Energy Considerations in Electrical Systems
1.) Just as we decided that the force-per-unit-charge E available at a point
(whether there was a charge there to experience the force or not) is a more
primary and, hence, useful parameter than is a statement of the force on a
specific body, the potential-energy-per-unit-charge available at a point
(whether there is a charge there to experience the energy or not) is a more
useful parameter than is the potential energy of a specific body. Below is a
definition, then a summary of how energy is used in electrical systems.
The amount of potential-energy-per-unit-charge available at a point is
defined as:
U
VA = A
q
Called the electrical potential, its units are joules/coulomb, or the volt.
Note: This means if you know the voltage at a point and want to know how much
potential energy a charge q has at the point, U A = qVA.
1.)
Relationships
The work a constant force
F does on a body as the
body moves a distance d:
! !
WF = F • d
!
The work-per-unit-charge
available between two points due
to a constant electric field E:
! !
W F•d ! !
=
= E•d
q
q
Note: This means if you know the E field, the
displacement (as a vector) and the charge
experiencing the work, the work on the charge
becomes:
! !
W = qE • d
The work a variable force
F does on a body as the
body moves along a path:
! !
WF = ! F • d r
!
The work-per-unit-charge
available between two points
due to a variable electric field E:
!
! !
W
F !
= ! • dr = ! E • dr
q
q
2.)
The work a conservative
force with known potential
energy function does on a
body traveling through the
field
!
The work-per-unit-charge available
between two points due to a static
electric field (note that static electric
fields generate conservative forces):
W
"U
=!
= !"V
q
q
WF = !"U
In other words, the change in voltage is related
to how much energy a field gives or takes
away from a charge moving through the field,
as:
W = !q"V
Combining the two
ways to determine
work, we get:
! !
F
! • d r = "#U
!
Combining the two ways to
determine work-per-unitcharge, we get:
! !
E
! • d r = "#V
3.)
Notice that the relationship:
! !
! F • d r = "#U
Notice that the relationship:
! !
E
! • d r = "#V
gives us a clever way to
derive potential-energy
functions!
gives us a clever way to
derive electrical-potential
functions!
For a constant force and a
straight-line path between
two points, we can write:
! !
F • d = !"U
For a constant electric field and
a straight-line path between
two points, we can write:
! !
E • d = !"V
4.)
So if you are asked to use the conservation of energy, for instance,
and you want to know how much potential-energy a charge has when
at a point whose electrical-potential (i.e., its voltage) is V, you get the
potential-energy by using:
VA =
UA
q
!
U A = qVA
5.)
Classes of Problems
B
1.) For the electric field lines shown to the right:
D
a.) Which point(s) is/are at higher electrical
potential? E-field lines go from higher electrical potential to lower.
A
C
b.) For which point(s) would an electron have a
larger potential energy? An electron will travel from higher to lower potential energy.
As it will also
travel opposite the E-field lines, the large U happens at A or B.
c.) Where, if any, are there points with the same
electrical potential? A and B, and C and D.
d.) How much work would be required for a
positive charge to travel from point A to B?
None as the points are on an equipotential line.
e.) The voltage at A is 8 volts. The electric field
magnitude is 12 v/m. The distance between A and
C is .7 meters. What is the !voltage at C?
!
E • d = !"V
A
! = 60 o
C
6.)
2.) A 2 coulomb charge is moved through a +6 volt potential difference.
a.) How much work is done by the field in the process?
W = !q"V
b.) Is the field giving energy to the charge or removing energy from
the charge?
c.) Let’s make the charge equal to -2 coulombs. Assuming the only
force acting is that of the field AND the charge was released from rest,
1
how fast will it be going by the end of the run? ( !q ) VA = 2 mv2 + ( !q ) VA
3.) (from Mr White): Two parallel plates are charged to a voltage of 50
volts.
a.) If the separation between
the plates is .05 meters, what’s the
! !
E
•
d
=
!"V
electric field strength?
b.) What’s the acceleration of an electron in the field?
!
!
( !q ) E = ma
7.)
4.) Derive an expression for the electrical potential generated by a
point charge that is positive at some defined point. How will this differ
if the charge is negative?
5.) What is the electrical potential of:
a.) A sphere that has Q’s worth of charge shot through it uniformly;
b.) A sphere with Q’s worth of charge located at its center;
c.) Inside a sphere that is a conductor?
6.) Derive an expression for the electrical potential generated by a
group of point charges at some defined point.
8.)
7.) For a hoop of radius R with
charge Q on it (sketches provided
by master White):
a.) Derive an expression for the
electrical potential generated a
distance x units down the hoop’s
central axis.
b.) Derive an expression for the
electric field as it exists at x using
the del operator.
Go to the file on the del operator.
9.)
7.) For a hoop of radius R with charge Q on
it (sketches provided by master White):
a.) Derive an expression for the
electrical potential generated a distance x
units down the hoop’s central axis.
A differential bit of charge dq produces a differential electrical potential dV equal to:
dq
dV = k
r
dq
=k
1/2
2
(x + R2 )
Integrating yields:
V = ! dV =
=k
(x
2
k
dq
r!
Q
+ R2
)
Note that this is exactly the same field as
would have been produced by a point charge
a distance r units from the charge.
1/2
10.)
7.) For an insulating solid sphere of
radius R and with Q’s worth of charge
uniformly shot through the structure
(sketch provided by master White):
a.) Derive an expression for V(r)
where r > R (i.e., outside the structure).
b.) Derive an expression for V(r)
where R < r (i.e., inside the structure).
c.) Derive an expression for the
electric field inside the structure.
d.) What is the electric field and
electrical potential at the center of the
sphere?
e.) Graph both the electric field and electrical potential field-strengths
as a function of r for this system.
11.)