The van de Graaff
Electrostatic Generator
Credit: Edward Hayden, Feb 17, 2010.
A willing honours student touches the holds
on to the metal sphere. The charge makes
his hair repel and stand up.
1
Conductors:
Electric Potential and Charge
A metal sphere (radius r1) has an electrical
potential that is equal everywhere on its surface
How to relate electric field, electric potential
and electric charge?
radius R
Gaussian
surface
1. Closed surface : begin with Gauss' Law
Imagine drawing a Gaussian surface just outside
the sphere
2. From Gauss' Law the flux is:
Metal sphere
3. The electric field is:
E =
E=
1 Q
4 0 R 2
. d s= 1 Q
4. The electric potential is: V = E
4 0 R
R
Q
=E 4 R 2 0
V=
1 Q
4 0 R
2
Conductors:
Electric Potential and Charge
A big metal sphere and a small metal sphere in
electrical contact (e.g. with a wire joining them) will
be at the same electrical potential V.
radius r1
radius r2
For the big sphere:
For the small sphere:
1 Q1
V=
4 0 r 1
1 Q2
V=
4 0 r 2
The big sphere has more charge than the small one.
Q1 Q2
=
r1 r2
3
Conductors:
Electric Potential and Electric Field
A big metal sphere and a small metal sphere in
electrical contact (e.g. with a wire joining them) will
be at the same electrical potential V.
radius r1
radius r2
For the big sphere:
1 Q1
V=
4 0 r 1
1 Q1 V
=
E 1=
2
4 0 r 1 r 1
The field on the big sphere is smaller !
The smaller the radius of curvature the higher the field.
Hence, lightning rods are made sharp.
For the small sphere:
1 Q2
V=
4 0 r 2
V
E 2=
r2
E 1 r 1 =E 2 r 2
4
Conductors:
Electric Potential and Electric Field
Why the spark ?
Air is normally a very good electric insulator.
But when the electric field is high, this induces
breakdown (called “dielectric breakdown”) of the
air. The “fresh air” smell during a thunderstorm
comes from ozone after dielectric breakdown.
Earth ground
Negative
Electric
potential
The left sphere is the “lightning rod” :
the smaller it is, the easier it is to generate sparks.
5
Electric Potential due to Continuous Charge
Distributions
Lets return to non-conducting objects.
We have considered electric potential due to a point charge
and in a uniform electric field.
How about a distribution of charges ?
For a continuous distribution of charges,
consider an infinitesimal (very small) charge dq
dq
Volume V
dV =
r
dq
1 dq
=k e
4 0 r
r
P
V =k e V
dq
r
6
Electric Potential due to Continuous Charge
Distributions: A uniformly charge ring.
Example 20.5 in book. Do in class
7
Electric Potential due to Continuous Charge
Distributions: A uniformly charge sphere.
V outside
Radius R
D
C
r
V =k e
Q
r
B
r/ R
V inside
V =k e
Q
3r / R2 2R
r/ R
V sphere
Note: this different from a conductor,
where the charge inside is zero!
[Example 20.6 in book. Do in class]
r/ R
8
Capacitance
Connect two conducting surfaces to the two terminals of a battery.
There will be charge buildup of equal magnitude and opposite sign on the two
surfaces.
Now disconnect the battery.
The terminals stay charged. This capacity to store charge is called capacitance.
-Q -
V
+
+
+
+
+ +Q
+
+
+
Capacitance is defined as:
Q
C=
V
Reading: All of Chapter 20.
9
Capacitance: The Parallel Plate Capacitor
We found the electric field between parallel charged plates was constant :
E=
0
or
E=
Q
0 A
Then the potential difference is
Qd
Q d
dx=
V =V B V A =A E . d s =
0 A 0
0 A
B
-Q -
A
B
x=0
x=d
d
V
+
+
+
+
+ +Q
+
+
+
The capacitance is:
C=
A
Q
= 0
V
d
Reading: All of Chapter 20.
10