what voltage would be needed to obtain this energy?

Learning Objectives

By the terminate of this department, you will be able to:

• Define electrical potential and electric potential energy.
• Describe the relationship between potential difference and electrical potential energy.
• Explain electron volt and its usage in submicroscopic process.
• Make up one's mind electric potential energy given potential departure and corporeality of charge.

Effigy 1. A accuse accelerated by an electric field is analogous to a mass going downwards a hill. In both cases potential energy is converted to another course. Work is done by a force, but since this force is conservative, we can write W= –ΔPE.

When a gratuitous positive accuse q is accelerated by an electric field, such as shown in Figure 1, it is given kinetic energy. The process is analogous to an object beingness accelerated past a gravitational field. It is equally if the accuse is going down an electrical hill where its electric potential energy is converted to kinetic free energy. Permit us explore the work done on a accuse q by the electrical field in this process, so that we may develop a definition of electric potential free 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 such as friction. When a force is conservative, it is possible to define a potential energy associated with the force, and information technology is usually easier to deal with the potential free energy (because information technology depends only on position) than to calculate the piece of work direct.

We use the letters PE to announce electric potential free energy, which has units of joules (J). The modify in potential energy, ΔPE, is crucial, since the piece of work washed by a conservative force is the negative of the change in potential free energy; that is, Due west = –ΔPE. For instance, work Westward done to accelerate a positive accuse from residuum is positive and results from a loss in PE, or a negative ΔPE. There must be a minus sign in front end of ΔPE to make Westward positive. PE can be found at any indicate by taking i bespeak every bit a reference and calculating the work needed to move a accuse to the other betoken.

Potential Energy

Westward = –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. There must be a minus sign in forepart of ΔPE to make W positive. PE tin be found at any point by taking ane signal as a reference and calculating the work needed to movement a accuse to the other point.

Gravitational potential free energy and electric 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 force directly. It is much more common, for instance, to use the concept of voltage (related to electrical potential energy) than to deal with the Coulomb force direct.

Calculating the work directly is generally difficult, since Due west =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 contained of test accuse, we define electrical potential 5 (or simply potential, since electric is understood) to be the potential free energy per unit charge [latex]5=\frac{\text{PE}}{q}\\[/latex].

Electric Potential

This is the electric 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 modify in potential energy ΔPE is crucial, and so nosotros are concerned with the divergence in potential or potential difference Δ5 between ii points, where

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

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

[latex]1\text{5}=i\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 alter in potential energy of a charge q moved from A to B, divided by the charge. Units of potential difference are joules per coulomb, given the name volt (5) after Alessandro Volta.

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

The familiar term voltage is the common name for potential difference. Keep in mind that whenever a voltage is quoted, it is understood to exist the potential divergence between two points. For instance, every battery has two terminals, and its voltage is the potential deviation betwixt them. More fundamentally, the point you choose to be nothing volts is arbitrary. This is analogous to the fact that gravitational potential energy has an arbitrary nothing, such as sea level or perhaps a lecture hall floor.

In summary, the relationship betwixt potential difference (or voltage) and electrical potential energy is given past [latex]\Delta{5}=\frac{\Delta\text{PE}}{q}\\[/latex] and ΔPE =qΔ5.

Potential Divergence and Electrical Potential Free energy

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

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

The second equation is equivalent to the offset.

Voltage is not the same as energy. Voltage is the energy per unit accuse. Thus a motorcycle bombardment and a motorcar battery can both have the same voltage (more precisely, the same potential departure between battery terminals), yet one stores much more free energy than the other since ΔPE =qΔV. The auto battery tin can move more than charge than the motorcycle battery, although both are 12 V batteries.

Instance i. Calculating Energy

Suppose y'all take a 12.0 Five motorcycle battery that can move 5000 C of charge, and a 12.0 V motorcar battery that can move 60,000 C of charge. How much energy does each evangelize? (Assume that the numerical value of each charge is accurate to three significant figures.)

Strategy

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

So to find the energy output, we multiply the charge moved past the potential deviation.

Solution

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

[latex]\begin{assortment}{lll}\Delta\text{PE}_{\text{cycle}}&=&\left(5000\text{ C}\right)\left(12.0\text{ V}\right)\\\text{ }&=&\left(5000\text{ C}\right)\left(12.0\text{ J/C}\correct)\\\text{ }&=&6.00\times10^iv\text{ J}\stop{array}\\[/latex]

Similarly, for the car battery, q = 60,000 C and

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

Give-and-take

While voltage and energy are related, they are non the aforementioned matter. The voltages of the batteries are identical, but the free 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 equally when headlights dim because of a low car battery. The energy supplied past the battery is still calculated every bit in this example, but non all of the energy is bachelor for external use.

Notation that the energies calculated in the previous instance are accented values. The modify in potential energy for the bombardment 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 concenter them to their positive terminals (B) as shown in Figure 2. The change in potential is ΔV=V _B– V _A = +12 V and the accuse q is negative, so that ΔPE =qΔ5 is negative, meaning the potential energy of the battery has decreased when q has moved from A to B.

Figure ii. A battery moves negative accuse from its negative terminal through a headlight to its positive terminal. Advisable combinations of chemicals in the battery separate charges so that the negative terminal has an excess of negative charge, which is repelled by it and attracted to the excess positive charge on the other terminal. In terms of potential, the positive final is at a college voltage than the negative. Inside the battery, both positive and negative charges movement.

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

When a 12.0 V machine battery runs a unmarried 30.0 W headlight, how many electrons pass through it each second?

Strategy

To find the number of electrons, we must first find the charge that moved in i.00 s. The charge moved is related to voltage and energy through the equation ΔPE = qΔFive. A 30.0 W lamp uses xxx.0 joules per 2d. Since the battery loses free energy, nosotros have ΔPE = –xxx.0 J and, since the electrons are going from the negative final to the positive, we see that ΔV= +12.0V.

Solution

To find the charge q moved, we solve the equation ΔPE = qΔ5: [latex]q=\frac{\Delta\text{PE}}{\Delta{V}}\\[/latex].

Entering the values for ΔPE and ΔFive, we become

[latex]q=\frac{-30.0\text{ J}}{+12.0\text{ V}}=\frac{-xxx.0\text{ J}}{+12.0\text{ J/C}}-2.fifty\text{ C}\\[/latex]

The number of electrons n_east is the total charge divided past the accuse per electron. That is,

[latex]\text{n}_{\text{e}}=\frac{-2.50\text{ C}}{-one.lx\times10^{-xix}\text{ C/e}^{-}}=1.56\times10^{19}\text{ electrons}\\[/latex]

Word

This is a very large number. It is no wonder that we exercise not ordinarily observe private electrons with and then many existence nowadays in ordinary systems. In fact, electricity had been in use for many decades before it was adamant that the moving charges in many circumstances were negative. Positive accuse moving in the opposite direction of negative accuse often produces identical furnishings; this makes it difficult to determine which is moving or whether both are moving.

The Electron Volt

Figure 3. A typical electron gun accelerates electrons using a potential divergence between two metallic plates. The energy of the electron in electron volts is numerically the same as the voltage between the plates. For case, a 5000 V potential divergence produces 5000 eV electrons.

The free energy per electron is very small in macroscopic situations like that in the previous example—a tiny fraction of a joule. Merely on a submicroscopic calibration, such energy per particle (electron, proton, or ion) can be of great importance. For example, even a tiny fraction of a joule can be nifty enough for these particles to destroy organic molecules and impairment living tissue. The particle may do its damage past direct collision, or it may create harmful x rays, which tin also inflict damage. It is useful to take an energy unit related to submicroscopic effects. Effigy 3 shows a situation related to the definition of such an free energy unit. An electron is accelerated between two charged metal plates equally 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 set tube, for example. (Notation that downhill for the electron is uphill for a positive accuse.) Since free energy is related to voltage past ΔPE =qΔFive, we tin think of the joule equally a coulomb-volt.

On the submicroscopic scale, it is more than convenient to ascertain an energy unit called the electron volt (eV), which is the energy given to a key accuse accelerated through a potential divergence of 1 V. In equation form,

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

Electron Volt

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

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

An electron accelerated through a potential difference of 1 5 is given an energy of one 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 accuse accelerated through 100 5 volition be given 200 eV of energy. These simple relationships between accelerating voltage and particle charges brand the electron volt a unproblematic and convenient free energy unit of measurement in such circumstances.

Making Connections: Free energy Units

The electron volt (eV) is the nearly common energy unit of measurement for submicroscopic processes. This will exist particularly noticeable in the capacity on modern physics. Energy is and then important to and so many subjects that there is a trend to define a special free energy unit for each major topic. There are, for case, calories for nutrient energy, kilowatt-hours for electrical free energy, and therms for natural gas energy.

The electron volt is unremarkably employed in submicroscopic processes—chemical valence energies and molecular and nuclear binding energies are amongst the quantities often expressed in electron volts. For case, well-nigh 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 free energy of 30 keV (30,000 eV) and information technology can break up as many every bit 6000 of these molecules (30,000 eV ÷ 5 eV per molecule= 6000 molecules). Nuclear disuse energies are on the society of one MeV (1,000,000 eV) per upshot and can, thus, produce pregnant biological damage.

Conservation of Free energy

The total free energy of a arrangement is conserved if there is no net addition (or subtraction) of work or estrus transfer. For conservative forces, such every bit the electrostatic forcefulness, conservation of energy states that mechanical energy is a constant.

Mechanical energy is the sum of the kinetic energy and potential free energy of a system; that is, KE+PE = constant. A loss of PE of a charged particle becomes an increase in its KE. Hither PE is the electric potential free energy. Conservation of free energy is stated in equation course as KE + PE = constant or KE_i + PE _i = KE_f + PE_f, where i and f stand for initial and final atmospheric condition. As nosotros have institute many times before, considering energy can give us insights and facilitate problem solving.

Example 3. Electrical Potential Energy Converted to Kinetic Energy

Calculate the concluding speed of a free electron accelerated from rest through a potential deviation of 100 V. (Assume that this numerical value is accurate to three significant figures.)

Strategy

We accept a system with merely conservative forces. Assuming the electron is accelerated in a vacuum, and neglecting the gravitational force (nosotros volition check on this assumption later), all of the electric potential free energy is converted into kinetic energy. Nosotros can identify the initial and final forms of free energy to be KE_i = 0, [latex]KE_{f}=\frac{1}{ii}mv^two\\[/latex], PE_i =qV, and PE_f = 0.

Solution

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

Entering the forms identified above, we obtain [latex]qV=\frac{mv^2}{2}\\[/latex].

We solve this for v :

[latex]\displaystyle{5}=\sqrt{\frac{2qV}{m}}\\[/latex]

Entering values for q,Five, andgrand gives

[latex]\begin{array}{lll}{v}&=&\sqrt{\frac{2\left(-1.threescore\times10^{-19}\text{ C}\right)\left(-100\text{ J/C}\right)}{ix.11\times10^{-31}\text{kg}}}\\\text{ }&=&5.93\times10^half-dozen\text{ m/s}\stop{array}\\[/latex]

Discussion

Annotation that both the charge and the initial voltage are negative, as in Figure three. From the discussions in Electric Accuse and Electric Field, we know that electrostatic forces on pocket-size particles are generally very large compared with the gravitational force. The large concluding speed confirms that the gravitational forcefulness is indeed negligible here. The large speed also indicates how piece of cake it is to advance electrons with small voltages considering of their very small mass. Voltages much college 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 exist taken into business relationship. That is why a low voltage is considered (accurately) in this example.

Section Summary

• Electrical potential is potential energy per unit of measurement charge.
• The potential difference between points A and B, Five _B − V _A, defined to be the change in potential energy of a charge q moved from A to B, is equal to the change in potential energy divided by the charge, Potential difference is commonly called voltage, represented past the symbol ΔV:[latex]\Delta 5=\frac{\Delta\text{PE}}{q}\\[/latex] and ΔPE = qΔV.
• An electron volt is the energy given to a cardinal accuse accelerated through a potential difference of i V. In equation form,
  [latex]\begin{array}{lll}\text{1 eV}& =& \left(1.60\times {\text{10}}^{\text{-19}}\text{C}\right)\left(1 Five\right)=\left(one.60\times {\text{ten}}^{\text{-nineteen}}\text{C}\correct)\left(1 J/C\right)\\ & =& 1.60\times {\text{ten}}^{\text{-xix}}\text{J.}\end{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 constant.

Conceptual Questions

1. Voltage is the common discussion for potential divergence. Which term is more descriptive, voltage or potential difference?
2. If the voltage between ii points is aught, can a examination charge be moved between them with zilch net work being done? Can this necessarily exist washed without exerting a forcefulness? Explain.
3. What is the human relationship between voltage and energy? More precisely, what is the human relationship between potential divergence and electric potential energy?
4. Voltages are e'er measured between ii points. Why?
5. How are units of volts and electron volts related? How do they differ?

Problems & Exercises

1. Notice the ratio of speeds of an electron and a negative hydrogen ion (one having an actress electron) accelerated through the aforementioned voltage, assuming not-relativistic concluding speeds. Accept the mass of the hydrogen ion to be 1.67 × 10⁻²⁷ kg.
2. An evacuated tube uses an accelerating voltage of forty kV to accelerate electrons to hit a copper plate and produce 10 rays. Non-relativistically, what would be the maximum speed of these electrons?
3. A bare helium nucleus has two positive charges and a mass of 6.64 × ten⁻²⁷ kg. (a) Calculate its kinetic energy in joules at 2.00% of the speed of low-cal. (b) What is this in electron volts? (c) What voltage would exist needed to obtain this free energy?
4. Integrated Concepts. Singly charged gas ions are accelerated from balance through a voltage of 13.0 V. At what temperature volition the boilerplate kinetic energy of gas molecules exist the same every bit that given these ions?
5. Integrated Concepts.The temperature near the center of the Lord's day is thought to be 15 1000000 degrees Celsius (1.5 × 10^vii ºC). Through what voltage must a singly charged ion exist accelerated to have the same free energy as the boilerplate kinetic energy of ions at this temperature?
6. Integrated Concepts. (a) What is the average power output of a heart 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 commodities strikes a tree, moving twenty.0 C of accuse through a potential deviation of 1.00 × 10² MV. (a) What energy was dissipated? (b) What mass of water could be raised from 15ºC to the humid indicate and then boiled past this energy? (c) Hash out the damage that could be caused to the tree past the expansion of the boiling steam.
8. Integrated Concepts. A 12.0 V bombardment-operated bottle warmer heats 50.0 m of drinking glass, 2.fifty × 10ⁱⁱ yard of baby formula, and 2.00 × 10² 1000 of aluminum from 20.0ºC to 90.0ºC. (a) How much charge is moved by the battery? (b) How many electrons per 2d flow if it takes five.00 min to warm the formula? (Hint: Assume that the specific heat of babe formula is nearly the same every bit the specific oestrus of water.)
9. Integrated Concepts. A battery-operated automobile utilizes a 12.0 V system. Detect the accuse the batteries must be able to motility in order to accelerate the 750 kg car from rest to 25.0 1000/s, brand it climb a ii.00 × 10² chiliad high hill, and then crusade information technology to travel at a constant 25.0 one thousand/due south by exerting a five.00 × x² N strength for an hour.
10. Integrated Concepts. Fusion probability is greatly enhanced when advisable nuclei are brought close together, but mutual Coulomb repulsion must be overcome. This tin can be washed using the kinetic free energy of high-temperature gas ions or by accelerating the nuclei toward i some other. (a) Calculate the potential energy of two singly charged nuclei separated by 1.00 × 10⁻¹² m past finding the voltage of one at that distance and multiplying past 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?
11. Unreasonable Results. (a) Notice the voltage near a 10.0 cm diameter metallic sphere that has eight.00 C of excess positive charge on it. (b) What is unreasonable most this event? (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 y'all determine the energy that must be supplied by the battery, and then calculate the amount of charge it must be able to move in order to supply this energy. Among the things to be considered are the energy needs and battery voltage. You may need to expect ahead to interpret manufacturer's battery ratings in ampere-hours as energy in joules.

Glossary

electric potential: potential energy per unit charge

potential difference (or voltage): change in potential energy of a charge moved from one point to some other, divided by the charge; units of potential deviation are joules per coulomb, known equally volt

electron volt: the free 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 system; this sum is a constant

Selected Solutions to Bug & Exercises

one. 42.eight

4. 1.00 × ten^v M

half dozen. (a) 4 × 10⁴ West; (b) A defibrillator does not cause serious burns because the skin conducts electricity well at high voltages, like those used in defibrillators. The gel used aids in the transfer of energy to the body, and the peel doesn't blot the energy, but rather lets it pass through to the heart.

8. (a) 7.twoscore × 10ⁱⁱⁱ C; (b) one.54 × 10²⁰ electrons per 2d

ix. 3.89 × 10^vi C

11. (a) 1.44 × 10¹² V; (b) This voltage is very high. A ten.0 cm diameter sphere could never maintain this voltage; it would discharge; (c) An eight.00 C charge is more charge than can reasonably exist accumulated on a sphere of that size.

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