely, the same potential divergence between battery terminals), even so one stores much more energy than the other since ΔPE =qΔFive. The machine battery tin can move more charge than the motorcycle battery, although both are 12 5 batteries.

Example ane. Calculating Energy

Suppose y'all have a 12.0 V motorcycle battery that tin motility 5000 C of accuse, and a 12.0 5 motorcar bombardment that tin can move 60,000 C of charge. How much energy does each deliver? (Presume that the numerical value of each charge is accurate to three significant figures.)

Strategy

To say nosotros take a 12.0 V battery means that its terminals have a 12.0 Five potential difference. When such a battery moves charge, it puts the accuse through a potential difference of 12.0 5, and the charge is given a modify in potential free energy equal to ΔPE =qΔV.

And so to observe the energy output, we multiply the accuse moved past the potential deviation.

Solution

For the motorcycle battery, q = 5000 C and ΔV= 12.0 Five. The total free energy delivered past the motorcycle battery is

[latex]\brainstorm{assortment}{lll}\Delta\text{PE}_{\text{wheel}}&=&\left(5000\text{ C}\right)\left(12.0\text{ 5}\correct)\\\text{ }&=&\left(5000\text{ C}\right)\left(12.0\text{ J/C}\right)\\\text{ }&=&half-dozen.00\times10^4\text{ J}\end{assortment}\\[/latex]

Similarly, for the machine battery, q = threescore,000 C and

[latex]\begin{array}{lll}\Delta\text{PE}_{\text{car}}&=&\left(60,000\text{ C}\right)\left(12.0\text{ V}\right)\\\text{ }&=&7.20\times10^v\text{ J}\end{array}\\[/latex]

Discussion

While voltage and energy are related, they are non the same thing. The voltages of the batteries are identical, but the energy supplied by each is quite different. Annotation too that every bit a battery is discharged, some of its free energy is used internally and its terminal voltage drops, such as when headlights dim because of a low automobile battery. The energy supplied by the battery is still calculated every bit in this example, but non all of the energy is available for external utilise.

Note that the energies calculated in the previous example are accented values. The alter in potential energy for the battery is negative, since it loses free energy. These batteries, like many electrical systems, really move negative accuse—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 Figure ii. The change in potential is ΔFive=V _B– V _A = +12 V and the charge q is negative, so that ΔPE =qΔV is negative, meaning the potential energy of the battery has decreased when q has moved from A to B.

Effigy 2. A bombardment moves negative charge from its negative terminal through a headlight to its positive terminal. Appropriate combinations of chemicals in the battery separate charges so that the negative final has an excess of negative charge, which is repelled by it and attracted to the excess positive charge on the other final. In terms of potential, the positive terminal is at a college voltage than the negative. Within the battery, both positive and negative charges move.

Instance 2. How Many Electrons Move through a Headlight Each Second?

When a 12.0 Five machine battery runs a unmarried 30.0 W headlight, how many electrons pass through it each 2d?

Strategy

To find the number of electrons, we must outset find the charge that moved in one.00 s. The charge moved is related to voltage and energy through the equation ΔPE = qΔ5. A 30.0 W lamp uses 30.0 joules per second. Since the battery loses free energy, nosotros accept ΔPE = –30.0 J and, since the electrons are going from the negative terminal to the positive, we come across that ΔV= +12.0V.

Solution

To find the charge q moved, we solve the equation ΔPE = qΔFive: [latex]q=\frac{\Delta\text{PE}}{\Delta{V}}\\[/latex].

Inbound the values for ΔPE and ΔV, we get

[latex]q=\frac{-30.0\text{ J}}{+12.0\text{ Five}}=\frac{-30.0\text{ J}}{+12.0\text{ J/C}}-2.50\text{ C}\\[/latex]

The number of electrons n_eastward is the total charge divided by the accuse per electron. That is,

[latex]\text{n}_{\text{e}}=\frac{-2.fifty\text{ C}}{-ane.60\times10^{-xix}\text{ C/east}^{-}}=1.56\times10^{xix}\text{ electrons}\\[/latex]

Word

This is a very large number. It is no wonder that we practise non unremarkably observe individual electrons with then many being present in ordinary systems. In fact, electricity had been in employ for many decades before information technology was adamant that the moving charges in many circumstances were negative. Positive charge moving in the reverse direction of negative accuse often produces identical effects; this makes it hard to decide which is moving or whether both are moving.

The Electron Volt

Figure iii. A typical electron gun accelerates electrons using a potential divergence between ii metal plates. The energy of the electron in electron volts is numerically the same as the voltage between the plates. For instance, a 5000 V potential divergence produces 5000 eV electrons.

The energy per electron is very small in macroscopic situations like that in the previous instance—a tiny fraction of a joule. But on a submicroscopic scale, such free energy per particle (electron, proton, or ion) can be of neat importance. For example, fifty-fifty a tiny fraction of a joule can be peachy plenty for these particles to destroy organic molecules and harm living tissue. The particle may practice its damage by direct standoff, or information technology may create harmful 10 rays, which tin besides inflict impairment. It is useful to have an free energy unit related to submicroscopic effects. Figure 3 shows a situation related to the definition of such an free energy unit of measurement. An electron is accelerated between 2 charged metal plates as information technology might be in an former-model television tube or oscilloscope. The electron is given kinetic free energy that is later converted to some other form—light in the television tube, for instance. (Note that downhill for the electron is uphill for a positive charge.) Since free energy is related to voltage by ΔPE =qΔV, we can call back of the joule as a coulomb-volt.

On the submicroscopic scale, it is more than convenient to define an energy unit called the electron volt (eV), which is the energy given to a primal charge accelerated through a potential difference of 1 5. In equation form,

[latex]\brainstorm{array}{lll}1\text{eV}&=&\left(1.60\times10^{-19}\text{ C}\correct)\left(1\text{ V}\correct)=\left(1.60\times10^{-xix}\text{ C}\right)\left(1\text{ J/C}\right)\\\text{ }&=&1.threescore\times10^{-xix}\text{ J}\end{array}\\[/latex]

Electron Volt

On the submicroscopic scale, it is more than 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 5. In equation grade,

[latex]\begin{array}{lll}1\text{eV}&=&\left(1.60\times10^{-nineteen}\text{ C}\right)\left(1\text{ V}\right)=\left(one.60\times10^{-19}\text{ C}\right)\left(one\text{ J/C}\correct)\\\text{ }&=&1.lx\times10^{-19}\text{ J}\end{assortment}\\[/latex]

An electron accelerated through a potential difference of 1 Five is given an energy of 1 eV. It follows that an electron accelerated through fifty V is given 50 eV. A potential difference of 100,000 V (100 kV) will give an electron an free 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 unproblematic relationships between accelerating voltage and particle charges brand the electron volt a uncomplicated and user-friendly energy unit in such circumstances.

Making Connections: Energy Units

The electron volt (eV) is the nigh common energy unit for submicroscopic processes. This will exist especially noticeable in the chapters on mod physics. Energy is so important to so many subjects that in that location is a trend to define a special energy unit for each major topic. In that location are, for example, calories for food energy, kilowatt-hours for electrical energy, and therms for natural gas energy.

The electron volt is usually employed in submicroscopic processes—chemical valence energies and molecular and nuclear bounden energies are amidst the quantities often expressed in electron volts. For example, about v eV of energy is required to pause upwardly certain organic molecules. If a proton is accelerated from remainder through a potential departure of 30 kV, information technology is given an energy of 30 keV (30,000 eV) and it can intermission up as many every bit 6000 of these molecules (30,000 eV ÷ five eV per molecule= 6000 molecules). Nuclear decay energies are on the social club of i MeV (1,000,000 eV) per issue and can, thus, produce significant biological impairment.

Conservation of Energy

The full energy of a system is conserved if there is no net addition (or subtraction) of piece of work or heat transfer. For bourgeois forces, such as the electrostatic strength, conservation of free 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, KE+PE = abiding. A loss of PE of a charged particle becomes an increase in its KE. Here PE is the electrical potential energy. Conservation of free energy is stated in equation form every bit KE + PE = constant or KE_i + PE _i = KE_f + PE_f, where i and f stand up for initial and final conditions. As nosotros have institute many times before, considering energy tin requite us insights and facilitate trouble solving.

Example 3. Electrical Potential Energy Converted to Kinetic Energy

Calculate the final speed of a free electron accelerated from rest through a potential difference of 100 V. (Presume that this numerical value is authentic to three meaning figures.)

Strategy

We have a organisation with only conservative forces. Bold the electron is accelerated in a vacuum, and neglecting the gravitational force (nosotros volition check on this supposition later), all of the electrical potential free energy is converted into kinetic free energy. We can identify the initial and final forms of free energy to be KE_i = 0, [latex]KE_{f}=\frac{one}{2}mv^2\\[/latex], PE_i =qV, and PE_f = 0.

Solution

Conservation of energy states that KE_i + PE _i = KE _f + PE _f .

Inbound the forms identified in a higher place, nosotros obtain [latex]qV=\frac{mv^two}{two}\\[/latex].

We solve this for v :

[latex]\displaystyle{v}=\sqrt{\frac{2qV}{m}}\\[/latex]

Inbound values for q,V, andm gives

[latex]\brainstorm{array}{lll}{v}&=&\sqrt{\frac{2\left(-i.sixty\times10^{-19}\text{ C}\right)\left(-100\text{ J/C}\right)}{9.11\times10^{-31}\text{kg}}}\\\text{ }&=&5.93\times10^6\text{ m/due south}\terminate{array}\\[/latex]

Discussion

Note that both the charge and the initial voltage are negative, equally in Figure 3. From the discussions in Electric Accuse and Electric Field, we know that electrostatic forces on small particles are generally very large compared with the gravitational force. The big concluding 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 pocket-size 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 case.

Department Summary

• Electric potential is potential energy per unit charge.
• The potential divergence between points A and B, 5 _B − V _A, defined to exist the change in potential energy of a accuse q moved from A to B, is equal to the change in potential energy divided past the charge, Potential divergence is commonly chosen voltage, represented by the symbol Δ5:[latex]\Delta V=\frac{\Delta\text{PE}}{q}\\[/latex] and ΔPE = qΔV.
• An electron volt is the energy given to a fundamental charge accelerated through a potential difference of ane V. In equation form,
  [latex]\begin{array}{lll}\text{i eV}& =& \left(ane.60\times {\text{10}}^{\text{-19}}\text{C}\right)\left(1 V\right)=\left(1.60\times {\text{10}}^{\text{-19}}\text{C}\right)\left(1 J/C\right)\\ & =& 1.threescore\times {\text{10}}^{\text{-19}}\text{J.}\terminate{array}\\[/latex]
• Mechanical free energy is the sum of the kinetic energy and potential energy of a system, that is, KE + PE. This sum is a abiding.

Conceptual Questions

1. Voltage is the common word for potential deviation. Which term is more than descriptive, voltage or potential difference?
2. If the voltage between two points is zero, can a test charge be moved betwixt them with zero internet work being washed? Tin can this necessarily exist done without exerting a force? Explain.
3. What is the relationship betwixt voltage and free energy? More precisely, what is the relationship between potential departure and electric potential energy?
4. Voltages are ever measured betwixt 2 points. Why?
5. How are units of volts and electron volts related? How exercise they differ?

Problems & Exercises

1. Notice the ratio of speeds of an electron and a negative hydrogen ion (1 having an actress electron) accelerated through the same voltage, assuming not-relativistic final speeds. Take the mass of the hydrogen ion to be 1.67 × x⁻²⁷ kg.
2. An evacuated tube uses an accelerating voltage of 40 kV to accelerate electrons to hit a copper plate and produce ten rays. Non-relativistically, what would exist the maximum speed of these electrons?
3. A blank helium nucleus has two positive charges and a mass of 6.64 × 10⁻²⁷ kg. (a) Calculate its kinetic energy in joules at ii.00% of the speed of light. (b) What is this in electron volts? (c) What voltage would exist needed to obtain this energy?
4. Integrated Concepts. Singly charged gas ions are accelerated from rest through a voltage of thirteen.0 Five. At what temperature will the boilerplate kinetic free energy of gas molecules be the same as that given these ions?
5. Integrated Concepts.The temperature nigh the heart of the Sunday is thought to exist 15 meg degrees Celsius (i.5 × 10⁷ ºC). Through what voltage must a singly charged ion be accelerated to have the aforementioned energy as the boilerplate kinetic energy of ions at this temperature?
6. Integrated Concepts. (a) What is the average power output of a center defibrillator that dissipates 400 J of energy in 10.0 ms? (b) Considering the high-power output, why doesn't the defibrillator produce serious burns?
7. Integrated Concepts. A lightning bolt strikes a tree, moving 20.0 C of charge through a potential difference of i.00 × 10² MV. (a) What free energy was dissipated? (b) What mass of water could be raised from 15ºC to the boiling indicate and and so boiled by this energy? (c) Discuss the damage that could be caused to the tree by the expansion of the boiling steam.
8. Integrated Concepts. A 12.0 V bombardment-operated bottle warmer heats 50.0 g of drinking glass, 2.50 × ten² g of baby formula, and ii.00 × 10² g of aluminum from 20.0ºC to 90.0ºC. (a) How much charge is moved by the bombardment? (b) How many electrons per 2d flow if it takes 5.00 min to warm the formula? (Hint: Assume that the specific rut of baby formula is almost the same as the specific heat of h2o.)
9. Integrated Concepts. A bombardment-operated car utilizes a 12.0 V system. Detect the charge the batteries must exist able to motion in order to accelerate the 750 kg car from remainder to 25.0 1000/south, make it climb a 2.00 × 10² m high hill, and then cause it to travel at a constant 25.0 m/s by exerting a five.00 × 10² N forcefulness for an 60 minutes.
10. Integrated Concepts. Fusion probability is greatly enhanced when appropriate nuclei are brought shut together, only mutual Coulomb repulsion must be overcome. This tin 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 1.00 × x⁻¹² m by finding the voltage of one at that distance and multiplying by the charge of the other. (b) At what temperature volition atoms of a gas have an average kinetic energy equal to this needed electrical potential energy?
11. Unreasonable Results. (a) Observe the voltage near a x.0 cm bore metal sphere that has 8.00 C of excess positive charge on it. (b) What is unreasonable most this result? (c) Which assumptions are responsible?
12. Construct Your Own Problem. Consider a battery used to supply energy to a cellular telephone. Construct a problem in which you determine the energy that must be supplied past the battery, and then calculate the amount of charge it must exist able to movement in gild to supply this energy. Among the things to be considered are the energy needs and battery voltage. Y'all may need to expect alee to interpret manufacturer's battery ratings in ampere-hours as free energy in joules.

Glossary

electrical potential: potential energy per unit charge

potential difference (or voltage): change in potential energy of a charge moved from ane signal to some other, divided by the accuse; units of potential difference are joules per coulomb, known as volt

electron volt: the energy given to a fundamental accuse accelerated through a potential difference of one volt

mechanical free energy: sum of the kinetic energy and potential energy of a organisation; this sum is a constant

Selected Solutions to Issues & Exercises

1. 42.8

4. 1.00 × ten⁵ K

half-dozen. (a) 4 × 10^four W; (b) A defibrillator does not cause serious burns considering the pare conducts electricity well at high voltages, like those used in defibrillators. The gel used aids in the transfer of free energy to the body, and the skin doesn't blot the energy, only rather lets it pass through to the middle.

eight. (a) 7.40 × 10^three C; (b) 1.54 × 10²⁰ electrons per 2d

9. three.89 × 10^half-dozen C

11. (a) 1.44 × 10¹² Five; (b) This voltage is very high. A 10.0 cm diameter sphere could never maintain this voltage; it would discharge; (c) An 8.00 C charge is more charge than tin can reasonably exist accumulated on a sphere of that size.

Source: https://courses.lumenlearning.com/physics/chapter/19-1-electric-potential-energy-potential-difference/

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