Equipotential surfaces
E
σ
A
ds
B
E
V1
The potential, V, is constant
on an equipotential surface.
Electric field lines are
everywhere perpendicular
to an equipotential surface.
PHYS 1101, Winter 2009, Prof. Clarke
V2
V3
V4
Four equipotential surfaces
(black) seen in cross section are
all perpendicular to the electric
field lines (violet).
1
Clicker question 1
In the figure to the right, the potential from charge
q at point A, VA, is given by:
kq
VA =
r
q
A
r
What is the potential, VP, at point P in the figure below?
q
q
r
r
b) VP = 0
r
4kq
c) VP =
r
P
r
q
kq
a) VP =
r
q
PHYS 1101, Winter 2009, Prof. Clarke
d) need to know all the angles precisely to tell.
2
Clicker question 1
In the figure to the right, the potential from charge
q at point A, VA, is given by:
kq
VA =
r
q
A
r
What is the potential, VP, at point P in the figure below?
q
q
r
r
b) VP = 0
r
4kq
c) VP =
r
P
r
q
kq
a) VP =
r
q
PHYS 1101, Winter 2009, Prof. Clarke
d) need to know all the angles precisely to tell.
3
Clicker question 2
In the figure to the right, the potential from charge
q at point A, VA, is given by:
kq
VA =
r
q
A
r
What is the potential, VP, at point P in the figure below?
q
q
r
q
r
q
r
r
P
kq
a) VP =
r
b) VP = 0
4kq
c) VP =
r
d) need to know all the angles precisely to tell.
PHYS 1101, Winter 2009, Prof. Clarke
4
Clicker question 2
In the figure to the right, the potential from charge
q at point A, VA, is given by:
kq
VA =
r
q
A
r
What is the potential, VP, at point P in the figure below?
q
q
r
q
r
q
r
r
P
kq
a) VP =
r
b) VP = 0
4kq
c) VP =
r
d) need to know all the angles precisely to tell.
PHYS 1101, Winter 2009, Prof. Clarke
5
Clicker question 3
Point P lies at a distance z from the centre of a thin ring along its symmetry
axis. If the ring has charge q and radius a, what is the potential at point P?
y
kq
r
kq
b) VP =
cosθ =
r
kq
c) VP =
sinθ =
r
kq
d) VP =
z
a) VP =
q
a
x
r
z
θ
P
PHYS 1101, Winter 2009, Prof. Clarke
kqz
r2
kqa
r2
6
Clicker question 3
Point P lies at a distance z from the centre of a thin ring along its symmetry
axis. If the ring has charge q and radius a, what is the potential at point P?
y
kq
r
kq
b) VP =
cosθ =
r
kq
c) VP =
sinθ =
r
kq
d) VP =
z
a) VP =
q
a
x
r
z
θ
P
kqz
r2
kqa
r2
Every point around the ring is the same distance r from P.
PHYS 1101, Winter 2009, Prof. Clarke
7
V along symmetry axis of a charged disc
q
da
a
r
z
R
P
dq
PHYS 1101, Winter 2009, Prof. Clarke
k
8
A conductor in electrostatic equilibrium
–
–
+
+
–
E=0
+
–
– V = constant +
–
+ +
PHYS 1101, Winter 2009, Prof. Clarke
E
9
Capacitance of two concentric cylinders
+Q
–Q
–
b
r
a
–
–
–Q
r + +a +
–
–
+
+
+ + +
E
b –
–
+Q
–
L
Side view of a cylindrical
capacitor of length L with
charge +Q on the inside
cylinder, –Q on the outside.
PHYS 1101, Winter 2009, Prof. Clarke
End view with electric field
vectors shown. Dashed
line is a Gaussian
(equipotential) surface.
10
Spherical and parallel-plate capacitors
+Q
–Q
–Q
area A
b
a
d
+Q
half of a spherical capacitor
PHYS 1101, Winter 2009, Prof. Clarke
parallel-plate capacitor
11
Parallel-plate capacitors and dielectrics
+ + + + + + + + + +q
d
+q
E
V0 d
– – – – – – – – – –q
area A
–q
+q
Gaussian surface S
d
d
κ=1
V d
+q
+ +
+
+
+
+
+
+
+
–
–
–
–
– qp
κ>1
–q
E
+q
– +
– – +
– – +
– – +
– – – qp
dielectric κ
area A
+q
+ + + + + +
+
+
+
–
–
E0
Fe
E
– – – – – +
– – +
– – –q
area A
L
PHYS 1101, Winter 2009, Prof. Clarke
κ
+ + + + + + + + + +q
E0
P1
–
–
–
–
κ b
d P P2
E
+
+
+
+
P3
E0
– – – – – – – – – –q
area A
12
Dielectric constants of some materials
material (20°C)
dielectric
constant κ
material (20°C)
dielectric
constant κ
dry air (1 atm)
1.00054
silicon
12
polystyrene
2.6
germanium
16
paper
3.5
ethanol
25
transformer oil
4.5
water (35°C)
75.0
pyrex
4.7
water
80.4
ruby mica
5.4
titania ceramic
130
porcelain
6.5
strontium titanate
310
PHYS 1101, Winter 2009, Prof. Clarke
13