Tuesday, February 23, 1999
Announcements: none

Lecture notes:

• At t=0, we close the switch, than electrons are removed from the top plate (by the battery) to the bottom plate.
• At an instant just after t=0, the amount of charges on the capacitor plates are q. At this instant, the potential difference across the plate is : V¹ = q/C.
• Why is ti not V? Because it is not at Q yet. (Q being the final amount of charge)
• The work necessary (supplied by the battery) to transfer an increment of charge -dq (from the top plate to the bottom plate) is dW = V¹dq = q/c dq
• This work is is needed to transfer charge dq from the bottom to the top plate.
• Therefore, the total amount of work required to charge the capacitor from q=0 to q=Q is
• W= I(integral) dW = I(Integral from 0 to Q) q/C dq = Q²/ 2C
• This work by an external source (the battery) is stored as potential energy in the capacitor
•  U = W = Q²/ 2C = (1/2)CV²
• Q = CV
• The potential energy stored in the capacitor, after the switch was closed for a long time is, ( in building up the electric field)   U = (1/2)CV²

•  U = (1/2)CV² = (1/2)(e_o A/d)V² = (1/2)(e_o A/d)E²d² = (1/2)(e_o Ad)E² (where Ad is the volume of the capacitor)
• u = U/V = (1/2) e_o E²  where u is the energy density in an electric field

• A dielectric is a nonconducting material such as paper, oil, or glass
•  When a dielectric is inserted into the capacitor, the voltage drops
• C_o = C_vacuum = Q/V
• C_dielectric = Q/V¹ = C_d
• C_d > C_o
• C_d = kC_o  where k is the dielectric constant
• Some values of k ...
• k = 1.0000 in a vacuum
• k = 1.00054 in air
• k = 3.5 for paper
• k = 80 for water
• For a parallel plate capacitor Cd = ke_oA/d
• We can be more general:
• In a region completely filled with a dielectric material, all electrostatic equations containing e_o are to be replaced by ke_o.
• e.g. For a point charge in a vacuum, E = q/ 4pe_or². In a dielectric, E¹ = E/k = q/ 4pke_or².

• If E=0; polar molecules have random orientations, nonpolar molecules has no dipole moment.
• If E does not equal 0, the dipole moment of polar molecules tends to align with the electric field. Nonpolar molecules will develop "induced" dipole moment.
• Electric field displaces negative charges from positive charges
• Induces a surface charge field.