1. Electrical Potential Energy
In Chapter 15, we saw that the gravitational and electrical (Coulomb) forces have similar forms
This similarity also leads to a similarity between the potential energies associated with each force
Ue depends on magnitude and sign of a pair of charges
Ue is positive (negative) when q1 and q2 have the same (opposite) sign
Remember: potential energy is a scalar quantity
Gravity
Electrical
(can be obtained directly through calculus)
Gravity
Electrical

2. Electrical Potential Energy
Comparison of Ug and Ue as a function of separation distance:
If 2 charges have opposite (same) signs, the potential energy of the pair increases (decreases) with separation distance
Charges always move from high to low potential energy
Positive (negative) charges move in the same (opposite) direction as the electric field Ug Ue Ue r r r
q1q2 < 0
q1q2 > 0

3. CQ 1: A positively charged particle starts at rest 25 cm from a second positively charged particle which is held stationary throughout the experiment. The first particle is released and accelerates directly away from the second particle. When the first particle has moved 25 cm, it has reached a velocity of 10 m/s. What is the maximum velocity that the first particle will reach?
10 m/s
14 m/s
20 m/s
Since the first particle will never escape the electric field of the second particle, it will never stop accelerating, and will reach an infinite velocity.

4. Electric Potential
Electric potential is defined as the electric potential energy per unit charge
Scalar quantity with units of volts (1 V = 1 J/C)
Sometimes called simply “potential” or “voltage”
Electric potential is characteristic of the field only, independent of a test charge placed in that field
Potential energy is a characteristic of a charge-field system due to an interaction between the field and a charge placed in the field
When a positive (negative) charge is placed in an electric field, it moves from a point of high (low) potential to point of lower (higher) potential
Higher potential
Lower potential

5. Electric Potential
When a point charge q moves between 2 points A and B, it moves through a potential difference:
The potential difference is the change in electric potential energy per unit charge:
The electric force on any charge (+ or –) is always directed toward regions of lower electric potential energy (just like gravity)
For a positive (negative) charge, lower potential energy means lower (higher) potential
Helpful detail: E points in the direction of decreasing V
Electric potential created by a point charge:
Depends only on q and r
Potential vs. Potential Energy

6. Example Problem #16.17 P
3.87 cm 1 2 3
The three charges shown in the figure are at the vertices of an isosceles triangle. Let q = 7.00 nC, and calculate the electric potential at the midpoint of the base.
Solution (details given in class):
–11.0 kV

7. Potential Differences in Biological Systems
Axons (long extensions) of nerve cells (neurons)
In resting state, fluid inside has a potential that is –85 mV relative to the fluid outside (due to differences in +/– ion concentrations)
A nerve impulse causes the outer membrane to become permeable to + Na ions for about 0.2 ms
This changes polarity of inside fluid to +
Potential difference across cell membrane changes from about –85 mV to +60 mV
Restoration of resting potential involves the diffusion of K and pumping of Na ions out of cell (“active transport”)
As much as 20% of the resting energy requirements of the body are used for active transport of Na ions

8. Potential Differences in Medicine
Polarity changes across membranes of muscle cells
Muscle cells have a layer of – ions on the inside of the membrane and + ions on the outside
Just before each heartbeat, + ions are pumped into the cells, neutralizing the potential difference (depolarization)
Cells become polarized again when the heart relaxes
Electrocardiogram (EKG)
Measures potential difference between points on chest as a function of time
Polarization and depolarization of cells in heart causes potential differences that are measured by electrodes
Electroencephalogram (EEG) and Electroretinogram (ERG)
Measures potential differences caused by electrical activity in the brain (EEG) and retina (ERG)

9. Potentials and Charged Conductors
We know that: DU = –W (from last semester) and DU = qDV
Combining these two equations:
No work is required to move a charge between two points at the same electric potential
For a charged conductor in equilibrium:
No work is done by E if charge is moved between points A and B
Since W = 0, VB – VA = 0 at surface
Since E = 0 inside a conductor, no work is required to move a charge inside conductor (thus DV = 0 inside as well)
Conclusion: Electric potential is constant everywhere inside a conductor and is equal to its (constant) value at the surface

10. CQ 2: Two charged metal plates are placed one meter apart creating a constant electric field between them. A one Coulomb charged particle is placed in the space between them. The particle experiences a force of 100 Newtons due to the electric field. What is the potential difference between the plates?
1 V
10 V
100 V
1000 V

11. CQ 3: How much work is required to move a positively charged particle along the 15 cm path shown, if the electric field E is 10 N/C and the charge on the particle is 8 C? (Note: ignore gravity)
0.8 J
8 J
12 J
1200 J

12. Equipotential Surfaces
An equipotential surface has the same potential at every point on the surface
Similar to topographic map, which shows lines of constant elevation
Since DV = 0 for each surface, W = along the surface
Thus electric field lines are perpendicular to the equipotential surfaces at all points
E points in the direction of the maximum decrease in DV (E points from high to low potential)
Similar to a topographic contour map (slope is steepest perpendicular to lines of constant elevation)
Electric field is thus sometimes called the potential gradient (meaning grade or slope)

13. Equipotential Surfaces
On a contour map a hill is steepest where the lines of constant elevation are close together
If equipotential surfaces are drawn such that the potential difference between adjacent surfaces is constant, then the surfaces are closer together where the field is stronger

14. Examples of Equipotential Surfaces

15. CQ 4: Interactive Example Problem: Drawing Equipotential Lines
Which equipotential plot best represents the electric field pattern shown?
Plot 1
Plot 2
Plot 3
Plot 4
(PHYSLET Physics Exploration 25.1, copyright Pearson Prentice Hall, 2004)

16. Capacitance
A capacitor is a device that stores electrical potential energy by storing separated + and – charges
2 conductors separated by vacuum, air, or insulation
+ charge put on one conductor, equal amount of – charge put on the other conductor
A battery or power supply typically supplies the work necessary to separate the charge
Simplest form of capacitor is the parallel plate capacitor
2 parallel plates, each with same area A, separated by distance d
Charge +Q on one plate, –Q on the other
If plates are close together, electric field will be uniform (constant) between the plates
Charging A Capacitor

17. Capacitance
For a uniform electric field, the potential difference between the plates is (see Example Problem #16.6) DV = Ed
E is proportional to the charge, and DV is proportional to E  therefore the charge is proportional to DV
The constant of proportionality between charge and DV is called capacitance
“Capacity” to hold charge for a given DV
1 F is very large unit: typical values of C are mF, nF, or pF
Capacitance depends on the geometry of the plates and the material between the plates
Units: C / V = Farad (F)
(for plates separated by air)

18. Capacitors in Circuits and Applications
Capacitors are used in a variety of electronic circuits
Example of “circuit diagram” consisting of capacitors and a battery shown at right
Many practical uses of capacitors
Some computer keyboard keys have capacitors with a variable plate spacing below them
Microphones using capacitors with one moving plate to create an electrical signal
Constant potential difference kept between plates by a battery
As plate spacing changes, charge flows onto and off of plates
The moving charge (current) is amplified to generate signal
Tweeters (speakers for high-frequency sounds) are microphones in reverse
Millions of microscopic capacitors used in each RAM computer memory chip
Charged and discharged capacitors correspond to 1 and 0 states

19. CQ 5: Interactive Example Problem: Fun With Capacitors
If a constant electric potential is maintained between the plates of the capacitor, what happens to the charge on the capacitor?
The charge gets smaller.
The charge gets larger.
The charge stays the same.
The capacitor discharges.
(PHYSLET Physics Exploration 26.2, copyright Pearson Prentice Hall, 2004)

20. Combinations of Capacitors
Capacitors can be combined in circuits to give a particular net capacitance for the entire circuit
Parallel Combination
Potential difference across each capacitor is the same and equal to DV of the battery
Qtot = Q1 + Q2 + Q3 + …
Total (equivalent) capacitance:
Series Combination
Magnitude of charge is the same on all plates
DV (battery) = DV1 + DV2 + DV3 + …
Total (equivalent) capacitance:

21. Example Problem
Capacitors C1 = 4.0 mF and C2 = 2.0 mF are charged as a series combination across a 100–V battery. The two capacitors are disconnected from the battery and from each other. They are then connected positive plate to positive plate and negative plate to negative plate. Calculate the resulting charge on each capacitor.
Solution (details given in class):
1.8  102 mC (4.0 mF capacitor)
89 mC (2.0 mF capacitor)

22. Example Problem #16.35
Find (a) the equivalent capacitance of the capacitors in the circuit shown, (b) the charge on each capacitor, and (c) the potential difference across each capacitor.
Solution (details given in class):
2.67 mF
24.0 mC (each 8.00-mF capacitor), 18.0 mC (6.00-mF capacitor), 6.00 mC (2.00-mF capacitor)
3.00 V (each capacitor)

23. Energy Stored in a Charged Capacitor
It’s easy to tell that a capacitor stores (releases) energy when it charges (discharges)
The energy stored by the capacitor = work required to charge the capacitor (typically performed by a battery or power supply)
As more and more charge is transferred between the plates, the charge, voltage, and work done by battery increases (DW = DVDQ)
Total work done = total energy stored:
Defibrillators typically release about 1.2 kJ of stored energy from capacitor with DV ≈ 5 kV

24. Capacitors with Dielectrics
A dielectric is an insulating material
Rubber, plastic, glass, nylon
When a dielectric is inserted between the conductors of a capacitor, the capacitance increases
Capacitance increases for a parallel-plate capacitor in which a dielectric fills the entire space between the plates
k = dielectric constant (ratio of capacitance with dielectric to capacitance without dielectric)
For any given plate separation d, there is a maximum electric field (dielectric strength) that can be produced in the dielectric before it breaks down and conducts
See Table 16.1 for values of k and dielectric strength for various materials

25. Capacitors with Dielectrics
The molecules of the dielectric, when placed in the electric field of a capacitor, become polarized
Centers of positive and negative charges become preferentially oriented in the field (see figure below at left)
Creates a net positive (negative) charge on the left (right) side of the dielectric (see figure below at right)
This helps attract more charge to the conducting plates for a given DV
Since plates can store more charge for a given voltage, the capacitance must increase (remember C = Q / DV )

26. Capacitors with Dielectrics
To increase capacitance while keeping the physical size reasonable, plates are often made of a thin conducting foil that is rolled into a cylinder
Dielectric material is sandwiched in between
High-voltage capacitor commonly consists of interwoven metal plates immersed in silicone oil
Very large capacitances can be achieved with an electrolytic capacitor at relatively low voltages
Insulating metal oxide layer forms on the conducting foil and serves as a (very thin) dielectric