Equilibrium in electrostatic field: Earnshaw’s theorem

There are NO points of stable equilibrium in any electrostatic field!

How to prove it? Gauss’s Law will help!

P

Imaginary surface surrounding P

If the equilibrium is to be a stable one, we require that if we move the charge away from P in any direction, there should be a restoring force directed opposite to the displacement. The electric field at all nearby points must be pointing inward – toward the point P. But that is in violation of Gauss’ law if there is no charge at P.

Thomson’s atom1899

If charges cannot be held stably, there cannot be matter made up of static point charges (electrons and protons) governed only by the laws of electrostatics. Such a static configuration would collapse!

Hw quiz

An infinitesimally thin, insulating, uniformly charged horizontal sheet has a small charged object “floating” above it. If the object has mass m and charge Q, find σ, the charge per unit area on the sheet. Assume the sheet is of infinite extent.

A sphere of radius A has a charge Q uniformly spread throughout its volume. Find the difference in the electric potential, in other words, the voltage difference, between the center and a point 2A from the center.

There is a conducting spherical shell, inner radius A and outer radius B. A charge Q1 is put at the center. If you now put charge -2Q1 on the shell, find the charge density everywhere.

CapacitorsConsider two large metal plates which are parallel to each other and separated by a distance small compared with their width.

Area A

The field between plates is 0

E

VL

y

LdydyEbottomVtopVLL

y00 00

)]()([

A

QLL

A

ALbottomVtopV

000

)]()([

0AQL

V

The capacitance is:

L

A

AQLQ

V

QC 0

0

Cylindrical Capacitor

L

A

AQLQ

V

QC 0

0

Capacitors in series: ...1111

321

CCCCtot

Capacitors in parallel: ...321 CCCCtot

22

2

1

2

1Q

CCVW

faradC ][

If you reduce the amount of charge on a capacitor by a factor of 2, how does the capacitance change?

If a 4-F capacitor and an 8-F capacitor are connected in parallel, which has the larger potential difference across it? Which has the larger charge?

A parallel-plate capacitor has a capacitance of 1.0 F. If the plates are 1 mm apart, what is the area of the plates?

Most capacitors have a non-conducting material, or dielectric, between their conducting plates. When we insert an uncharged sheet of dielectric between the plates, experiments show that the potential difference decreases to a smaller value V.

dielectricwithoutV

QC

00

dielectricwithV

QC

0CC

When the space between plates is completely filled by the dielectric, the ratio

0C

CK is called dielectric constant.

Moore’s Law (1965): every 2 years the number of transistors on a chip is doubled

Smaller, Denser, Cheaper