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Dr. Gregory J. Mazzaro Spring 2017 ELEC 318 – Electromagnetic Fields Lecture 4(d) Electrostatic Boundary Conditions THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA 171 Moultrie Street, Charleston, SC 29409 Boundaries between Material Media boundary conditions: -- refer to the behavior of fields (E, H) and flux densities (D, B) at the surfaces where material media meet -- electrostatic media are characterized by e = permittivity (dielectric constant) s = conductivity -- in general, across a boundary, E1 E2 For a boundary between two media given in spherical coordinates by R = 3 m , determine the components of E which are normal and tangential to the boundary at P(3, p/2, p/4) if ˆ 2sin θˆ 6cos 4 ˆ V m E 4R R V ˆ V Et 2 θˆ 6 ˆ , En 12 R m m 2 Tangential Electric Field Intensity E E1t E1n w h 2 To describe the behavior of electric fields tangential to the boundary, we use the fact that electrostatic fields are irrotational : abcd w h 2 E dl 0 E2t w E2 n h 2 E2 n h 2 E1t w E2 n h 2 E2 n h 2 0 E2t w E1t w 0 E1t E2t Across a boundary between material media, tangential electric field intensity is continuous. 3 Normal Electric Flux Density D eE To describe the behavior of electric fields normal to the boundary, we use Gauss’ Law: S n̂ D1 D1n D dS Qenc D1t D2t s = charge per unit area, at the boundary D2n S h 2 s h 2 D2 Qenc s S nˆ D1n nˆ S D2n nˆ S s D1n D2 n s For a charge-free boundary (s = 0), normal electric flux density is continuous. 4 Example: Charge-Free Boundary With reference to this figure (at right), determine E1 if E2 is given by E2 5xˆ 9yˆ 3zˆ V m and the two materials are characterized by e1 2e 0 , e 2 8e 0 Assume that s = 0 at the boundary. E1t E2t , D1n D2n s 5 Example: Charge at the Boundary With reference to this figure (at right), determine E1 if E2 is given by E2 5xˆ 9yˆ 3zˆ V m and the two materials are characterized by e1 2e 0 , e 2 8e 0 Assume that s = 35.4 pC/m2 at the boundary. E1t E2t , D1n D2n s 7 Example: Dielectric/Conductor With reference to this figure (at right), determine E1 and E2 if s = 35.4 pC/m2 at the boundary and material 1 has a dielectric constant of er1 = 2 , and material 2 is a perfect conductor . conductor E1t E2t , D1n D2n s 9 Dr. Gregory J. Mazzaro Spring 2017 ELEC 318 – Electromagnetic Fields Lecture 4(x) Electrostatic Fields: Additional Examples THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA 171 Moultrie Street, Charleston, SC 29409 Example: Linear Superposition Three infinitely-long lines of charge, each with density 445 pC/m, are parallel to the z axis. One is on the z-axis (x = 0, y = 0). The second is at x = 0, y = –3 m. The third is at x = 0, y = 3 m. Determine the electric field intensity at P(x = 4 m, y = 3 m, z = 6 m), in free space. Prior result: For a single line charge… E l rˆ 2pe 0 r 12 Example: Surface Charge, Coulomb’s z Calculate the electric field intensity at any point P above an infinite sheet of constant charge density S in the x-y plane by calculating the field along the z-axis for a disc of radius R0 and taking the limit of this result as R0 y x rˆ cos xˆ sin yˆ ˆ sin xˆ cos yˆ 14 Example: Surface Charge, Gauss’ Law z Calculate the electric field intensity at any point P above an infinite sheet of constant charge density S in the x-y plane using Gauss’ Law. Q S D dS y x 16