Ch. 16 Electrical Energy and
Capacitance
Concept questions: 2, 7, 8, 16
Problems: 2, 5, 7, 10, 11, 13, 16, 19,
22, 23, 25, 30, 32, 40, 47, 66
Electric potential
From physics 1, work done by a conservative
force depends only on the initial and final
positions. (path independent)
Produces Potential Energy
Coulomb force and gravitational force are both
proportional to 1/r2.
They are both conservative forces.
Gravity
FG
m1m2
G 2
r
gravitational pot. energy
Electric
FE
q1q2
k 2
r
electric pot. energy
Work and potential energy
Work done on a force to move an object is the
product of the component of the force parallel
to the displacement and the displacement.
W = F d cos
in figure 16.1
Wab = F x = qEx(xf – xi)
This is the work that moves a charge from A to B.
Work and Energy
• Work Energy theorem: Wnet = KE
• From Ch 5, the work done by a cons. force is
equal to the negative of the change in
potential energy associated with the force.
Gravity example:
yi
y
yf
W = mg y
PEg = mg (yf – yi)
For a charge in an electric field:
PE = -WAB = - qEx x
In fig. 16, as the charge moves from left to
right, positive work is done on the charge, the
charge loses some electrical potential energy.
(Note that this equation is valid when the field is constant.)
Similar to how we treated gravitational
potential energy.
Conservation of Energy
KE + PE = 0
If a charge was moving or released in an
electric field, it will gain or lose electric
potential energy. The change will be equal to
the change in kinetic energy.
field lines
+
starts here,
released from rest
x
+
KE = - PE
KE = qE x
½ mv2 = qE x
Electric Potential
Earlier we defined:
F = qE
E = F/q
Do the same thing for potential energy
PE = q V
V = PE/q
Potential Difference
Electric potential difference is defined as the change in
electric potential energy between two points, divided by
the charge.
V = Vb – Va = PE/q
SI units Joule/Coulomb, or volt
J/C = V
Potential difference is a scalar quantity.
1 Joule is needed to move 1 Coulomb of charge through a
potential difference of 1 Volt.
For uniform electric field in x-direction:
PE = - qE x
(eq. 16.1)
combined with
PE/q = V
(eq. 16.2)
Gives:
V = -E x
Potential difference (Volts) has same units as
electric field times distance (N/C) m
1 N/C = 1 V/m
If you release a positive charge from rest, it
will accelerate from regions of high potential
to low potential.
If you release a negative charge from rest, it
will accelerate from regions of low potential to
high potential.
V
x
TV tube example.
A proton is injected between two parallel plates
with a speed o 1x106 m/s. The plates are 5 cm
apart.
a) what must be the potential difference if the
proton is to exit with a speed of 3x106 m/s?
b) What is the magnitude of the electric field
between the two plates.
Page 537-538
KE + PE = KE + q V = 0
V
V
KE
q
1.67 x10 27 kg
6
2
6
2
[(
3
x
10
m
/
s
)
(
1
x
10
m
/
s
)
]
19
2(1.6 x10 C )
V = -4.18x104 V
E = - V/ x = (4.18x104 V)/(0.05m) = 8.35x105N/C
Electrical Potential fro Point Charges
The electric field of a point charge extends
throughout space. So does the electric potential.
The electric potential can be taken to be zero
anywhere. Usually picked to be infinitely far
away from the charge.
V
q
k
r
see graphs of electric field and potential as
function of distance.
If you have two or more charges, you can find
the electric potential via the superposition
principle.
Find the potential from each charge and add
up these values.
Potential is scalar so no need to worry about
vectors this time.
See example of dipole.
Potential energy of a pair of charges
If two charges are a distance r apart you can find the
potential energy of the pair. This is equivalent to
finding the work needed to bring the pair together
from infinity (very far apart).
Work = PE = q V
when far apart PE = 0
PE = PEf (when separated by r)
PE q 2V1
q1q2
k
r
or PE = q1V2
Potentials and Charged Conductors
W = - PE
PE = q(Vb – Va)
W = -q(Vb – Va)
No net work is needed to move a charge
between two points that are at the same
electric potential.
If Vb – Va= 0
Then W = 0
Remember that for a conductor in equilibrium, all the
charge is on the surface and there is no electric field
inside the conductor. The electric field is perpendicular
to the surface of the conductor.
Because the electric field is normal to the surface, it
takes no work to move a charge along the surface.
Because of this, all points on the surface of a charges
conductor in equilibrium are at the same potential.
Also since there is no charge and thus no
electric field inside the conductor:
No net work is needed to move a charge
around in the interior and from the interior to
the surface.
the electric potential is constant everywhere
inside a conductor and equal to the same
value at the surface.
Doesn’t have to be zero, just constant.
Electron Volt
Conveniently sized unit of energy.
An electron volt is the kinetic energy that an
electron gains when accelerated through a
potential difference of 1V.
1V = 1 J/C
charge of electron = 1.6x10-19 C
1eV = 1.6x10-19C V = 1.6x10-19 J
This is convenient because 1 J is a lot of
energy to give an electron.
What would happen to an electron with 1 J of
kinetic energy?
KE = ½ mv2
1 J = ½ (9.11x10-31kg)v2
v = 1.5 x 1015 m/s
The electrons would be moving about 5 million
times faster then the speed of light.
Equipotential Surfaces
A surface on which all points are at the same
potential. No work is needed to move a charge at
constant speed on an equipotential surface.
Think of them as the contour lines on a topographical map.
The electric field is always perpendicular to an
equipotential surface.
Van de graaf
Van de graaf accelerator
Using principles from chapters 15 and 16 to
accelerate charged particles.
Capacitance
capacitor – electrical device used in many circuits
that is used to store electrical energy to used
later. Consists of two conductors separated from
each other.
Example: parallel plate capacitor. Two parallel
metal plates separated by distance, d, and
connected to positive and negative terminals of a
battery. One plate loses electrons and receives a
charge of +Q. The electrons are transferred
through the battery to the other plate which
obtains a charge of - Q.
Capacitance
Capacitance of a capacitor is the ratio of the
magnitude of the charge on either conductor
to the magnitude of the potential difference
between the conductors.
Capacitance (C) = Q/ V
SI unit of capacitance: farad (F)
farad = coulomb per volt
F = C/V
The charge on a capacitor is Q = C V
Capacitance depends on the geometric
arrangement of the conductors.
For parallel plates, with surface charge
density,
C=
q
V
A
Ed
A
(
)d
A
d
0
0
for a parallel plate capacitor, the capacitance
depends only on the size of the plates and
their separation.
We could also find the capacitance of
capacitors made of concentric cylinders or
spheres. The capacitances will still be
determined by the geometry.
-Q
Concentric spheres:
+Q
b
Va-b = kQ/a – kQ/b
a
b
a
Va-b = kQ
ab
use k = 1/(4
o)
ab
C = Q/ V = 4 0
b a
Combinations of Capacitors
We can combine capacitors in different ways
to produce different values of capacitance.
symbol of capacitor
symbol for battery
+ -
combine capacitors is
parallel
series
Parallel Capacitors
The capacitors share 2 points in common. (see figure 16.17)
Both capacitors are each hooked up across the battery, so
they have the same potential difference V across them.
If each capacitor can hold charges Q1 and Q2, the total charge
stored by both is
Q = Q1+Q2
If we replaced the capacitors with an equivalent capacitor, it
must hold the same charge and have the potential difference.
Q1 = C1 V and Q2 = C2 V must add up to Q = Ceq V
Ceq V = C1 V + C2 V
V is the same for all of the capacitors, so:
Ceq= C1+ C2
For two capacitors, we found:
Ceq = C1 + C2
We can extend this to more capacitors.
Ceq = C1 + C2 + C3 +C4 + ……
The total capacitance of capacitors is parallel
is the sum of the capacitances.
Thus the equivalent capacitance is larger than
any of the individual capacitances.
(Electrical devices in parallel have the same
potential difference across each device.)
Capacitors in Series
Capacitors in series share 1 common point.
For capacitors in series, the magnitude of the
charges must be the same on all plates.
see figure 16.19
After the series capacitors are fully charged,
an equivalent capacitor must have a charge of
–Q on the plate nearest the + terminal of the
battery and a charge of +Q on the plate near
the negative terminal.
Apply V = Q/Ceq, for each capacitor:
V1 = Q/C1 and V2 = Q/C2
V = V1 + V2
Q/Ceq = Q/C1 + Q/C2
1/ Ceq = 1/C1 + 1/C2
This happens, since the potential difference
across any number of capacitors in series
equals the sum of the potential differences
across the individual capcitors.
For two capacitors
1/ Ceq = 1/C1 + 1/C2
For any number of capacitors in series:
1/Ceq = 1/C1 + 1/C2 + 1/C3 + 1/C4 + ……
The equivalent capacitance of a series
combination is always smaller than any of the
individual capacitances in the combination.
Energy Stored in a capacitor
• capacitors store electrical energy
• That energy is the same as the work required to
move charge, Q, onto the plates of the capacitor.
• When a capacitor discharges, it releases the
V
energy. (sparks)
V=Q/C
W= V Q
W=½Q V
Q
charge
Energy stored is equal to the work
Energy stored = W = ½ Q V
Using the substitution Q = C V, we can also show:
Energy stored= ½ C( V)2 = Q2/(2C)
Capacitors with dielectrics
• dielectric – an insulating material such as plastic
• can be inserted into a capacitor to change the
capacitance without changing the geometry of
the capacitor
• If dielectric completely fills the capacitor, the
capacitance is multiplied by the dielectric
constant ( )
1
•
= 1 for vacuum
• Capacitor without dielectric:
V = Q0/C0
• By inserting the dielectric between the plates,
the voltage across the plates is reduced by
factor .
V = V0/
• Because > 1, then V is less than V0
• The charge on the plates is fixed.
C
C
Q0
V
C0
Q0
V0
Q0
V0
Parallel plate capacitor
• capacitance without dielectric:
C
0
A
d
• capacitance with dielectric:
C
0
A
d
Dielectric strength
• dielectric strength – the maximum electric field
that can be produced in a dielectric before it
breaks down and begins to conduct
• For air, the dielectric strength is about 3x106 V/m
• If the electric field is increased above this value,
a spark will be thrown.
Sparks in air
dielectric strength of air is 3x106 V/m
A spark 1 meter long requires a potential
difference of 3x106 V.
A 1cm long spark requires 3x104 V
So even a small spark can involve voltages in
the thousands.