Chapter 7 Dielectrics: II Continuity Conditions at The interface Potential Energy Forces on Conductors in the presence of Dielectrics a Forces on dielectrics Displacement Current Frequency and Temperature dependence, Anisotropy
Chapter 7 Dielectrics: II ◼ Continuity Conditions at The Interface ◼ Potential Energy ◼ Forces on Conductors In The Presence Of Dielectrics ◼ Forces On Dielectrics ◼ Displacement Current ◼ Frequency and Temperature Dependence, Anisotropy
7.1 Continuity Conditions at the Interface Between Two Media At the interface between the two media, the quantities V, E, and d must satisfy certain boundary conditions 1. The potential v At the boundary btwn two media, V must be contin uous,otherwise aV/ar would be oo, and e would be For a charge distribution in a finite region the potential V is normally set to zero at infinity
2. The Normally Component of D (See Fig 7-1) Consider a short gaussian cylinder drawn about a bound- ary surface The cross-section area of the cylinder is s The boundary surface carries a free surface charge o Then the flux of d emerging from the flat cylinder is equal to the charge enclosed Thus the bdry condition of d is (DnI -Dn2
S D D Figure 7-1 Gaussian cylinder on the interface between two media I and 2. The difference D, I-D,, between the normal components of D is equal to the surfac density of free charge o
Examples If the boundary surface free charge of=0, the en m1=1m2, the normal component of d is continuous this is usu ally the case for two dielectric media If the boundary is between a conductor and a dielectric, D=0in the conductor, and Dn =of in the dielectric where o f is the free charge density on the surface of the conductor
3. the Tangential Component of e As we know in the static case. the electric field is curless VXE=0, i.e. along any closed path c the integral IcE.dI=0 Taking a closed path around the boundary as shown in Fig 7-2, we have E11-Et2L=0,→E1=Et2
L Figure 7-2 Closed path of integration crossing the interface between two media I and 2. Whatever be the surface charge density a, the tangential components of E on either side of the interface are equal: En= en
4. Bending Of Lines of e From the boundary conditions it follows that the vectors D and e change directions at the boundary between two media.(SeeF「g73) Figure 7-3 Lines of D or of E crossing the interface between two media I and 2 The lines change direction in such a way that e, i tan 02=6, tan 0
For an interface with the free charge density of =0 from(Dnl -Dm2) - Of we have Di cos 01- D2 cos 82=0 or r1e0E1 cos 61= Er2E0 E2 cos 02 From the tangential component Et1= Et2, we have E1 sin 01- E2 sin B2 Combining these two equations yields tan T tan e2 Er2 The medium with the larger relative permittivity cr has a larger angle 0 from the normal
Figure 74(a) Lines of D(arrows)and equipotentials for a point charge in air near a dielectric. They are not shown near the charge because they get too close together. Rotating the figure around the horizontal axis generates equipotential spaces
