Chapter 17 Electric Potential and Electric Energy; Capacitance
17.1 Electric Potential and Potential Difference
Electric Potential Energy – the amount of energy a charge has.
If we measure the change in electric potential energy, we measure how much negative work is done when a
charge moves from one point to another.
A positive charge or negative charge has its largest potential energy near its source (usually a positive or
negative plate). As it flows, the potential energy is transferred into kinetic energy.
Electric Potential – the potential energy per unit charge (measured in Volts (symbol, V))
V = PE/q
Potential Difference – the difference in potential energy per charge between two points
Voltage – another term used to describe potential difference.
The change in potential energy is equal to the charge times the potential difference.
ΔPE = qV
If we look at electric potential energy in comparison to gravitational potential energy, the potential difference
would correspond to the height of the object and the charge would correspond to the mass of the object.
If we talk about a battery, we can calculate how much charge an object requires when using the battery. For
example, if someone leaves their headlights on, the headlights may require 5 Coulombs of charge over a certain
amount of time. We can use what we learned about energy to calculate how much energy that drains from the
battery. The charge is 5C and the battery is a 12V battery, so 5C*12V = 60 Joules of energy. This is the
amount of potential energy drained from the battery. This is why an alternator is crucial in a car. He recharges
the charge within a battery, so that the battery isn’t drained every time it’s used.
17.2 Relation Between Electric Potential and Electric Field
We know that in the case of two parallel plates, the gain or loss of potential energy requires work. W = qV =
Fd = qEd
If we simplify, we can say that V = Ed or E = V/d
The electric field in a given direction at any point in space is equal to the rate at which the electric potential
changes over distance in that direction. The field lines are drawn and represented by the amount of change in
potential energy per charge divided by the amount of distance. This describes why the field lines follow certain
patterns when they are drawn.
E = V/d (Electric field formula for a uniform field)
17.3 Equipotential Lines
Equipotential lines – lines drawn (always perpendicular to the field lines). Each line drawn represents an area
around the charge with the same amount of electric potential.
Think of a side profile of a mountain. As the height of the mountain increases, the gravitational potential
energy increases.
As the elevation of the mountain increases, we can place lines along points of same elevation. You can see that
the lines on the topography map change by 100 feet. This means that every time the elevation of the mountain
increases by 100 feet a new line of elevation is drawn.
It’s the same way with electrical potential energy. If a point charge (proton) is placed inside of a field, it
requires more potential energy to move closer to a positive charge, and loses potential energy if it moves in the
direction of a negative charge. Consider and electric dipole (one positive charge next to one negative charge).
If we look at the voltage like elevation, then we can make horizontal circles at each given elevation (see below).
The voltage at +Q is infinite (has infinite limit) and –Q is negative infinite.
If we draw a top view (topographic view), we get the plot below. Once again if electric potential is like
elevation, then the green lines around the positive charge would be coming out of the paper (mountain) and the
green lines around the negative charge would be going into the paper (canyon). The straight green line directly
between the charges is where the voltage is equal to zero.
Now if we draw red lines perpendicular to the green lines, then the electric field takes shape. Note that the field
lines always go from high electric potential to low electric potential (out of the positive (top of mountain) and
into the negative (bottom of canyon)).
Now let’s consider a parallel plate capacitor. If we have one positive plate charged with 50 Volts and the
negative plate is charged with -10 Volts, then we could draw a equipotential line every 10 Volts. The electric
field lines would be perpendicular to the equipotential lines. Remember that for a uniform field (like a parallel
plate capacitor) the field strength is equal to the voltage/distance E = V/d
We can compare the voltage and distance from each plate, plot V vs d and the slope will equal E.
17.4 The Electron Volt, a Unit of Energy
Electron Volt (eV) – since a joule is large unit and the energy of one electron is so small, scientists came up
with the electron volt to talk about energy at a small scale.
1 eV = (1.6 x 10-19C)*(1.0V) = 1.6 x 10-19J
17.5 Electric Potential Due to Point Charges
This section is difficult to teach without the use of calculus. Since this is an algebra based course, we can
examine this from a conceptual standpoint. We’ll leave out the calculus derivation, but explore how we get
from one equation to the next.
What if we want to find the electric potential at a distance R from a single point charge +Q? We could place a
positive point charge (proton) in the field a distance R from the charge +Q. Because of the potential at that
point, the charge would move from position R away from charge +Q towards an infinite radius (this is the
assumption in calculus). The reason calculus chooses an infinite radius is because the electric potential and
electric field strength at infinite is zero. If we calculate the potential difference between a point charge at R and
r = ∞, then we’re left with the potential at R because the potential at r = ∞ is zero.
Now, to get around the calculus, we will use some incorrect physics to attempt to describe what’s going on.
Recall the relationship we found between electric potential and electric field for a parallel plate capacitor.
V=Ed is the equation we used, but there’s one fundamental problem with looking at this equation (here comes
the bad physics); this equation is used only for uniform fields. A point charge does not have a uniform electric
field. If you move closer or farther from the point charge, the field will get stronger or weaker, respectively.
Here’s the big BUT… if we pretend that d represents the radius R as presented above, we’d have the equation
V=ER. Now we also know the field strength is E=kQ/r2. If we substitute this equation into the electric
potential equation, then we get…
V = kQ/r = Q/(4π єor) (potential for a single point charge)
If we take the area under the curve of the graph below, we’ll get the potential difference. If we take the radius
to infinite where the field is zero, the area will represent the electric potential of the field at radius r.
As a footnote in this section Giancoli mentions a very important relationship about potential energies between
point charges. If we combine ΔPE = qV and V = kQ/r, we get…
PE = kQ1Q2/r (potential energy between two point charges)
17.7 Capacitance
Capacitor – a device that can store electric charge, and consists of two conducting objects (usually plates or
sheets) placed near each other but not touching. One plate obtains a positive charge and the other obtains a
negative charge (this creates a potential difference).
They are used to store up energy for later use.
Example: In our computers, RAM or memory is simply a capacitor that remembers ones and zeroes (binary
code) for later use.
Sometimes the plates are parallel to one another and other times the plates are placed next to each other with an
insulator in between and then rolled.
Symbol for capacitor -
and the unit is a farad (F)
The amount of charge on a plate can be calculated by Q = CV where C represents the capacitances of the
capacitor or the amount of charge it can hold per volt.
C = єo(A/d)
parallel- plate capacitor
where A is the area of the plate, d is the distance between the plates, and єo is the permittivity of free space or
8.85 x 10-12 C2/Nm2
17.8 Dielectrics
Dielectric – an insulating sheet (such as paper or plastic) placed between the two plates of a capacitor.
Why would we use an insulating sheet? There are a few reasons. The first is that we can apply a larger voltage
to the capacitor without charge leaking across the field (there’s an insulator in the way). Plates can be placed
closer together without touching which decreases d and increases the capacitance. Finally, the insulating
surface can become polar. This provides extra storage space for charge. The side of the insulator near the
positive plate becomes negatively charged, and the opposite for the negative side. See diagram below:
If a dielectric is placed in the field of a capacitor, then we have a dielectric constant, K, to describe the increase
in capacitance. The K value varies depending on the insulating material. Look on a table for the K value if
there is a dielectric in the capacitor. For a parallel plate capacitor, the equation is
C = Kєo(A/d)
parallel- plate capacitor with dielectric
17.9 Storage of Electric Energy
A charged capacitor stores electric energy and the energy stored in that capacitor is equal to the work done on
the capacitor. When a battery is hooked up to a capacitor, it doesn’t charge instantly. It takes time. The charge
doesn’t necessary want to move to one plate, so it requires work to do so. The more charge that’s on that plate,
the more work is required because the incoming charge wants to be there less and less. In the end the amount of
work is equal to the charge on the plate times the voltage across the capacitor (W=QV). This is not the equation
we use to describe a capacitor charging because as charge is being added to the plate, the voltage is increasing.
If the voltage starts at zero, and ends at some value, we must use an average voltage to calculate the amount of
work done on the capacitor. We can simply take the final voltage and divide by 2 to get the average.
W = QVf/2
Since the work is equal to the amount of energy stored in the capacitor, we can now say…
U = potential energy = 1/2QV
where V is the potential difference and Q is the charge on each plate.
Since Q = CV, we can rewrite the equation above as…
U = 1/2QV = 1/2CV2 = 1/2Q2/C (Energy stored in a capacitor)
This is a graph of a capacitor charging and being discharged. Observe how the potential changes as it charges.