Thursday, February 19, 1999
Announcements:
none
Lecture notes:
These notes are a continuation of last lecture's discussion of dielectrics in an electric field.
 

• The electric field induced by the dielectric points in the opposite direction of the electric field due to the parallel plate capacitor.

• E_o is the applied field
• E_i is due to the induced dipole moment.
• E is the net field
• Net field: E = E_o + E_i

• Gauss' Law for Vacuum: E_o = q/e_o A
• Gauss' Law with a dielectric E = q- q_i/e_o A where q_i is induced charge
• We know that by inserting a dielectric into a capacitor,  E = E_o /k = q/ ke_o A = q - q_i/e_o A
• Therefore  q- q_i = q/k
• e_o I (integral) k E (dot product) dA = q
• i.e. in a dielectric replace e_o  with ke_o

• Consider the case when a battery is connected in a loop and conducting wire

• The battery will continuously move the charge around the loop
• We can say that a constant i  is being set up in the loop
• Although it is the electrons that are being pulled toward the positive terminal and pushed away form the negative terminal, by convention the direction of current, i, is determined to be the opposite of the direction of  the flow of electrons
• We imagine that current describes the flow of positive charges.
• What is the speed of the charges?
• There are two different speeds...
• Without a battery, the electrons inside a conductor are moving extremely rapidly but randomly but randomly. There is no net flow of charge.
• v_random = 10⁶m/sec
• i.e. at a cross section of a wire, the total number of electrons flowing from left to right equals the number from right to left
• With a battery: To set up an electric field, there is a net drift of the charges
• v_drift = 10^-4 m/s
• Note: we have said earlier that there can be no electric field inside a conductor. This is true only if the conductor is in static equilibrium

• Charge moving in the conductor
• Dx = Vd Dt

• n = number of "mobile" charges per unit volume
• DQ = amount of charges within Dx  = (number of charges) x (charge per carrier)
• DQ = (n) (A Dx ) e
• If  Dx  = UdDt, then in a time interval of   Dt, all the charges within   Dx   will cross the boundary on the right
• DQ = (n) (A Ud ) e
• DQ /Dt = (n) (A Ud ) e = i
• Unit of current: coul/sec = Ampere
• i = nAUde
•  i/A = nUde = J = current density
• J is proportional to Ud; they both have vector characteristics
• Note: n is a property of the material; for good conductors n is approxiamately one charge carrier per atom
• Both J and i increase with Ud

• For most materials; J= sE
• Current density is proportional to the electric field ( usually established by a battery)
• s is called the conductivity of the material
• Vb-Va = V = EL
• J=  sE = sV/L
• V=  (L/ s)J = ( L/  sA) i
• or similarly V/i = L/sA = R (resistance)
• r = 1/s; where r is resistivity
• Thus... R= r (L/A) and V/i = R
• W = Ohm = Volt/ Ampere = V/A

• In the expression V/i = R,  if R is a constant and independent of V, (equivalently in J=  sE if s is independent of E), then these two expressions describe a device that satifies Ohm's Law
• 
• Note that many electrical and electronic devices do not obey Ohm's Law
• i.e. In the expression V= iR, the value of R depends on V (and on i!)