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== 7.2#h(0.6em)Electric Potential and Potential Difference


  Recall that earlier we defined electric field to be a quantity independent of the test charge in a given system, which would nonetheless allow us to calculate the force that would result on an arbitrary test charge. (The default assumption in the absence of other information is that the test charge is positive.) We briefly defined a field for gravity, but gravity is always attractive, whereas the electric force can be either attractive or repulsive. Therefore, although potential energy is perfectly adequate in a gravitational system, it is convenient to define a quantity that allows us to calculate the work on a charge independent of the magnitude of the charge. Calculating the work directly may be difficult, since #math.equation(block: false, alt: "W equals F → times d →")[$W = arrow("F") · arrow("d")$] and the direction and magnitude of #math.equation(block: false, alt: "F →")[$arrow("F")$] can be complex for multiple charges, for odd-shaped objects, and along arbitrary paths. But we do know that because #math.equation(block: false, alt: "F → equals q E →")[$arrow("F") = q arrow("E")$], the work, and hence #math.equation(block: false, alt: "Δ U ,")[$"Δ" U ,$] is proportional to the test charge #emph[q]. To have a physical quantity that is independent of test charge, we define #strong[electric potential] #emph[V] (or simply potential, since electric is understood) to be the potential energy per unit charge:


  #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Electric Potential]


    The electric potential energy per unit charge is


    #math.equation(block: true, alt: "V equals the fraction U over q .")[$V = frac(U, q) .$]
]


  Since #emph[U] is proportional to #emph[q], the dependence on #emph[q] cancels. Thus, #emph[V] does not depend on #emph[q]. The change in potential energy #math.equation(block: false, alt: "Δ U")[$"Δ" U$] is crucial, so we are concerned with the difference in potential or potential difference #math.equation(block: false, alt: "Δ V")[$"Δ" V$] between two points, where


  #math.equation(block: true, alt: "Δ V equals V sub B minus V sub A equals the fraction Δ U over q .")[$"Δ" V = V_(B) − V_(A) = frac("Δ" U, q) .$]
  #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Electric Potential Difference]


    The #strong[electric potential difference] between points #emph[A] and #emph[B], #math.equation(block: false, alt: "V sub B minus V sub A ,")[$V_(B) − V_(A) ,$] is defined to be the change in potential energy of a charge #emph[q] moved from #emph[A] to #emph[B], divided by the charge. Units of potential difference are joules per coulomb, given the name #strong[volt (V)] after Alessandro #strong[Volta].


    #math.equation(block: true, alt: "1 V equals 1 J/C")[$1 #h(0.2em) "V" = 1 #h(0.2em) "J/C"$]
]


  The familiar term #strong[voltage] is the common name for electric 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. It is worthwhile to emphasize the distinction between potential difference and electrical potential energy.


  #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Potential Difference and Electrical Potential Energy]


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


    #math.equation(block: true, alt: "Δ V equals the fraction Δ U over q or Δ U equals q Δ V .")[$"Δ" V = frac("Δ" U, q) #h(0.2em) "or" #h(0.2em) "Δ" U = q "Δ" V .$]
]


  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 because #math.equation(block: false, alt: "Δ U equals q Δ V .")[$"Δ" U = q "Δ" V .$] The car battery can move more charge than the motorcycle battery, although both are 12-V batteries.


  #examplebox("Example 1")[Calculating Energy][
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 #math.equation(block: false, alt: "Δ U equals q Δ V .")[$"Δ" U = q "Δ" V .$] To find the energy output, we multiply the charge moved by the potential difference.


    Solution
For the motorcycle battery, #math.equation(block: false, alt: "q equals 5000 C")[$q = 5000 #h(0.2em) "C"$] and #math.equation(block: false, alt: "Δ V equals 12.0 V")[$"Δ" V = 12.0 #h(0.2em) "V"$]. The total energy delivered by the motorcycle battery is


    #math.equation(block: true, alt: "Δ U sub cycle equals open parenthesis 5000 C close parenthesis open parenthesis 12.0 V close parenthesis equals open parenthesis 5000 C close parenthesis open parenthesis 12.0 J/C close parenthesis equals 6.00 times 10 to the power 4 J .")[$"Δ" U_("cycle") = ( 5000 #h(0.2em) "C" ) ( 12.0 #h(0.2em) "V" ) = ( 5000 #h(0.2em) "C" ) ( 12.0 #h(0.2em) "J/C" ) = #h(0.2em) 6.00 #h(0.2em) × #h(0.2em) 10^(4) #h(0.2em) "J" "."$]
    Similarly, for the car battery, #math.equation(block: false, alt: "q equals 60,000 C")[$q = 60,000 #h(0.2em) "C"$] and


    #math.equation(block: true, alt: "Δ U sub car equals open parenthesis 60,000 C close parenthesis open parenthesis 12.0 V close parenthesis equals 7.20 times 10 to the power 5 J .")[$"Δ" U_("car") = ( 60,000 #h(0.2em) "C" ) ( 12.0 #h(0.2em) "V" ) = 7.20 #h(0.2em) × #h(0.2em) 10^(5) #h(0.2em) "J" "."$]
    Significance
Voltage and energy are related, but they are not the same thing. The voltages of the batteries are identical, but the energy supplied by each is quite different. A car battery has a much larger engine to start than a motorcycle. 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 depleted 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.
]


  #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
How much energy does a 1.5-V AAA battery have that can move 100 C?


    
    #solutionbox[
#math.equation(block: true, alt: "Δ U equals q Δ V equals open parenthesis 100 C close parenthesis open parenthesis 1.5 V close parenthesis equals 150 J")[$"Δ" U = q "Δ" V = ( 100 #h(0.2em) "C" ) ( 1.5 #h(0.2em) "V" ) = 150 #h(0.2em) "J"$]
]
]


  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 (#emph[A]) through whatever circuitry is involved and attract them to their positive terminals (#emph[B]), as shown. The change in potential is #math.equation(block: false, alt: "Δ V equals V sub B minus V sub A equals plus 12 V")[$"Δ" V = V_(B) − V_(A) = + 12 #h(0.2em) "V"$] and the charge #emph[q] is negative, so that #math.equation(block: false, alt: "Δ U equals q Δ V")[$"Δ" U = q "Δ" V$] is negative, meaning the potential energy of the battery has decreased when #emph[q] has moved from #emph[A] to #emph[B].


  #figure(figph[The figure shows a headlight connected to terminals of a 12V battery. The charge q flows out from terminal A of the battery and back into terminal B of the battery.], alt: "The figure shows a headlight connected to terminals of a 12V battery. The charge q flows out from terminal A of the battery and back into terminal B of the battery.", caption: [A battery 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 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 terminal is at a higher voltage than the negative terminal. Inside the battery, both positive and negative charges move.])


  #examplebox("Example 2")[How Many Electrons Move through a Headlight Each Second?][
When a 12.0-V car battery powers a single 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 moves in 1.00 s. The charge moved is related to voltage and energy through the equations #math.equation(block: false, alt: "Δ U equals q Δ V .")[$"Δ" U = q "Δ" V .$] A 30.0-W lamp uses 30.0 joules per second. Since the battery loses energy, we have #math.equation(block: false, alt: "Δ U equals −30 J")[$"Δ" U = −30 #h(0.2em) "J"$] and, since the electrons are going from the negative terminal to the positive, we see that #math.equation(block: false, alt: "Δ V equals +12.0 V .")[$"Δ" V = "+12.0" #h(0.2em) "V" "."$]


    Solution
To find the charge #emph[q] moved, we solve the equation #math.equation(block: false, alt: "Δ U equals q Δ V :")[$"Δ" U = q "Δ" V :$]


    #math.equation(block: true, alt: "q equals the fraction Δ U over Δ V .")[$q = frac("Δ" U, "Δ" V) .$]
    Entering the values for #math.equation(block: false, alt: "Δ U")[$"Δ" U$] and #math.equation(block: false, alt: "Δ V")[$"Δ" V$], we get


    #math.equation(block: true, alt: "q equals the fraction −30.0 J over plus 12.0 V equals the fraction −30.0 J over plus 12.0 J/C equals −2.50 C .")[$q = frac(−30.0 #h(0.2em) "J", + 12.0 #h(0.2em) "V") = frac(−30.0 #h(0.2em) "J", + 12.0 #h(0.2em) "J/C") = −2.50 #h(0.2em) "C" "."$]
    The number of electrons #math.equation(block: false, alt: "n sub e")[$n_(e)$] is the total charge divided by the charge per electron. That is,


    #math.equation(block: true, alt: "n sub e equals the fraction −2.50 C over −1.60 times 10 to the power −19 C/e to the power minus equals 1.56 times 10 to the power 19 electrons .")[$n_(e) = frac(−2.50 #h(0.2em) "C", −1.60 #h(0.2em) × #h(0.2em) attach(#h(0.2em) 10, t: −19) #h(0.2em) "C/e"^(−)) = 1.56 #h(0.2em) × #h(0.2em) attach(#h(0.2em) 10, t: 19) #h(0.2em) "electrons" "."$]
    Significance
This is a very large number. It is no wonder that we do not ordinarily observe individual electrons with so many being present in ordinary systems. In fact, electricity had been in use for many decades before it was determined that the moving charges in many circumstances were negative. Positive charge moving in the opposite direction of negative charge often produces identical effects; this makes it difficult to determine which is moving or whether both are moving.
]


  #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
How many electrons would go through a 24.0-W lamp each second from a 12-volt car battery?


    
    #solutionbox[
–2.00 C, #math.equation(block: false, alt: "n sub e equals 1.25 times 10 to the power 19 electrons")[$n_(e) = 1.25 #h(0.2em) × #h(0.2em) 10^(19) #h(0.2em) "electrons"$]
]
]


  
    
=== The Electron-Volt


    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.


    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 gains kinetic energy that is later converted into another form—light in the television tube, for example. (Note that in terms of energy, “downhill” for the electron is “uphill” for a positive charge.) Since energy is related to voltage by #math.equation(block: false, alt: "Δ U equals q Δ V")[$"Δ" U = q "Δ" V$], we can think of the joule as a coulomb-volt.


    #figure(figph[Part a shows an electron gun with two metal plates and an electron between the plates. The metal plates are connected to terminals of a battery and have opposite charges with a potential difference V subscript AB. Part b shows the photo of an electron gun.], alt: "Part a shows an electron gun with two metal plates and an electron between the plates. The metal plates are connected to terminals of a battery and have opposite charges with a potential difference V subscript AB. Part b shows the photo of an electron gun.", caption: [A typical electron gun accelerates electrons using a potential difference between two separated metal plates. By conservation of energy, the kinetic energy has to equal the change in potential energy, so #math.equation(block: false, alt: "K E equals q V")[$K E = q V$]. The energy of the electron in electron-volts is numerically the same as the voltage between the plates. For example, a 5000-V potential difference produces 5000-eV electrons. The conceptual construct, namely two parallel plates with a hole in one, is shown in (a), while a real electron gun is shown in (b).])


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Electron-Volt]


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


      #math.equation(block: true, alt: "1 eV equals open parenthesis 1.60 times 10 to the power −19 C close parenthesis open parenthesis 1 V close parenthesis equals open parenthesis 1.60 times 10 to the power −19 C close parenthesis open parenthesis 1 J/C close parenthesis equals 1.60 times 10 to the power −19 J .")[$1 #h(0.2em) "eV" = ( 1.60 #h(0.2em) × #h(0.2em) 10^(−19) #h(0.2em) "C" ) ( 1 #h(0.2em) "V" ) = ( 1.60 #h(0.2em) × #h(0.2em) 10^(−19) #h(0.2em) "C" ) ( 1 #h(0.2em) "J/C" ) = 1.60 #h(0.2em) × #h(0.2em) 10^(−19) #h(0.2em) "J" "."$]
]


    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 gains 50 eV. A potential difference of 100,000 V (100 kV) gives 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 gains 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.


    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 acquires an energy of 30 keV (30,000 eV) and can break up as many as 6000 of these molecules #math.equation(block: false, alt: "open parenthesis 30,000 eV divided by 5 eV per molecule equals 6000 molecules).")[$\( 30,000 #h(0.2em) "eV" ÷ 5 #h(0.2em) "eV" #h(0.2em) "per molecule" = 6000 #h(0.2em) "molecules)."$] Nuclear decay energies are on the order of 1 MeV (1,000,000 eV) per event and can thus produce significant biological damage.


  
  
    
=== Conservation of Energy


    The total energy of a system is conserved if there is no net addition (or subtraction) due to 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, #math.equation(block: false, alt: "K plus U equals constant .")[$K + U = "constant" "."$] A loss of #emph[U] for a charged particle becomes an increase in its #emph[K]. Conservation of energy is stated in equation form as


    #math.equation(block: true, alt: "K plus U equals constant")[$K + U = "constant"$]
    or


    #math.equation(block: true, alt: "K sub i plus U sub i equals K sub f plus U sub f")[$K_("i") + U_("i") = K_("f") + U_("f")$]
    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.


    #examplebox("Example 3")[Electrical Potential Energy Converted into Kinetic Energy][
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 #math.equation(block: false, alt: "K sub i equals 0 , K sub f equals the fraction 1 over 2 m v squared , U sub i equals q V , U sub f equals 0 .")[$K_("i") = 0 , K_("f") = frac(1, 2) m v^(2) , U_("i") = q V , U_("f") = 0 .$]


      Solution
Conservation of energy states that


      #math.equation(block: true, alt: "K sub i plus U sub i equals K sub f plus U sub f .")[$K_("i") + U_("i") = K_("f") + U_("f") .$]
      Entering the forms identified above, we obtain


      #math.equation(block: true, alt: "q V equals the fraction m v squared over 2 .")[$q V = frac(m v^(2), 2) .$]
      We solve this for #emph[v]:


      #math.equation(block: true, alt: "v equals the square root of the fraction 2 q V over m .")[$v = sqrt(frac(2 q V, m)) .$]
      Entering values for #emph[q], #emph[V], and #emph[m] gives


      #math.equation(block: true, alt: "v equals the square root of the fraction 2 open parenthesis −1.60 times 10 to the power −19 C close parenthesis open parenthesis minus 100 J/C close parenthesis over 9.11 times 10 to the power −31 kg equals 5.93 times 10 to the power 6 m/s .")[$v = sqrt(frac(2 ( −1.60 #h(0.2em) × #h(0.2em) 10^(−19) #h(0.2em) "C" ) ( − 100 #h(0.2em) "J/C" ), 9.11 #h(0.2em) × #h(0.2em) 10^(−31) #h(0.2em) "kg")) = 5.93 #h(0.2em) × #h(0.2em) 10^(6) #h(0.2em) "m/s" "."$]
      Significance
Note that both the charge and the initial voltage are negative, as in . From the discussion of 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. These higher voltages produce electron speeds so great that effects from special relativity must be taken into account and hence are reserved for a later chapter (Relativity). That is why we consider a low voltage (accurately) in this example.
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
How would this example change with a positron? A positron is identical to an electron except the charge is positive.


      
      #solutionbox[
It would be going in the opposite direction, with no effect on the calculations as presented.
]
]


  
  
    
=== Voltage and Electric Field


    So far, we have explored the relationship between voltage and energy. Now we want to explore the relationship between voltage and electric field. We will start with the general case for a non-uniform #math.equation(block: false, alt: "E →")[$arrow("E")$] field. Recall that our general formula for the potential energy of a test charge #emph[q] at point #emph[P] relative to reference point #emph[R] is


    #math.equation(block: true, alt: "U sub P equals − ∫ R P F → times d l → .")[$U_(P) = "−" display(∫_(R)^(P) arrow("F") · d arrow("l")) .$]
    When we substitute in the definition of electric field #math.equation(block: false, alt: "open parenthesis E → equals F → / q close parenthesis ,")[$( arrow("E") = arrow("F") "/" q ) ,$] this becomes


    #math.equation(block: true, alt: "U sub P equals − q ∫ R P E → times d l → .")[$U_(P) = "−" q display(∫_(R)^(P) arrow("E") · d arrow("l")) .$]
    Applying our definition of potential #math.equation(block: false, alt: "open parenthesis V equals U / q close parenthesis")[$( V = U "/" q )$] to this potential energy, we find that, in general,


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: true, alt: "V sub P equals − ∫ R P E → times d l → .")[$V_(P) = "−" display(∫_(R)^(P) arrow("E") · d arrow("l")) .$]
]


    From our previous discussion of the potential energy of a charge in an electric field, the result is independent of the path chosen, and hence we can pick the integral path that is most convenient.


    Consider the special case of a positive point charge #emph[q] at the origin. To calculate the potential caused by #emph[q] at a distance #emph[r] from the origin relative to a reference of 0 at infinity (recall that we did the same for potential energy), let #math.equation(block: false, alt: "P equals r")[$P = r$] and #math.equation(block: false, alt: "R equals ∞ ,")[$R = ∞ ,$] with #math.equation(block: false, alt: "d l → equals d r → equals r ^ d r")[$d arrow("l") = d arrow("r") = hat("r") d r$] and use #math.equation(block: false, alt: "E → equals the fraction k q over r squared r ^ .")[$arrow("E") = frac(k q, r^(2)) hat("r") .$] When we evaluate the integral


    #math.equation(block: true, alt: "V sub P equals − ∫ R P E → times d l →")[$V_(P) = "−" display(∫_(R)^(P) arrow("E") · d arrow("l"))$]
    for this system, we have


    #math.equation(block: true, alt: "V sub r equals − ∫ ∞ r the fraction k sub e q over r squared r ^ times r ^ d r ,")[$V_(r) = "−" display(∫_(∞)^(r) frac(k_(e) q, r^(2)) hat("r") · hat("r") d r) ,$]
    which simplifies to


    #math.equation(block: true, alt: "V sub r equals − ∫ ∞ r the fraction k sub e q over r squared d r equals the fraction k sub e q over r minus the fraction k sub e q over ∞ equals the fraction k sub e q over r .")[$V_(r) = "−" display(∫_(∞)^(r) frac(k_(e) q, r^(2)) #h(0.2em) d r) = frac(k_(e) q, r) − frac(k_(e) q, ∞) = frac(k_(e) q, r) .$]
    This result,


    #math.equation(block: true, alt: "V sub r equals the fraction k sub e q over r")[$V_(r) = frac(k_(e) q, r)$]
    is the standard form of the potential of a point charge. This will be explored further in the next section.


    To examine another interesting special case, suppose a uniform electric field #math.equation(block: false, alt: "E →")[$arrow("E")$] is produced by placing a potential difference (or voltage) #math.equation(block: false, alt: "Δ V")[$"Δ" V$] across two parallel metal plates, labeled #emph[A] and #emph[B] . Examining this situation will tell us what voltage is needed to produce a certain electric field strength. It will also reveal a more fundamental relationship between electric potential and electric field.


    #figure(figph[The figure shows electric field between two plates (A and B) with opposite charges. The plates are separated by distance d and have a potential difference V subscript AB. A positive charge q is located between the plates and moves from A to B.], alt: "The figure shows electric field between two plates (A and B) with opposite charges. The plates are separated by distance d and have a potential difference V subscript AB. A positive charge q is located between the plates and moves from A to B.", caption: [The relationship between #emph[V] and #emph[E] for parallel conducting plates is #math.equation(block: false, alt: "E equals the fraction minus Δ V over d")[$E = frac(− "Δ" V, d)$]. The field points toward lower potential #emph[V].])


    From a physicist’s point of view, either #math.equation(block: false, alt: "Δ V")[$"Δ" V$] or #math.equation(block: false, alt: "E →")[$arrow("E")$] can be used to describe any interaction between charges. However, #math.equation(block: false, alt: "Δ V")[$"Δ" V$] is a scalar quantity and has no direction, whereas #math.equation(block: false, alt: "E →")[$arrow("E")$] is a vector quantity, having both magnitude and direction. (Note that the magnitude of the electric field, a scalar quantity, is represented by #emph[E].) The relationship between #math.equation(block: false, alt: "Δ V")[$"Δ" V$] and #math.equation(block: false, alt: "E →")[$arrow("E")$] is revealed by calculating the work done by the electric force in moving a charge from point #emph[A] to point #emph[B]. But, as noted earlier, arbitrary charge distributions require calculus. We therefore look at a uniform electric field as an interesting special case.


    The work done by the electric field in  to move a positive charge #emph[q] from #emph[A], the positive plate, higher potential, to #emph[B], the negative plate, lower potential, is


    #math.equation(block: true, alt: "W equals − Δ U equals − q Δ V equals - q V sub A B .")[$W = "−" "Δ" U = "−" q "Δ" V = "-" q V_(A B) .$]
    Work is #math.equation(block: false, alt: "W equals F → times d → equals F d cos θ")[$W = arrow("F") · arrow("d") = F d #h(0.2em) "cos" #h(0.2em) θ$]; here #math.equation(block: false, alt: "cos θ equals 1")[$#h(0.2em) "cos" #h(0.2em) θ = 1$], since the path is parallel to the field. Thus, #math.equation(block: false, alt: "W equals F d")[$W = F d$]. Since #math.equation(block: false, alt: "F equals q E")[$F = q E$], we see that #math.equation(block: false, alt: "W equals q E d")[$W = q E d$].


    Substituting this expression for work into the previous equation gives


    #math.equation(block: true, alt: "q E d equals - q V sub A B .")[$q E d = "-" q V_(A B) .$]
    The charge cancels, so we obtain for the voltage between points #emph[A] and #emph[B]


    #math.equation(block: true, alt: "V sub A B equals - E d; E equals the fraction - V sub A B over d } open parenthesis uniform E -field only close parenthesis")[$V_(A B) = "-" E d \ E = frac("-" V_(A B), d) \} ( "uniform" #h(0.2em) E "-field only" )$]
    where #emph[d] is the distance from #emph[A] to #emph[B], or the distance between the plates in . Note that this equation implies that the units for electric field are volts per meter. We already know the units for electric field are newtons per coulomb; thus, the following relation among units is valid:


    #math.equation(block: true, alt: "1 N / C equals 1 V / m .")[$1 #h(0.2em) "N" #h(0.2em) "/" #h(0.2em) "C" = 1 #h(0.2em) "V" #h(0.2em) "/" #h(0.2em) "m" .$]
    Furthermore, we may extend this to the integral form. Substituting  into our definition for the potential difference between points #emph[A] and #emph[B], we obtain


    #math.equation(block: true, alt: "V sub A B equals V sub B minus V sub A equals − ∫ R B E → times d l → plus ∫ R A E → times d l →")[$V_(A B) = V_(B) − V_(A) = "−" display(∫_(R)^(B) arrow("E") · d arrow("l")) + display(∫_(R)^(A) arrow("E") · d arrow("l"))$]
    which simplifies to


    #math.equation(block: true, alt: "V sub B minus V sub A equals − ∫ A B E → times d l → .")[$V_(B) − V_(A) = "−" display(∫_(A)^(B) arrow("E") · d arrow("l")) .$]
    The potential difference is negative (#emph[V] is lower at #emph[B] than at #emph[A]) when the displacement is in the same direction as the field. In other words, the electric field points toward lower electric potential. We are often only interested in the magnitude of the electric field, in which case you may see #math.equation(block: false, alt: "Δ V equals E d")[$"Δ" V = E d$] instead of #math.equation(block: false, alt: "| Δ V | equals | E d |")[$| "Δ" V | = | E d |$] and it is understood that all of the variables in the equation represent the magnitudes of the quantities. As a demonstration, from this we may calculate the potential difference between two points (#emph[A] and #emph[B]) equidistant from a point charge #emph[q] at the origin, as shown.


    #figure(figph[The figure shows a charge q equidistant from two points, A and B.], alt: "The figure shows a charge q equidistant from two points, A and B.", caption: [The arc for calculating the potential difference between two points that are equidistant from a point charge at the origin.])


    To do this, we integrate around an arc of the circle of constant radius r between #emph[A] and #emph[B], which means we let #math.equation(block: false, alt: "d l → equals r φ ^ d φ ,")[$d arrow("l") = r hat(φ) d φ ,$] while using #math.equation(block: false, alt: "E → equals the fraction k q over r squared r ^")[$arrow("E") = frac(k q, r^(2)) hat("r")$]. Thus,


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: true, alt: "Δ V sub A B equals V sub B minus V sub A equals − ∫ A B E → times d l →")[$"Δ" V_(A B) = V_(B) − V_(A) = "−" display(∫_(A)^(B) arrow("E") · d arrow("l"))$]
]


    for this system becomes


    #math.equation(block: true, alt: "V sub B minus V sub A equals − ∫ A B the fraction k sub e q over r squared r ^ times r φ ^ d φ .")[$V_(B) − V_(A) = "−" display(∫_(A)^(B) frac(k_(e) q, r^(2))) hat("r") · r hat(φ) d φ .$]
    However, #math.equation(block: false, alt: "r ^ times φ ^ equals 0")[$hat("r") · hat(φ) = 0$] and therefore


    #math.equation(block: true, alt: "V sub B minus V sub A equals 0 .")[$V_(B) − V_(A) = 0 .$]
    This result, that there is no difference in potential along a constant radius from a point charge, will come in handy when we map potentials.


    #examplebox("Example 4")[What Is the Highest Voltage Possible between Two Plates?][
Dry air can support a maximum electric field strength of about #math.equation(block: false, alt: "3.0 times 10 to the power 6 V/m .")[$3.0 #h(0.2em) × #h(0.2em) 10^(6) #h(0.2em) "V/m" "."$] Above that value, the field creates enough ionization in the air to make the air a conductor. This allows a discharge or spark that reduces the field. What, then, is the maximum voltage between two parallel conducting plates separated by 2.5 cm of dry air?


      Strategy
We are given the maximum electric field #emph[E] between the plates and the distance #emph[d] between them. We can use the equation #math.equation(block: false, alt: "Δ V equals E d")[$"Δ" V = E d$] (recall that these represent absolute values) to calculate the maximum voltage.


      Solution
The potential difference or voltage between the plates is


      #math.equation(block: true, alt: "Δ V equals E d .")[$"Δ" V = E d .$]
      Entering the given values for #emph[E] and #emph[d] gives


      #math.equation(block: true, alt: "Δ V equals open parenthesis 3.0 times 10 to the power 6 V/m close parenthesis open parenthesis 0.025 m close parenthesis equals 7.5 times 10 to the power 4 V")[$"Δ" V = ( 3.0 #h(0.2em) × #h(0.2em) 10^(6) #h(0.2em) "V/m" ) ( 0.025 #h(0.2em) "m" ) = 7.5 #h(0.2em) × #h(0.2em) 10^(4) #h(0.2em) "V"$]
      or


      #math.equation(block: true, alt: "Δ V equals 75 kV .")[$"Δ" V = 75 #h(0.2em) "kV" "."$]
      (The answer is quoted to only two digits, since the maximum field strength is approximate.)


      Significance
One of the implications of this result is that it takes about 75 kV to make a spark jump across a 2.5-cm (1-in.) gap, or 150 kV for a 5-cm spark. This limits the voltages that can exist between conductors, perhaps on a power transmission line. A smaller voltage can cause a spark if there are spines on the surface, since sharp points have larger field strengths than smooth surfaces. Humid air breaks down at a lower field strength, meaning that a smaller voltage will make a spark jump through humid air. The largest voltages can be built up with static electricity on dry days .


      #figure(figph[The first photo shows a spark chamber and the second photo shows its operation.], alt: "The first photo shows a spark chamber and the second photo shows its operation.", caption: [A #strong[spark chamber] is used to trace the paths of high-energy particles. Ionization created by the particles as they pass through the gas between the plates allows a spark to jump. The sparks are perpendicular to the plates, following electric field lines between them. The potential difference between adjacent plates is not high enough to cause sparks without the ionization produced by particles from accelerator experiments (or cosmic rays). This form of detector is now archaic and no longer in use except for demonstration purposes.])
]


    #examplebox("Example 5")[Field and Force inside an Electron Gun][
An electron gun  has parallel plates separated by 4.00 cm and gives electrons 25.0 keV of energy. (a) What is the electric field strength between the plates? (b) What force would this field exert on a piece of plastic with a #math.equation(block: false, alt: "0.500 - μ C")[$0.500 "-" μ "C"$] charge that gets between the plates?


      Strategy
Since the voltage and plate separation are given, the electric field strength can be calculated directly from the expression #math.equation(block: false, alt: "E equals the fraction Δ V over d")[$E = frac("Δ" V, d)$]. Once we know the electric field strength, we can find the force on a charge by using #math.equation(block: false, alt: "F → equals q E → .")[$arrow("F") = q arrow("E") .$] Since the electric field is in only one direction, we can write this equation in terms of the magnitudes, #math.equation(block: false, alt: "F equals q E")[$F = q E$].


      Solution


      + The expression for the magnitude of the electric field between two uniform metal plates is
#math.equation(block: true, alt: "E equals the fraction Δ V over d .")[$E = frac("Δ" V, d) .$]
Since the electron is a single charge and is given 25.0 keV of energy, the potential difference must be 25.0 kV. Entering this value for #math.equation(block: false, alt: "V sub A B")[$V_(A B)$] and the plate separation of 0.0400 m, we obtain
#math.equation(block: true, alt: "E equals the fraction 25.0 kV over 0.0400 m equals 6.25 times 10 to the power 5 V/m .")[$E = frac(25.0 #h(0.2em) "kV", 0.0400 #h(0.2em) "m") = 6.25 #h(0.2em) × #h(0.2em) 10^(5) #h(0.2em) "V/m" "."$]
+ The magnitude of the force on a charge in an electric field is obtained from the equation
#math.equation(block: true, alt: "F equals q E .")[$F = q E .$]
Substituting known values gives 
#math.equation(block: true, alt: "F equals open parenthesis 0.500 times 10 to the power −6 C close parenthesis open parenthesis 6.25 times 10 to the power 5 V/m close parenthesis equals 0.313 N .")[$F = ( 0.500 #h(0.2em) × #h(0.2em) 10^(−6) #h(0.2em) "C" ) ( 6.25 #h(0.2em) × #h(0.2em) 10^(5) #h(0.2em) "V/m" ) = 0.313 #h(0.2em) "N" "."$]


      Significance
Note that the units are newtons, since #math.equation(block: false, alt: "1 V/m equals 1 N/C")[$1 #h(0.2em) "V/m" = 1 #h(0.2em) "N/C"$]. Because the electric field is uniform between the plates, the force on the charge is the same no matter where the charge is located between the plates.
]


    #examplebox("Example 6")[Calculating Potential of a Point Charge][
Given a point charge #math.equation(block: false, alt: "q equals plus 2.0 nC")[$q = + 2.0 #h(0.2em) "nC"$] at the origin, calculate the potential difference between point #math.equation(block: false, alt: "P sub 1")[$P_(1)$] a distance #math.equation(block: false, alt: "a equals 4.0 cm")[$a = 4.0 #h(0.2em) "cm"$] from #emph[q], and #math.equation(block: false, alt: "P sub 2")[$P_(2)$] a distance #math.equation(block: false, alt: "b equals 12.0 cm")[$b = 12.0 #h(0.2em) "cm"$] from #emph[q], where the two points have an angle of #math.equation(block: false, alt: "φ equals 24 °")[$φ = 24 "°"$] between them .


      #figure(figph[The figure shows two points P subscript 1 and P subscript 2 at distances a and b from the origin and having an angle phi between them.], alt: "The figure shows two points P subscript 1 and P subscript 2 at distances a and b from the origin and having an angle phi between them.", caption: [Find the difference in potential between #math.equation(block: false, alt: "P sub 1")[$P_(1)$] and #math.equation(block: false, alt: "P sub 2")[$P_(2)$].])


      Strategy
Do this in two steps. The first step is to use #math.equation(block: false, alt: "V sub B minus V sub A equals − ∫ A B E → times d l →")[$V_(B) − V_(A) = "−" display(∫_(A)^(B) arrow("E") · d arrow("l"))$] and let #math.equation(block: false, alt: "A equals a equals 4.0 cm")[$A = a = 4.0 #h(0.2em) "cm"$] and #math.equation(block: false, alt: "B equals b equals 12.0 cm ,")[$B = b = 12.0 #h(0.2em) "cm" ,$] with #math.equation(block: false, alt: "d l → equals d r → equals r ^ d r")[$d arrow("l") = d arrow("r") = hat("r") d r$] and #math.equation(block: false, alt: "E → equals the fraction k sub e q over r squared r ^ .")[$arrow("E") = frac(k_(e) q, r^(2)) hat("r") .$] Then perform the integral. The second step is to integrate #math.equation(block: false, alt: "V sub B minus V sub A equals − ∫ A B E → times d l →")[$V_(B) − V_(A) = "−" display(∫_(A)^(B) arrow("E") · d arrow("l"))$] around an arc of constant radius #emph[r], which means we let #math.equation(block: false, alt: "d l → equals r φ ^ d φ")[$d arrow("l") = r hat(φ) d φ$] with limits #math.equation(block: false, alt: "0 less than or equal to φ less than or equal to 24 ° ,")[$0 ≤ φ ≤ 24 "°" ,$] still using #math.equation(block: false, alt: "E → equals the fraction k sub e q over r squared r ^")[$arrow("E") = frac(k_(e) q, r^(2)) hat("r")$]. Then add the two results together.


      Solution
For the first part, #math.equation(block: false, alt: "V sub B minus V sub A equals − ∫ A B E → times d l →")[$V_(B) − V_(A) = "−" display(∫_(A)^(B) arrow("E") · d arrow("l"))$] for this system becomes #math.equation(block: false, alt: "V sub b minus V sub a equals − ∫ a b the fraction k sub e q over r squared r ^ times r ^ d r")[$V_(b) − V_(a) = "−" display(∫_(a)^(b) frac(k_(e) q, r^(2)) hat("r") · hat("r") d r)$] which computes to


      #math.equation(block: true, alt: "| Δ V |, equals | − ∫ a b the fraction k sub e q over r squared d r | equals k sub e q [ the fraction 1 over a minus the fraction 1 over b ]; equals open parenthesis 8.99 times 10 to the power 9 Nm squared /C squared close parenthesis open parenthesis 2.0 times 10 to the power −9 C close parenthesis [ the fraction 1 over 0.040 m minus the fraction 1 over 0.12 m ] equals 300 V .")[$| "Δ" V | & = | "−" display(∫_(a)^(b) frac(k_(e) q, r^(2)) d r) | = k_(e) q [ frac(1, a) − frac(1, b) ] \  & = ( 8.99 #h(0.2em) × #h(0.2em) 10^(9) #h(0.2em) "Nm"^(2) "/C"^(2) ) ( 2.0 #h(0.2em) × #h(0.2em) 10^(−9) #h(0.2em) "C" ) [ frac(1, 0.040 #h(0.2em) "m") − frac(1, 0.12 #h(0.2em) "m") ] = 300 #h(0.2em) "V" "."$]
      For the second step, #math.equation(block: false, alt: "V sub B minus V sub A equals − ∫ A B E → times d l →")[$V_(B) − V_(A) = "−" display(∫_(A)^(B) arrow("E") · d arrow("l"))$] becomes #math.equation(block: false, alt: "Δ V equals − ∫ 0 24 ° the fraction k sub e q over r squared r ^ times r φ ^ d φ")[$"Δ" V = "−" display(∫_(0)^(24 "°") frac(k_(e) q, r^(2)) hat("r") · r hat(φ) d φ)$], but #math.equation(block: false, alt: "r ^ times φ ^ equals 0")[$hat("r") · hat(φ) = 0$] and therefore #math.equation(block: false, alt: "Δ V equals 0 .")[$"Δ" V = 0 .$] Adding the two parts together, we get 300 V.


      Significance
We have demonstrated the use of the integral form of the potential difference to obtain a numerical result. Notice that, in this particular system, we could have also used the formula for the potential due to a point charge at the two points and simply taken the difference.
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
From the examples, how does the energy of a lightning strike vary with the height of the clouds from the ground? Consider the cloud-ground system to be two parallel plates.


      
      #solutionbox[
Given a fixed maximum electric field strength, the potential at which a strike occurs increases with increasing height above the ground. Hence, each electron will carry more energy. Determining if there is an effect on the total number of electrons lies in the future.
]
]


    Before presenting problems involving electrostatics, we suggest a problem-solving strategy to follow for this topic.


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Electrostatics]


      + Examine the situation to determine if static electricity is involved; this may concern separated stationary charges, the forces among them, and the electric fields they create.
+ Identify the system of interest. This includes noting the number, locations, and types of charges involved.
+ Identify exactly what needs to be determined in the problem (identify the unknowns). A written list is useful. Determine whether the Coulomb force is to be considered directly—if so, it may be useful to draw a free-body diagram, using electric field lines.
+ Make a list of what is given or can be inferred from the problem as stated (identify the knowns). It is important to distinguish the Coulomb force #emph[F] from the electric field #emph[E], for example.
+ Solve the appropriate equation for the quantity to be determined (the unknown) or draw the field lines as requested.
+ Examine the answer to see if it is reasonable: Does it make sense? Are units correct and the numbers involved reasonable?
]


  
  
    
=== Summary


    - Electric potential is potential energy per unit charge.
- The potential difference between points #emph[A] and #emph[B], #math.equation(block: false, alt: "V sub B minus V sub A ,")[$V_(B) − V_(A) ,$] that is, the change in potential of a charge #emph[q] moved from #emph[A] to #emph[B], is equal to the change in potential energy divided by the charge.
- Potential difference is commonly called voltage, represented by the symbol #math.equation(block: false, alt: "Δ V")[$"Δ" V$]: #linebreak() 
#math.equation(block: false, alt: "Δ V equals the fraction Δ U over q or Δ U equals q Δ V .")[$"Δ" V = frac("Δ" U, q) #h(0.2em) "or" #h(0.2em) "Δ" U = q "Δ" V .$]
- An electron-volt is the energy given to a fundamental charge accelerated through a potential difference of 1 V. In equation form, #linebreak() 
#math.equation(block: false, alt: "1 eV equals open parenthesis 1.60 times 10 to the power −19 C close parenthesis open parenthesis 1 V close parenthesis")[$1 #h(0.2em) "eV" = ( 1.60 #h(0.2em) × #h(0.2em) 10^(−19) #h(0.2em) "C" ) ( 1 #h(0.2em) "V" )$] #linebreak() #math.equation(block: false, alt: "equals open parenthesis 1.60 times 10 to the power −19 C close parenthesis open parenthesis 1 J/C close parenthesis equals 1.60 times 10 to the power −19 J .")[$= ( 1.60 #h(0.2em) × #h(0.2em) attach(#h(0.2em) 10, t: −19) #h(0.2em) "C" ) ( 1 #h(0.2em) "J/C" ) = 1.60 #h(0.2em) × #h(0.2em) 10^(−19) #h(0.2em) "J" "."$]


  
  
    
=== Conceptual Questions


    
      Discuss how potential difference and electric field strength are related. Give an example.


    
    
      What is the strength of the electric field in a region where the electric potential is constant?


    
    #solutionbox[
The electric field strength is zero because electric potential differences are directly related to the field strength. If the potential difference is zero, then the field strength must also be zero.
]


    
      If a proton is released from rest in an electric field, will it move in the direction of increasing or decreasing potential? Also answer this question for an electron and a neutron. Explain why.


    
    
      Voltage is the common word for potential difference. Which term is more descriptive, voltage or potential difference?


    
    #solutionbox[
Potential difference is more descriptive because it indicates that it is the difference between the electric potential of two points.
]


    
      If the voltage between two points is zero, can a test charge be moved between them with zero net work being done? Can this necessarily be done without exerting a force? Explain.


    
    
      What is the relationship between voltage and energy? More precisely, what is the relationship between potential difference and electric potential energy?


    
    #solutionbox[
They are very similar, but potential difference is a feature of the system; when a charge is introduced to the system, it will have a potential energy which may be calculated by multiplying the magnitude of the charge by the potential difference.
]


    
      Voltages are always measured between two points. Why?


    
    
      How are units of volts and electron-volts related? How do they differ?


    
    #solutionbox[
An electron-volt is a volt multiplied by the charge of an electron. Volts measure potential difference, electron-volts are a unit of energy.
]


    
      Can a particle move in a direction of increasing electric potential, yet have its electric potential energy decrease? Explain


    
  
  
    
=== Problems


    
      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 #math.equation(block: false, alt: "1.67 times 10 to the power −27 kg .")[$1.67 #h(0.2em) × #h(0.2em) 10^(−27) #h(0.2em) "kg" "."$]


    
    #solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 2 m sub e v e 2, equals, q V , the fraction 1 over 2 m sub H v H 2 equals q V , so that; the fraction m sub e v e 2 over m sub H v H 2, equals, 1 or the fraction v sub e over v sub H equals 42.8")[$frac(1, 2) m_(e) v_(e)^(2) & = & q V , #h(0.2em) frac(1, 2) m_("H") v_("H")^(2) = q V , #h(0.2em) "so that" \ frac(m_(e) v_(e)^(2), m_("H") v_("H")^(2)) & = & 1 #h(0.2em) "or" #h(0.2em) frac(v_(e), v_("H")) = 42.8$]
]


    
      An evacuated tube uses an accelerating voltage of 40 kV to accelerate electrons to hit a copper plate and produce X-rays. Non-relativistically, what would be the maximum speed of these electrons?


    
    
      Show that units of V/m and N/C for electric field strength are indeed equivalent.


    
    #solutionbox[
#math.equation(block: true, alt: "1 V equals 1 J/C; 1 J equals 1 N times m → 1 V/m equals 1 N/C")[$1 #h(0.2em) "V" = 1 #h(0.2em) "J/C;" #h(0.2em) 1 #h(0.2em) "J" = 1 #h(0.2em) "N" · "m" → 1 #h(0.2em) "V/m" #h(0.2em) = 1 #h(0.2em) "N/C"$]
]


    
      What is the strength of the electric field between two parallel conducting plates separated by 1.00 cm and having a potential difference (voltage) between them of #math.equation(block: false, alt: "1.50 times 10 to the power 4 V")[$1.50 #h(0.2em) × #h(0.2em) 10^(4) #h(0.2em) "V"$]?


    
    
      The electric field strength between two parallel conducting plates separated by 4.00 cm is #math.equation(block: false, alt: "7.50 times 10 to the power 4 V/m")[$7.50 #h(0.2em) × #h(0.2em) 10^(4) #h(0.2em) "V/m"$]. (a) What is the potential difference between the plates? (b) The plate with the lowest potential is taken to be zero volts. What is the potential 1.00 cm from that plate and 3.00 cm from the other?


    
    #solutionbox[
a. #math.equation(block: false, alt: "V sub A B equals 3.00 kV")[$V_(A B) = 3.00 #h(0.2em) "kV"$]; b. #math.equation(block: false, alt: "V sub A B equals 750 V")[$V_(A B) = 750 #h(0.2em) "V"$]
]


    
      The voltage across a membrane forming a cell wall is 80.0 mV and the membrane is 9.00 nm thick. What is the electric field strength? (The value is surprisingly large, but correct.) You may assume a uniform electric field.


    
    
      Two parallel conducting plates are separated by 10.0 cm, and one of them is taken to be at zero volts. (a) What is the electric field strength between them, if the potential 8.00 cm from the zero volt plate (and 2.00 cm from the other) is 450 V? (b) What is the voltage between the plates?


    
    #solutionbox[
a. #math.equation(block: false, alt: "V sub A B equals E d → E equals 5.63 kV/m")[$V_(A B) = E d → E = 5.63 #h(0.2em) "kV/m"$]; #linebreak() 
b. #math.equation(block: false, alt: "V sub A B equals 563 V")[$V_(A B) = 563 #h(0.2em) "V"$]
]


    
      Find the maximum potential difference between two parallel conducting plates separated by 0.500 cm of air, given the maximum sustainable electric field strength in air to be #math.equation(block: false, alt: "3.0 times 10 to the power 6 V/m")[$3.0 #h(0.2em) × #h(0.2em) 10^(6) #h(0.2em) "V/m"$].


    
    
      An electron is to be accelerated in a uniform electric field having a strength of #math.equation(block: false, alt: "2.00 times 10 to the power 6 V/m .")[$2.00 #h(0.2em) × #h(0.2em) 10^(6) #h(0.2em) "V/m" "."$] (a) What energy in keV is given to the electron if it is accelerated through 0.400 m? (b) Over what distance would it have to be accelerated to increase its energy by 50.0 GeV?


    
    #solutionbox[
a. #math.equation(block: false, alt: "Δ K, equals, q Δ V and V sub A B equals E d , so that; Δ K, equals, 800 keV;")[$"Δ" K & = & q "Δ" V #h(0.2em) "and" #h(0.2em) V_(A B) = E d , #h(0.2em) "so that" \ "Δ" K & = & 800 #h(0.2em) "keV;"$] #linebreak() 
b. #math.equation(block: false, alt: "d equals 25.0 km")[$d = 25.0 #h(0.2em) "km"$]
]


    
      Use the definition of potential difference in terms of electric field to deduce the formula for potential difference between #math.equation(block: false, alt: "r equals r sub a")[$r = r_(a)$] and #math.equation(block: false, alt: "r equals r sub b")[$r = r_(b)$] for a point charge located at the origin. Here #emph[r] is the spherical radial coordinate.


    
    
      The electric field in a region is pointed away from the z-axis and the magnitude depends upon the distance #emph[s] from the axis. The magnitude of the electric field is given as #math.equation(block: false, alt: "E equals the fraction α over s")[$E = frac(α, s)$] where #math.equation(block: false, alt: "α")[$α$] is a constant. Find the potential difference between points #math.equation(block: false, alt: "P sub 1 and P sub 2")[$P_(1) #h(0.2em) "and" #h(0.2em) P_(2)$], explicitly stating the path over which you conduct the integration for the line integral.


      #figure(figph[The figure shows two points P subscript 1 and P subscript 2 at distances a and b from the origin and having an angle phi between them.], alt: "The figure shows two points P subscript 1 and P subscript 2 at distances a and b from the origin and having an angle phi between them.", caption: none)


    
    #solutionbox[
One possibility is to stay at constant radius and go along the arc from #math.equation(block: false, alt: "P sub 1")[$P_(1)$] to #math.equation(block: false, alt: "P sub 2")[$P_(2)$], which will have zero potential due to the path being perpendicular to the electric field. Then integrate from #emph[a] to #emph[b]: #math.equation(block: false, alt: "V sub a b equals α ln open parenthesis the fraction a over b close parenthesis")[$V_(a b) = α #h(0.2em) "ln" ( frac(a, b) )$] The potential at point #math.equation(block: false, alt: "P sub 1")[$P_(1)$] is higher than at #math.equation(block: false, alt: "P sub 2")[$P_(2)$] since the field points radially outward, and this is consistent with a positive potential difference going from distance #math.equation(block: false, alt: "a")[$a$] to a larger distance #math.equation(block: false, alt: "b")[$b$].
]


    
      Singly charged gas ions are accelerated from rest through a voltage of 13.0 V. At what temperature will the average kinetic energy of gas molecules be the same as that given these ions?