Electric fields in Material Space Sandra CruzPol Ph

Electric fields in Material Space Sandra Cruz-Pol, Ph. D. INEL 4151 ch 5 Electromagnetics I ECE UPRM Mayagüez, PR

Last Chapter: free space NOW: different materials

Some applications o o superconductors High permittivity dielectrics Transistors Electromagnets

We will study Electric charges: o Conductors or Insulators n Depends on Frequency and Temperature… o Boundary conditions Conductors Insulators (metals) (dielectrics) Semiconductors

Material @ 20 o. C Low frequency Silver Conductivity (S/m) Copper 5. 8 x 107 Gold 4. 1 x 107 Aluminum 3. 5 x 107 Carbon 3 x 104 Sea water 4 Silicon 4. 4 x 10 -4 Pure water 10 -4 Dry Earth 10 -5 Glass, Quartz 10 -12, 10 -17 Appendix B 6. 1 x 107 Conductorshave many free electrons available. Colder metals conduct better. (superconductivity) semiconductor Insulators at most lower frequencies.

Current Units: Amperes [A] Definition: is the electric charge passing through an area per unit time. Current Density, [A/m 2] Is the current thru a perpendicular surface:

Depending on how I is produced: There are different types of currents. o Convection- I flows thru isolator: liquid, gas, vacuum. n Doesn’t involve conductors, n doesn’t satisfies Ohm’s Law o Conduction- flows thru a conductor o Displacement (ch 9)

u Current in a filament o Convection current, [A] Dl DS rv o Convection density, A/m 2

Conduction Current o Requires free electrons, it’s inside conductor. o Suffers collisions, drifts from atom to atom Newton’s Law o Conduction current density is: where rv=ne

A Perfect conductor Has many charges that are free to move. o Therefore it can’t have an E field inside which would not let the charges move freely. o So, inside a conductor Charges move to the surface to make E=0

Resistance o If you force a Voltage across a conductor: o Then E is not 0 o The e encounter resistance to move E I S l V +- rc=1/s= resistivity of the material

Power in Watts =Rate of change of energy or force x velocity Joule’s Law

PE 5. 1 Find the current thru the cylindrical surface o For the current density

PE 5. 2 In a Van de Graaff generator, w=0. 1 m, u=10 m/s and the leakage paths have resistance 1014 W. o If the belt carries charge 0. 5 m. C/m 2, find the potential difference between the dome and the base. w= width of the belt u= speed of the belt

PE 5. 3 The free charge density in Cu is 1. 81 x 1010 C/m 3. . o For a current density of 8 x 106 A/m 2, find the electric field intensity and the drift velocity.

Polarization in dielectrics The effect of polarization on a dielectric is to have a surface bound charge of: and leave within it an accumulation of volume bound charge: rps and rpv are the polarization (bounded) surface and volume charge densities

Permittivity and Strength o Not really a constant!

Dielectric properties o Linear = e doesn’t change with E o Isotropic= e doesn’t change with direction o Homogeneous= e doesn’t change from point to point. Coulomb’s Law for any material:

PE 5. 6. A parallel plate capacitor with plate separation of 2 mm has a 1 k. V voltage applied to its plane. o If the space between its plates is filled with polystyrene, find E and P.

PE 5. 7. In a dielectric material, Ex= 5 V/m and o Find:

Continuity Equation o Charge is conserved.

For steady currents: o Change= output current –input current = 0

Substituting in: where Tr=e/s is called the Relaxation time

What is Relaxation Time? [s]

What is Relaxation Time? [s] Is the time it takes a charge placed in the interior of a material to drop to e-1 of its initial value. o Find Tr for silver o Find Tr for rubber:

Boundary Conditions o We have two materials o How the fields behave @ interface?

Boundary Conditions o We have two materials o How do the fields behave @ interface? We look at the tangential and the perpendicular component of the fields.

Cases for Boundary Conditions: 1. Dielectric- dielectric 2. Conductor- Dielectric 3. Conductor-Free Space

Dielectric-dielectric B. C. o Consider the figure below: E 1 n E 1 q 1 e 1 E 2 n e 2 a b E 1 t Dh E 2 t d c Dw

D 2 n Dielectric-dielectric B. C. o Consider the figure below: D 1 n D 1 DS D 1 t D 2 t e 1 e 2 r. S Dh

Dielectric-Dielectric B. C. In summary: E 1 n D 2 n E 1 q 1 E 1 t D 2 t e 1 e 2 Dh

Conductor-dielectric B. C. o Consider the figure below: En e 1 q 1 c d Et dielectric conductor E s 2=∞ E 2=0 Dh a Dw b

Conductor-dielectric B. C. o Consider the figure below: En e 1 DS q 1 Et dielectric conductor E s 2=∞ E 2=0 r. S Dh

Conductor-Free Space B. C. o Consider the figure below: En eo q 1 c d Et Free space conductor E s 2=∞ E 2=0 Dh a Dw b

PE 5. 9 A homogeneous dielectric (er=2. 5) fills region 1 (x<0), while region 2(x>0) is free space. o Find

5. 29 Lightning strikes a dielectric sphere of radius 2 -mm for which er=2. 5, s=5 x 10 -6 S/m and deposits uniformly a charge of 1 C. o Determine the initial volume charge density and the volume charge density 2 ms later. Answer: 29. 84 KC/m 3, 18. 98 k. C/m 3