Work and Voltage
We studied electric force and electric field…now
we can expand our discussion to include WORK
and POTENTIAL ENERGY just as we did with the
gravitational field:
Recall:
Wgrav = Fd
hi
= (mg)d
Let d = Dh, the change in height; then,
Wgrav = -mgDh
Why the minus sign?
hf
Work and Voltage
We insert the negative sign because we want the
WORK due to gravity to be POSITIVE when the
object falls.
Wgrav = -mgDh
Wgrav = -mg(hf-hi)
Wgrav = -mghf + mghi
hi
hf
Since hf is smaller than
hi, our answer is now
positive
Work and Voltage
For the exact same reason (hf is less than hi), the
object loses PE as it falls; this we expect.
Note that:
DPE = mgDh
= mg(hf – hi)
= mghf - mghi
This will be a negative answer
hi
hf
Work and Voltage
Therefore we found
W = -DPE
We can apply the same analysis to the electric
field…with one condition:
The following analysis applies only to POSITIVE
charges …
Work and Voltage
There are TWO cases to be analyzed
1.Spherical charge distribution
• Fairly specialized
+
The van de Graaf generator is
one example.
Work and Voltage
2. Uniform charge distribution
+
+
+
•Practical application: the
field inside electrical wires
metal plate
A wire may seem small to you, but if
the wire had a diameter that equaled
the distance from the Earth to the
Sun (93,000,000 mi), an electron
would be smaller than the period at
the end of this sentence.
Exposed end of wire
Work and Voltage
1. Spherical charge
distribution
Welec  Fd
2. Uniform charge
distribution
Welec  Fd
Work and Voltage
1. Spherical charge
distribution
Welec  Fd
Let d = r
Welec  Fr
2. Uniform charge
distribution
Welec  Fd
Work and Voltage
1. Spherical charge
distribution
Welec  Fd
2. Uniform charge
distribution
Welec  Fd
Let d = r
Welec  Fr
Now we can substitute in an expression for
FORCE (F)
Work and Voltage
1. Spherical charge
distribution
Welec  Fd
2. Uniform charge
distribution
Welec  Fd
Let d = r
Welec  Fr
What are two formulas for “F”?
Work and Voltage
2. Uniform charge
distribution
1. Spherical charge
distribution
Welec  Fd
Welec  Fd
Welec  qE d
Let d = r
Welec  Fr
Welec
 k q1q2
 2
 r

r

Work and Voltage
Electric Field
Force
F
k q1 q2
r
2
W = Fd
Work
k q1q2
W
r
F = |q|E
E
kq
r
2
W = qEd
Equations in red are
for uniform electric
fields
Equations in black are
for spherical charges
Work and Voltage
2. Uniform charge distribution
Given a uniform Electric
Field:
+
Welec = Fd
+
+
+A
B
for uniform electric fields only
Work and Voltage
2. Uniform charge distribution
Given a uniform Electric
Field:
+
Welec = qEd
+
+
+A
B
for uniform electric fields only
Work and Voltage
2. Uniform charge distribution
Distance is measured
from the charge source,
so B > A (unlike gravitation)
+
+
+
+A
B
Work and Voltage
2. Uniform charge distribution
Distance is measured
from the charge source,
so B > A (unlike gravitation)
+
+
+
+A
B
If we let the charge “fall” (Electric Field does work)
it will move from A to B and we get + work done:
Welec = qE(dB – dA)
Work and Voltage
2. Uniform charge distribution
If we push the charge
from B to A (recall B > A),
we get negative
work…meaning that
energy is stored.
final
+
+
+
+A
B
initial
Welec = qE(dA-dB) = negative
Work and Voltage
Work stored in Electric Field is negative.
Welec = qE(dA-dB)
Work done by the Electric Field is positive.
Welec = qE(dB – dA)
Work and Voltage
F
k q1 q2
r
2
W = Fd
E
F = qE
kq
r
2
W = qEd
Work
W
k q1 q2
r
Equations in red are
for uniform electric
fields
Equations in black are
for spherical charges
Work and Voltage
Since we previously determined that
W = -DPE
We can write:
-DPE = qEd
DPE = -qEd
When work is done, PE is lost
Work and Voltage
Now we define voltage:
DPE
DV 
q
Joules
DV 
 volts(V )
Coulomb
Voltage is the Energy available to
each coulomb of charge (it is NOT
the total energy – that’s “W”)
Work and Voltage
Why isn’t there a ‘gravitational voltage’?
DPE
Joules
DVgrav 

m
kilogram
mg (Dh)
DVgrav 
m
DVgrav  g (Dh)
What the
heck does
this mean?
Work and Voltage
Example:
For Earth, g = 9.8 N/kg. What is the
gravitational voltage at a height of 3.0 meters?
Work and Voltage
Example:
For Earth, g = 9.8 N/kg. What is the
gravitational voltage at a height of 3.0 meters?
DVgrav = gh
= (9.8 N/kg)(3.0 m)
= 29.4 Joules/kg
Meaning: every kg of mass will provide 29.4 Joules of
energy at this height
Work and Voltage
For a spherical
charge:
DPE
DV 
q
DV 
 k qq
r
q
kq
DV 
r
For a uniform
Electric field:
DVelec
DVelec
DVelec
DPE

q
 qEd

q
  Ed
Work and Voltage
F
k q1 q2
r
2
W = Fd
W
k q1 q2
r
F = qE
W = qEd
E
kq
r
2
V = -Ed
k q
V
r
W = work (J); V = voltage(Volts); E = Electric Field (N/C);
F = force (N); q = charge (Coul); d = distance(m)
Work and Voltage
So now we examine VOLTAGE for a uniform
Electric Field:
DVelec   Ed
+
+
+ High
Voltage
Low
Voltage
The negative sign means the voltage
gets more negative (decreases) as you
move away from the positive charges.
Work and Voltage
Example:
Find the voltage that
the + charge moves
through given an Electric
Field = 1000 N/C and a
distance, d = 0.10 m.
DV
+
= -Ed
= -(1000 N/C)(0.1 m)
= -100 Nm/C = -100 Joules/C
= -100 volts
+
+
+A
0.1 m
B
This tells us that
every Coulomb of
charge will lose 100
Joules of energy.
One-tenth of a
Coulomb will lose 10
Joules…etc.
Work and Voltage
We can relate Work and Voltage in the following
way:
W  qEd
W
 Ed
q
W
   Ed
q
Since DV = -Ed, we can write
W
  DV
q
W   q (DV )
Work and Voltage
F
k q1 q2
r
2
W = Fd
W
k q1 q2
r
F = qE
W = qEd
W = -qV
E
kq
r
2
V = -Ed
k q
V
r
W = work (J); V = voltage(Volts); E = Electric Field (N/C);
F = force (N); q = charge (Coul); d = distance(m)
Work and Voltage
Example 2:
+
+
A proton “falls” from
12 volts to 0 volts.
What work was done
(or what energy was
used)?
W = -q(DV)
W = -q(Vf – Vi)
W = -(1.6x10-19 C)(0 - 12 V)
W = +1.92 x 10-18 Joules
+
+A
B
High
Voltage
(12 V)
Low
Voltage
(0 V)
Work and Voltage
Example 3:
+
An electron (N.B.!)
moves from 0 to 12
volts. What energy
was used (or what
work was done)?
W = -q(DV)
+
+
A
-B
High
Voltage
(12 V)
Low
Voltage
(0 V)
W = -q(Vf – Vi)
W = -(-1.6x10-19 C)(12 - 0 V)
W = +1.92 x 10-18 Joules
The electron falls
“up”!
Work and Voltage
Other names for Voltage:
• Potential difference (NOT potential energy
difference)
• Electric potential
Work and Voltage
+
+
+
Voltage 1
Voltage 2
Voltage 3
In an electric field, the
voltage at the equal
distance from the source
is always the same: the
blue lines are called
“lines of equipotential”.
Work and Voltage
Voltage 2
For spherical sources,
the “lines of
equipotential” are also
spherical.
Voltage 1
+