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TRANSCRIPT
Electric Potential
&
Electric Potential Energy
The force on a test charge q0 in an E-field:
Τ¦πΉπ0,πππππ = π0πΈ
To move a charge around in an electric field, work is by the field (and by an external agent).
ππππππ = βπππ₯π‘. πππππ‘
Recall: πππππππ = Τ¦πΉπ0,πππππ β π Τ¦π
Soβ¦ ππππππ = π0 π΄π΅πΈ β π Τ¦π = βΞπ
where π Τ¦π is a differential displacement along some path from Point Ato Point B.
Since Ffield is a conservative force, what does this imply about how DUdepends on the path taken from Point A to Point B?
No path dependence!
A mechanical analogy may by helpfulβ¦
Because βoppositesβ attract and βlikesβ repel, assemblies of charge possess (electric) potential energy.
Weβll see that like universal gravitation, it is often convenient to define U = 0 when the charges are β apart.
Ug
K
β
+
UE
+β
K
Defining Potential
Again, consider a mechanical analogy: Gravitational Potential
We can define gravitational potential, βGPβ, as
the βgravitational potential energy per unit mass.β
That is, πΊπ =ππ
π=
ππβ
π= πβ
Note that GP depends only on location in space. There is no gravitational potential energy until a mass m is placed at a location where the gravitational potential is nonzero.
Twice the mass, then twice the Ug and twice the work to move it.
Ten times the mass, then ten times the Ug and ten times the work.
Ug = mghm
h
Defining Electric Potential
Similarly, moving a particle with twice the charge in an electric field
requires twice as much work.
Moving a particle with ten times the charge means ten times the work.
We define Electric Potential, V, as the
the βelectric potential energy per unit charge.β That is,
π =ππΈπ0
where V is measured in (Joules/Coulomb) and 1 (J/C) β‘ 1 βVoltβ
and depends only on the location in space.
Potential Difference
DV = Vfinal β Vinitial = (DUE/q0) = (-Wfield /q0)
Ξπ = βΰΆ±π΄
π΅
πΈ β π Τ¦π
Again, since the Ffield is conservative,
DV is path independent.
Example: Uniform E-field
βπ = ππ΅ β ππ΄ = βΰΆ±π΄
π΅
πΈ β π Τ¦π = βπΈπ
βπ = π0βπ = β π0πΈπ
If q0 > 0, then DU < 0 (particle moves to a lower U.)
d πΈA B
Uniform E-field (contβd)
βπ = ππΆ β ππ΄ = βΰΆ±π΄
πΆ
πΈ β π Τ¦π = βπΈ β πβ²
= βπΈπβ²πππ π = βπΈπ
For path AβB β C, DV = βEd + βπ΅
πΆπΈ β π Τ¦π = βEd
DVAC = DVAB implies VC = VB.
d πΈA B
C
dβq
0, since πΈ β₯ πΤ¦π along path BC.
Equipotential Contours/Surfaces
Paths along which the electric potential is constant are referred to as equipotential contours.
Question:
If E points to the right,
is V higher or lower on
the left side?
Answer: V is higher on the left side.
πΈ
100 V 80 V 60 V
90 V 70 V
Question
To move an electron from 90 V to 70 V does an
external agent do positive work, negative work or
zero work?
A) Wext > 0 B) Wext < 0
C) Wext = 0 D) Not enough information.
Answer: Wext > 0
Electric Potential from a Point Charge
βπ = ππ΅ β ππ΄ = π΄βπ΅πΈ β π Τ¦π , where πΈ =
ππ
π2ΖΈπ.
Then, βπ = π΄βπ΅ ππ
π2ΖΈπ β π Τ¦π
where ΖΈπ β π Τ¦π = ππ cosπ = ππ (the projection of πΤ¦π along ΖΈπ).
βπ = π΄βπ΅ ππ
π2ππ = ππ
1
ππ΅β
1
ππ΄.
Letting rA = β, then VA = 0, and
VB = DVB = ππ
ππ΅
A
Bπ Τ¦π
ΖΈπ
ππ΄
ππ΅
q
Some things to note:
β’ For a point charge, V at a location P in space depends
only on radial the coordinate r from the charge.
β’ Electric potential superposition. So, for two or more
points charges:
ππ = π1π + π2π+ π3π+ β¦ + πππ= Οπ=1π πππ = πΟπ=1
π ππ
ππ
β’ VP is an algebraic sum rather than a vector sum. Thus,
it is generally easier to calculate V than to calculate E.
Electric Potential Energy
Now bring a q2 from β and
place it a distance r12 from q1:
ππΈ = π2π1 =ππ1π2
π12for the 2-particle system.
If instead you bring q1 from β and place it a
distance r12 from q2, thenβ¦
ππΈ = π1π2 =ππ1π2
π12(the same.)
q1
q2
r12
Now, what happens when you then
bring in a third point charge and
place it at P near the first two?
Then βπ = ππ β ππ = π3ππ = π3(π1 + π2)
The total electrical potential energy of the 3-particle system is ππ = ππ + βπ = ππ + π3π1 + π3π2 or
ππ =ππ1π2π12
+ππ1π3π13
+ππ2π3π23
To calculate the total UE, you must sum over every pair of charges. UE does not obey superposition!
q1
q2
r12
q3
r23
r13
P
Electric Potential from a Dipole
At P: ππ = Οπ=12 ππ = π+ + πβ= π
π
π++
βπ
πβ= ππ
πβ+ π+
π+πβ
For π β« π:
πβ + π+ β πcosπ
and π+πβ β π2.
Thus, π = πππcosπ
π2= k
πcosπ
π2,
since p = qd.
Where r+ = rβ (as in the case of any point in the xz-plane),
V = 0.
β
+
x
y
d
r
r+
rβ
P
q