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

221

r

qqkF ee 2

21

r

mmGFg

Gravity Electrical

r

qqkU ee

21r

mmGU g

21

Gravity Electrical

(can be obtained directly through calculus)

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

rUe

rUe

r

q1q2 < 0 q1q2 > 0

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?

A) 10 m/s B) 14 m/sC) 20 m/sD) 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.

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

q

UV e

Higher potential

Lower potential

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

ABif VVVVV

VqU e

r

qkV e

Potential vs. Potential Energy

Example Problem #16.17

Solution (details given in class):

–11.0 kV

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.

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

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)

Potentials and Charged Conductors• We know that: U = –W (from last semester) and

U = qV • 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 V = 0 inside as well)

– Conclusion: Electric potential is constant everywhere inside a conductor and is equal to its (constant) value at the surface

AB VVqVqW

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?

A) 1 V B) 10 VC) 100 VD) 1000 V

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)

A) 0.8 JB) 8 JC) 12 JD) 1200 J

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 V = 0 for each surface, W = 0 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 V (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)

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

Examples of Equipotential Surfaces

CQ 4: Interactive Example Problem:Drawing Equipotential Lines

(PHYSLET Physics Exploration 25.1, copyright Pearson Prentice Hall, 2004)

Which equipotential plot best represents the electric field pattern shown?

A) Plot 1B) Plot 2C) Plot 3D) Plot 4

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

Capacitance• For a uniform electric field, the potential difference

between the plates is (see Example Problem #16.6) V = Ed– E is proportional to the charge, and V is proportional to E

therefore the charge is proportional to V

• The constant of proportionality between charge and V is called capacitance

– “Capacity” to hold charge for a given V – 1 F is very large unit: typical values of C are F, nF, or pF

• Capacitance depends on the geometry of the plates and the material between the plates

V

QC

Units: C / V = Farad (F)

d

AC 0 (for plates separated by air)

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

CQ 5: Interactive Example Problem:Fun With Capacitors

(PHYSLET Physics Exploration 26.2, copyright Pearson Prentice Hall, 2004)

If a constant electric potential is maintained between the plates of the capacitor, what happens to the charge on the capacitor?

A) The charge gets smaller.B) The charge gets larger.C) The charge stays the same.D) The capacitor discharges.

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 V of the battery

– Qtot = Q1 + Q2 + Q3 + …– Total (equivalent) capacitance:

• Series Combination– Magnitude of charge is the same on

all plates– V (battery) = V1 + V2 + V3 + … – Total (equivalent)

capacitance:

321eq CCCC

321eq

1111

CCCC

Example Problem

Solution (details given in class):

1.8 102 C (4.0 F capacitor)

89 C (2.0 F capacitor)

Capacitors C1 = 4.0 F and C2 = 2.0 F 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.

Example Problem #16.35

Solution (details given in class):(a) 2.67 F(b) 24.0 C (each 8.00-F capacitor), 18.0 C (6.00-F

capacitor), 6.00 C (2.00-F capacitor)(c) 3.00 V (each capacitor)

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.

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 (W = VQ)

• Total work done = total energy stored:

• Defibrillators typically release about 1.2 kJ of stored energy from capacitor with V ≈ 5 kV

C

QVCVQE

22

1

2

1 22

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– = 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 and dielectric strength for

various materials

d

AC 0

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 V

– Since plates can store more charge for a given voltage, the capacitance must increase (remember C = Q / V )

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