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The Electric Potential
28.1
Electric Potential Energy
A) Items to recall
A) Conservative forces
(1) Allow us to figure changes in energy levels
B) Any change in position is final minus initial
C) Work can be calculated by Force times distance, but what if force is non-constant
B) Uniform Fields
A) Our first analogy to gravitation
(1) The gravitational field near the surface of the earth is uniform 𝑔 = 9.8 m/s2
(2) An object moving in the direction of the gravitational field looses energy based on
its mass, the field strength, and the distance moved
(3) 𝑈𝑔𝑟𝑎𝑣 = 𝑈0 + 𝑚𝑔𝑦
B) Charges moving in an electric field will have similar losses and gains in electric potential
energy
(1) ∆𝑈𝑒𝑙𝑒𝑐 = 𝑈𝑓 − 𝑈𝑖 = −𝑊𝑒𝑙𝑒𝑐 (𝑖 → 𝑓) = 𝑞𝐸𝑠𝑓 − 𝑞𝐸𝑠𝑖
(2) 𝑈𝑒𝑙𝑒𝑐 = 𝑈0 + 𝑞𝐸𝑠
C) Decreases and Increases in Potential Energy
(1) Negative and positive charges influence gain or loss
(2) Direction of motion with or against field influence gain or loss
28.2
The Potential Energy of a Point Charge
A) Establishing the equation
𝑥
𝑥 𝐾𝑞1 𝑞2
𝑑𝑥
𝑥2
𝑖
A) 𝑊𝑒𝑙𝑒𝑐 = ∫𝑥 𝑓 𝐹1on2 𝑑𝑥 = ∫𝑥 𝑓
𝑖
B) ∆𝑈𝑒𝑙𝑒𝑐 = 𝑈𝑓 − 𝑈𝑖 = −𝑊𝑒𝑙𝑒𝑐 (𝑖 → 𝑓) =
C) 𝑈𝑒𝑙𝑒𝑐 =
𝐾𝑞1 𝑞2
𝑥
=
−1 𝑥𝑓
| =
𝑥 𝑥𝑖
𝐾𝑞1 𝑞2
𝐾𝑞 𝑞
− 𝑥1 2
𝑥𝑓
𝑖
= 𝐾𝑞1 𝑞2
−
𝐾𝑞1 𝑞2
𝑥𝑓
+
𝐾𝑞1 𝑞2
𝑥𝑖
1 𝑞1 𝑞2
4𝜋𝜖0 𝑟
The choice 𝑈0 = 0 is equivalent to saying that the potential energy of two charged
particles is zero when they are infinitely far apart
- Potential energy for two like charges is positive and for two opposite charges is
negative, the positive and negative charges want to be next to one another.
- The equation also works for spheres of charge since all the charge can be considered at
the center of the sphere
- “Far apart” or “far away” can mean 𝑈𝑒𝑙𝑒𝑐 ≈ 0
B) Electric Force Is a Conservative Force
A) Work done is independent of the path taken to get from initial to final position
C) Multiple Point Charges
-
A) 𝑈𝑒𝑙𝑒𝑐 = ∑𝑖<𝑗
28.3
𝐾𝑞𝑖 𝑞𝑗
𝑟𝑖𝑗
, stipulation 𝑖 < 𝑗, describes that each charge is counted only once
The Potential Energy of a Dipole
A) An electric dipole placed in a field has the opportunity to begin moving, rotationally, it
has potential energy
A) this was discussed previously when the torque on the dipole was found. 𝜏 = 𝑝⃑ × 𝐸⃑⃑
B) 𝑑𝑊𝑒𝑙𝑒𝑐 = −𝑝𝐸 sin 𝜙𝑑𝜙
𝜙
C) 𝑊𝑒𝑙𝑒𝑐 = −𝑝𝐸 ∫𝜙 𝑓 sin 𝜙 𝑑𝜙 = 𝑝𝐸 cos 𝜙𝑓 − 𝑝𝐸 cos 𝜙𝑖
𝑖
D) Δ𝑈dipole = 𝑈𝑓 − 𝑈𝑖 = −𝑊𝑒𝑙𝑒𝑐 (𝑖 → 𝑓) = −𝑝𝐸 cos 𝜙𝑓 + 𝑝𝐸 cos 𝜙𝑖
E) 𝑈dipole = −𝑝𝐸 cos 𝜙 = −𝑝⃑ ⋅ 𝐸⃑⃑
F) Torques, we want to find the force perpendicular to lever arm; Work, we want to find
force parallel to displacement
28.4
The Electric Potential
A) Property of a source charge
A) A source charge will create an electric field
B) Placing another charge in that field will allow
(1) There to be a force on both the charges
(2) There to be a potential energy established
C) The amount of electric potential energy per charge in the field
𝑈
𝑉≡
𝑞
Joule
D) Units are Coulomb = Volt
B) Using the Electric Potential
A) Often the difference in two electric potentials is more important than the value of a
single electric potential
(1) ∆𝑉 = 𝑉𝑓 − 𝑉𝑖
(2) This is called electric potential difference or voltage
B) Conservative forces are at work, so conservation of energy can help to determine many
factors about the motion of a charge through a potential difference
(1) 𝐾𝑓 + 𝑞𝑉𝑓 = 𝐾𝑖 + 𝑞𝑉𝑖
(2) Table 28.2
C) Example 28.6
28.5
The Electric Potential Inside a Parallel-Plate Capacitor
- Normally begin with values relative to a point charge, electric force, electric field,
electric flux, however, Electric Potential studies begin with a capacitor because the
electric field between the plates is uniform
I) Electric Potential Energy of the Capacitor
A) 𝑈𝑒𝑙𝑒𝑐 = 𝑈0 + 𝑞𝐸𝑠, where 𝑈𝑜 = 0
1) 𝑈𝑒𝑙𝑒𝑐 = 𝑞𝐸𝑠, ∴
2) 𝑉 = 𝐸𝑠
a) The electric potential exists at all points between the plates
b) Electric potential increases linearly from negative plate
c) 𝑠 is the distance from the negative electrode
B) Potential difference of a capacitor
1) ∆𝑉𝐶 = 𝑉+ − 𝑉− = 𝐸𝑑
2) Rearranging the equation allows us to get 𝐸 =
∆𝑉𝑐
,
𝑑
allowing us to discover new units
for electric field
a) 1 N/C = 1 V/m
C) The electric field within a capacitor is constant, but not the potential
𝑠
1) 𝑉 = 𝐸𝑠 = 𝑑 ∆𝑉𝐶
II) Equipotential Surfaces
A) Lines that detail points of like
potential
B) Analogous to lines on
topographical map
C) Lines of e.p. will be
perpendicular to electric field
lines
D) Usually established in natural
segments of voltage, i.e. 5V,
10V, 15V or 2V,4V,6V
III) Battery establishes the Charge on the Plates
28.6
I)
The Electric Potential of a point Charge
Establishing the equation
1 𝑞′𝑞
4𝜋𝜖0 𝑟
𝑈𝑞′ & 𝑞
1 𝑞
=
𝑞′
4𝜋𝜖0 𝑟
A) 𝑈𝑞′ & 𝑞 =
B) 𝑉 =
II)
Visualizing the Potential of a Point Charge
III)
**Note both similarities and differences of this figure and the drawings done in the special
section “Graphical representation of the electric potential inside a capacitor”**
The Electric Potential of a Charged Sphere
𝑄
A) 𝑉 = 𝐾 𝑟 , (sphere of charge, 𝑟 ≥ 𝑅)
B) 𝑉 =
28.7
I)
𝑅
𝑉
𝑟 0
(sphere charged to potential 𝑉0 )
The Electric Potential of Many Charges
Superposition
1
A) 𝑉 = ∑𝑖 4𝜋𝜖
II)
0
𝑞𝑖
𝑟𝑖
B) Principle of Superposition says the total is the sum of the individuals, benefit that
potential is a scalar quantity
Continuous Distribution of Charge
A) PSS 28.2