Electric Energy and Capacitance
Dr Jacob Dunningham,
School of Physics and Astronomy, University of Leeds
EM-L4-1
Review of Lecture 3
• Potential difference
∆V = −
Z b
a
~
~ · dl
E
• Potential due to a point charge
kq
V =
+ V0
r
(Frequently used : V0 = 0 which implies V = 0 at r = ∞)
~ from V :
• Calculating E
∂V
Ex = −
∂x
∂V
Er = −
∂r
etc
!
e.g.
Review of Lecture 3
~ = − ∂V î + ∂V ĵ + ∂V k̂
E
∂x
∂y
∂z
EM-L4-2
Overview of Lecture 4
The plan for todays lecture
• Electrostatic energy
U =
Z Q
0
V (q) dq
• Capacitance
1 Q2
C = Q/V ; U =
2 C
• Electric field energy density
ue =
Overview of Lecture 4
1
0 E 2
2
EM-L4-3
Electrostatic Energy
EM-L4-4
Electrostatic energy: point charges
The electrostatic energy of a system of point charges is the
work needed to bring them from infinite separation to their final
positions.
e.g. If we have a point charge q1 at point 1, the potential V2 at
point 2 a distance r1,2 away is
V2 =
kq1
r1,2
The energy to bring a charge q2 from infinity to point 2 is:
kq1q2
U2 = q2V2 =
r1,2
If we brought into another charge, q3 from infinity to point 3,
the electrostatic energy would be:
kq1q3
kq2q3
kq1q2
U2 =
+
+
r1,2
r1,3
r2,3
i.e. there is a component from each pair of particles.
Electrostatic energy
EM-L4-5
Electrostatic energy: continuous charges
Electrostatic energy to add a charge element dq in the presence
of potential V (q) generated by charge distribution q.
dU = V (q) dq
U =
Z Q
0
V (q) dq
Note: Each charge element dq requires a different amount of
energy V (q) dq because the potential V (q) will normally depend
on the amount of charge already accumulated.
Electrostatic energy
EM-L4-6
Example: Uranium nucleus
Estimate the electric potential energy (in MeV) stored in a uranium nucleus with A = 235 nucleons of which Z = 92 are protons. The radius of an atomic nucleus is approximately R ≈
10−15 m · A1/3.
Electrostatic energy
EM-L4-7
Example: Uranium nucleus
Estimate the electric potential energy (in MeV) stored in a uranium nucleus with A = 235
nucleons of which Z = 92 are protons. The radius of an atomic nucleus is approximately
R ≈ 10−15 m · A1/3 .
U =
Z Q
0
V (q) dq
where
kq
V (q) ≈
R
k Q
U ≈
V (q) dq
R 0
kQ2
k 1 2 Q
=
q
=
R 2
2R
0
Z
Numbers:
R = 10−15(235)1/3 ≈ 6.2 × 10−15 m
Q = 92e ≈ 92(1.6 × 10−19) ≈ 1.5 × 10−17 C
So:
U = 1.6 × 10−10 J.
= 1020 MeV
Electrostatic energy
EM-L4-8
Capacitance
EM-L4-9
Definition: capacitance
Q
C =
V
The Capacitance C is a measure of the capacity to
store a charge Q for a given potential difference V.
In most cases the capacitance depends only on the
size and shape of the conductor.
Capacitance
EM-L4-10
Unit of capacitance
The self-capacitance of a spherical conductor is:
C=
Q
R
Q
=
=
= 4π0R
V
kQ/R
k
Unit of capacitance: 1 Farad (F) – named
after Michael Faraday*
1C
1F=
1V
The permittivity of free space, 0, is
pF
= 8.85
m
m
F
−12
0 = 8.85 · 10
The radius of a spherical conductor with 1 F capacitance is 1440
times the radius of the Earth! (c.f. camera flash ∼ 100µF)
* Faraday had a fascinating life - well worth reading about
Capacitance
EM-L4-11
Example: parallel plate capacitor
Uniform electric field E = σ/0.
σd
Q·d
V =E·d=
=
0
0 · A
Q
0 A Q
0 A
C=
=
=
V
dQ
d
Capacitance C does not depend on potential V nor charge Q.
Capacitance
EM-L4-12
Real capacitors
In practice large capacitors are often made from rolled up metallic
foils separated by a thin plastic film insulator.
Capacitance
EM-L4-13
Electric field energy density
EM-L4-14
Energy stored in a capacitor
To add a charge element dq to a partially charged capacitor
requires an energy of
q
dU = V (q) dq = dq
C
Energy to then fully charge the capacitor to final charge Q is
U =
Field Energy density
1 Q2
1
1
dq =
= QV = C V2
C
2 C
2
2
Z Q
q
0
EM-L4-15
Electric field internal energy
The energy is stored in the electric field itself!
Example: parallel plate capacitor
1
U =
C V2
2 1 0 · A
=
(E d)2
2
d
1
=
0 E 2 (A d)
2
Electric field energy density ue: energy U stored in the electric
field per volume (A d)
1
ue =
0 E 2
2
This applies generally to any electric field.
Field Energy density
EM-L4-16
Summary
• Electrostatic energy
U =
Z Q
0
V (q) dq
• Capacitance
C = Q/V
• Electric field energy density
ue =
1
0 E 2
2
Recommended reading: Tipler, sections 24-1, 24-2, 24-3
Preparation: Tipler, sections 24-4, 24-5, 24-6, 25-1, 25-2
Summary
EM-L4-17