Chapter 7 Dielectrics: Il 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
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 eala At the interface between the two media, the quantities V, E, and d must satisfy certain boundary conditions 1. The potent At the boundary btwn two media, v must be contin uous, otherwise av/ax 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 f Then the flux of d emerging from the flat cylinder is equal to the charge enclosed (DnI-dn2)S=af Thus the bdry condition of D is (Dn1-Dn2)=af
S D D Figure 7-1 Gaussian cylinder on the interface between two media I and 2.The difference D-D, between the nomal components of D is equal to the surface density of free charge of
Examples If the boundary surface free charge of=0, then Dn1= Dn2 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=0 in 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 CE. dI Taking a closed path around the boundary as shown in Fig 7-2, we have Et1L-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 b on either side of the interface are equal: E =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. (See Fig 7-3) 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 0,=6: tan O
For an interface with the free charge density rom(DnI -Dn2)=af we have D1 cos 8 COS 02=0. Erico el cos 1= Er2eo e2 cos 82 From the tangential component Et1= Et2, we have 1 sIn 01=Bo sin 02 Combining these two equations yields tan tan The medium with the larger relative permittivity Er has a larger angle 0 from the normal
Figure 7-4(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 surfaces