Download Potential Difference & Electric Potential

Electrical Potential Energy
• In Chapter 15, we saw that the gravitational and
electrical (Coulomb) forces have similar forms
m1m2
Fg  G 2
r
Gravity
Fe  ke
q1 q2
r2
Electrical
• This similarity also leads to a similarity between the
potential energies associated with each force
m1m2
U g  G
r
q1q2
U e  ke
r
(can be obtained directly
through calculus)
Gravity
Electrical
– 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
Electrical Potential Energy
• Comparison of Ug and Ue as a function of separation
Ue
Ue
distance: Ug
r
r
r
q1q2 < 0
– 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
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)
B)
C)
D)
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.
Electric Potential
• Electric potential is defined as the electric
potential energy per unit charge
Ue
V
q
– 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
Electric Potential
• When a point charge q moves between 2 points A
and B, it moves through a potential difference:
V  V f  Vi  VB  VA
• The potential difference is the change in electric
potential energy per unit charge: U e  qV
• 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
V  ke
q
r
Example Problem #16.17
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.
3.87 cm
2
1
Solution (details given in class):
–11.0 kV
P
3
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 = qV
• Combining these two equations: W  qV  qVB  VA 
– 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
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)
B)
C)
D)
1V
10 V
100 V
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)
B)
C)
D)
0.8 J
8J
12 J
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
Which equipotential plot best represents the
electric field pattern shown?
A)
B)
C)
D)
Plot 1
Plot 2
Plot 3
Plot 4
(PHYSLET Physics Exploration 25.1, copyright Pearson Prentice Hall, 2004)
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
Charging A Capacitor
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
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
Q
C
V
Units: C / V = Farad (F)
– “Capacity” to hold charge for a given V
– 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
A
C  0
(for plates separated by air)
d
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
If a constant electric potential is maintained
between the plates of the capacitor, what
happens to the charge on the capacitor?
A)
B)
C)
D)
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)
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:
Ceq  C1  C2  C3  
• Series Combination
– Magnitude of charge is the same on
all plates
– V (battery) = V1 + V2 + V3 + …
– Total (equivalent) 1
1
1
1




capacitance:
C
C C C
eq
1
2
3
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)
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):
(a) 2.67 mF
(b) 24.0 mC (each 8.00-mF capacitor), 18.0 mC (6.00-mF
capacitor), 6.00 mC (2.00-mF capacitor)
(c) 3.00 V (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 = VQ)
• Total work done = total energy stored:
1
1
Q2
2
E  QV  C V  
2
2
2C
• Defibrillators typically release about 1.2 kJ of stored
energy from capacitor with V ≈ 5 kV
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
A
C  k 0
– k = dielectric constant (ratio of capacitance
d
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
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