Lecture 6 Capacitance and Capacitors Electrostatic Potential Energy Prof. Viviana Vladutescu

Capacitance A conductor in electrostatic field is equipotential and charges distribute themselves on the surface such way that E=0 inside the conductor Q on the surface is producing V

linear dependence k=C

Depends on –the geometry of the conductors -the dielectric constant of the medium between conductors Capacitance (of the isolated conducting body) - is the electric charge that is added to the body per unit increase in its electric potential (is a constant of proportionality)

Capacitors

Electrolytic capacitors

Determine capacitance 1- assume V ab Q (in terms of V ab ) use boundary conditions 2- assume Q V ab (in terms of Q)

Q V ab Step 1Chose coordinate system for given geometry Step 2 Assume +Q and –Q on the conductors Step 3 Q E from D=εE=ρ s or Step 4 E Step 5 C=Q/V ab

Example Step 1 Step 2 Step 3

Step 4 Step 5

V ab Q Step 1Chose coordinate system for given geometry Step 2 Assume V ab between plates Step 3 V ab E D (from Laplace’s equation) Step 4 Boundary conditions at each plate conductor –dielectric boundary: ρ s Q. Step 5 C=Q/V ab

Example z Step 1 Step 2 V ab Step 3 Laplace’s equation to find the potential everywhere in the dielectric

There is no φ and z variation

Step 4

Q on the inner conductor Step 5

Series Connected Capacitors Parallel Connected Capacitors

Electrostatic Potential Energy

Electric potential at a point in an electric field is the work required to bring a unit positive charge from infinity (at reference zero potential) to that point.

Now suppose we want to bring Q 3 at R 13 from Q 1 and R 23 from Q 2

-can be negative -represents only interaction energy

For a group of N discrete charges at rest For a continuous charge distribution of density ρ

Electrostatic energy in terms of field quantities Substitute ρ And by using

Electrostatic Energy Density

Equipotential surfaces are at right angles to the electric field. Otherwise a force would act and work would be done on the path A to B.For a uniform electric field, equipotentials form planes perpendicular to the field. Along AB, W = -q∆V = zero! Example