5
Electrical Potential V
Electrical Potential Energy
Unit Charge
WAB: work done against electrical forces going from A to B
W
VB - VA ≡ AB
q
S.I. Unit: 1 Volt ≡ 1 J.C-1
V ≡
Compare with gravitational potential ≡
gravitational PE
mass
Electric potential and Field
dWagainst E = − dWE
≡ − FE ds cos θ
FE ds cos θ
dWagainst E
∴
dV =
=
−
q
q
Eq ds cos θ
= −
q
dV = − E ds cos θ
E * compt of ds // E
E
F
ds cos θ
θ
θ
q
ds
ds
dV
Conversely:
E = − dr where dr is //E
Potential due to a point charge:
V
V+dV
Q
E
dr
dV = − E ds cos θ = − E dr
dr
dV = − kQ . 2
r
r
V = − kQ
⌠
 dr

⌡ r2
choose
rref = ∞
reference
= kQ
[ 1r ]
r
r=∞
1
V(r) = kQ r
dV
check:
E = − dr where dr is //E
1
E = kQ 2
r
qi
for many
1
charges V = 4πε o Σ ri
Equipotential Surfaces have constant value of V
E
ds cos θ
θ
dV = − E ds cos θ
If dV = 0, then either E = 0 or θ = 90°
∴ E perpendicular to equipotentials.
e.g. V(r) for point charge
ds
1V
2V
3V
4V
Q
5V
E
ds cos θ
θ
ds
dV = − E ds cos θ
Corollary 1: In equipotential surface, dV = 0.
∴ E perpendicular to surface.
~
Corollary 2: Ideal conductor is equipotential.
Corollary 3: Ex = -
∂V
∂V
∂V
, Ey =
E =∂x
∂y z
∂z
Example: uniform field in x direction
dV = − E ds cos θ
E * compt of ds // E
dV = − E dx
Field is approximately uniform between // plates
+ + + + + + + + 3 V2 V
1V
0V
-1V
_________
-2V
-3V
6
7
Example: V due to dipole
-q
+q
P
x
a a
x
V(r,θ) = V+ + Vkq kq
=  x - x 
 1 2
x -x
= kq x2 x 1
1 2
2a
if x>> a
≅ kq 2
x
p
∴
V= k 2
x
Example: (cf previous derivation)
dV = − E ds cos θ
on the axis
2p
dV
E = − dx = k 3
x
All info. about vector field E (r ) is contained in scalar V(r )
~ ~
~
(Not always possible for other fields).
Capacitance
q
remember V = ∑ Vi = k ∑ r i
i i
∴, for given geometry:
V ∝ Q.
Q
∴ define
C ≡ V
S.I. Unit: Farad ≡ Coulomb/Volt
how much charge something
stores at a given potential
Example: C of isolated sphere.
kQ
Q r
V = r
∴ C≡V =k
A limited analogy with water: Q → vol, I → flow, V → P
Vol
Capacitance of tank is
pressure
What is capacitance of the earth?
C = rearth/k
6.4x106m
=
9 109 Nm2C-2
= 710 µF
1/k is small
Coulomb is big, Farad is big
Of a person, isolated in space?
C ~ r/k
~ 0.3 m
=
9 109 Nm2C-2
= 3 10-10 F = 300 pF
doesn't take much charge
to produce lots of V
Field due to large plate, far from edge
Charge Q, area A → Uniform charge density Q/A
A
+Q
E
For infinite plate, from symmetry, the field is normal to the plane, and equal
on both sides
Coulomb's law and integration gives
Q
E = 2πkA
on both sides
Field between two (large) parallel plates
+Q
A
-Q
E-
E+
+Q
-Q
d
Fields cancel outside, reinforce between them
Q
Q
E = 4πkA = ε A
o
Need A >> d2 to neglect edge effects
V = − ∫ E.ds
E is uniform,
Q
∴
V = Ed = ε A d
o
Capacitors used for storing energy (particularly for brief applications,
AC→DC), for filtering and for timing applications.
Parallel place capacitors
Circuit symbol:
ε A
Q
C ≡ V = Q. o
Qd
ε A
C = o
d
εo is 8.85 10-12 Fm-1 permittivity of space
εο = 8.85 pFm-1: to get large C, d often of molecular size
e.g. Two plates, 1 m2, separated by 1 mm.
8.85 pFm-1
= 9 10-9 F = 9 nF
1 10-3 mm
Put ± 1 microcoulomb on the plates:
Q
1 10-6
V ≡ C =
= 110 V
9 10-9
E = V/d = 110 kV.m-1
C =
8
9
Dielectrics
put insulator in elec field:
++ + + + +++ +++ +
E vac
- - - - - - - - - - - -
++ + + ++++ +++ +
+
+
+
+
+
+
+
+
+
- - + + +
E vac
E dipole
- - - - - - - - - - - -
innate or
molecular dipoles induced
Edipole oppose Evac ∴ Eresultant < Evac
E
Dielectric constant
K ≡ E vac > 1
resultant
Intrinsic dipoles, K large
e.g. Kwater = 78
Nonpolar media K smaller, but > 1 K plastic ~ 3
Parallel plate capacitor with dielectric:
Q
C=V
Q
= Ed
KQ Aε KQ
=E d= o
Qd
vac
KεoA εA
∴
C=
= d
d
where ε ≡ Kεo is permittivity of medium.
Dielectric Breakdown
For any insulator ∃ a value of E
where molecules → ions , ∴ conducts.
This value called dielectric strength (Emax)
air: ~3 MVm-1
mica:
160 MVm-1
Example Using mica, how thick would a capacitor be to withstand 10 V?
How big would it have to be if it had a value of 1 Farad?
E c → d, C → A, A & d → size
V = Ed
E < Ec
V
V
d = E > E
c
10 V
=
= 6 10-8 m = 60 nm
160 MVm-1
εA Kε A
C = d = o
d
Cd
A = Kε
o
1 F 60 nm
=
= 2000 m2
3 8.85 pF.m-1
volume of mica = Ad = 140 ml
10
Energy Stored in C:
Charge up from q=0 to q=Q
v=0 to v=V
q
dW = vdq = C dq
Qq
1 q2 Q
W = ∫ dW = ∫ C dq = C [ 2 ] o
0
∴
Q2 1
W = 2C = 2 CV2
Energy density
W
1
2. 1
volume = 2 CV Ad =
V2
but
= E2
d2
W 1
so vol = 2 εE2
εA . 2 . 1
2d V Ad
Example:
d
R
L
A nerve cell is ~ cylindrical, L = 20 mm long and R = 5 µm diameter. It has
a fatty membrane d = 4 nm thick, separating salt solutions (conductors)
inside and out. What is its capacitance? What is its capacitance per unit
area? Usually, the inside is at − 90 mV with respect to the outside. What
charge does it store? How much energy? (Take Kfat = 3)
KεoA
εA
C = d = d
Kεo2πrL
=
= 4 nF
d
|q| = CV = 400 pC
U = 12 CV2 = 30 pJ
Example:
Capacitor, A = 10m2, d = 10nm, K = 3.0, diel. strength = 10 MV m-1.
What is max W stored?
1
1
Wmax = CV2max = C (Emaxd)2
2
2
1
A
= 2 Kεo d E2 max d2
= ......
= 130 µJ
Vmax
=
Emax d = 0.1V
Example: capacitor with K = 3, diel. strength = 108 Vm-1. How big must it
be to store 1 kW.hr ?
(1 kW.hour = 1 kW. 3600 s = 3.6 MJ)
W 1
(20): vol = εE2
(~ 1 MJ.m-3)
2
2Wmax
vol =
= 3 m3
KεοEmax2
Compare with petrol. What is ∆H per litre of petrol?
We can make a rough estiemate from things that we know.
Car travels 100 km in one hour , burns 6 litres of fuel and produces 20 kW at efficiency
of 20%
20 kW 1 hr
~ 20 kWhr/litre = 20,000 kWhr/m3
6 litres 20%
Storage efficiency:
nuclear > chemical > electrochemical > electric
nuc fuel chem fuel
battery
capacitor
But energy can be stored in & obtained from capacitors very quickly, easily
and efficiently.
e.g. Flash in cameras, power regulation, timing
Capacitors in parallel
C1
q1
q2
-q 1
-q 2
C // =
∴
C2
V
q
q
V = C1 = C2
1
2
q 1 + q2
q
q
= V1 + V2
V
C// = C1 + C2
Don't compare with resistors,
Compare with conductors G ≡ 1/R
Capacitors in series
q
-q q
C1
-q q
C2
C3
V = V1 + V2 + V3
q
q
q
q
∴
=
+
+
Cser C1 C2 C3
1
1
1
1
Cser = C1 + C2 + C3
-q
11
12
Example
C 2 = 20 µF
C1 = 8 µF
10 V
C3 = 30 µF
Switch moved from left to right. What are the final values of V1 V2 V3 ?
C1 = 8 µF
10 V
Switch to left: capacitor charges until V1 = ε
V1 = 10 V
q1 = C1V1
= C1ε = = 80 µC
q'2
q'1
C1
-q'1
-q'2
q'2
-q'2
C2
C
3
Switch to right:
charges on C2 & C3 are equal.
V'1 = V'2 + V'3
Unknowns q1', q2'
V1' = V2' + V3'
Need 2 equations
(1)
Where does the charge come from? Initial charge on C1.
charge
∴ q1 = q1' + q2'
(2)
conservation
From the definition of C, V = q/C, ∴
q 1 ' q 2'
q 2'
(2) →
C1 = C2 + C3
q1-q2'
q 2' q 2 '
C1 = C2 + C3
∴ q2' = ....
q1' = q1 − q2' = .....
q3' = q2'
q1', q2', q3' ⇒ V1, V2, V3