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1 5. Conductors and dielectrics EMLAB Contents 2 1. Current and current density 2. Continuity of current 3. Metallic conductors 4. Conductor properties and boundary conditions 5. The method of images 6. Semiconductors 7. Dielectric materials 8. Boundary conditions for dielectric materials EMLAB 3 Current and voltage EMLAB 4 EMLAB 5 5.1 Current and current density S Current I dQ dt n̂ J I nˆ S I Current density • Current is electric charges in motion, and is defined as the rate of movement of charges passing a given reference plane. • In the above figure, current can be measured by counting charges passing through surface S in a unit time. J I S I J S I J da S • In field theory, the interest is usually in event occurring at a point rather than within some large region. •For this purpose, current density measured at a point is used, which is current divided by the area. EMLAB Current density from velocity and charge density 6 With known charge density and velocity, current density can be calculated. S vt Charges with density ρ Q V ( volume) S t Q I I S , J v t S EMLAB Continuity equation : Kirchhoff ’s current law 7 J Kirchhoff ’s current law I Q J Charges going out through dS. dS n̂ 0 ; Steady current . t For steady state, charges do not accumulate at any nodes, thus ρ become constant. q J nˆ St Q J da dt d d V C d J d a d J d C V V t d dt V J t J t differential form I n n dQ t integral form EMLAB 8 Electrons in an isolated atom Electron energy level 1 atom - + - - - - - - - Tightly bound electron - More freely moving electron Energy levels and the radii of the electron orbit are quantized and have discrete values. For each energy level, two electrons are accommodated at most. EMLAB 9 Electrons in a solid Atoms in a solid are arranged in a lattice structure. The electrons are attracted by the nuclei. The amount of attractions differs for various material. Freely moving electron + Eext External E-field + + Tightly bound electron + + - + - - - + + + - + - - - + + + - + - - - + + - Electron energy level - - - To accommodate lots of electrons, the discrete energy levels are broadened. EMLAB 10 Insulator and conductor Insulator atoms + + Conductor atoms + - + - + - + - + - + - External E-field + - + - + - + - - - External E-field + + - - + + - - + + - - + + - - + + - - + + - - Empty energy level - Energy level of insulator atoms Occupied energy level - Energy level of conductor atom EMLAB Movement of electrons in a conductor 11 EMLAB 12 EMLAB Electron flow in metal : Ohm’s law 13 F qE eE + + - + + + - + - - + - E v eE + - + - - + - - + - + - + + - + Jv - + - • n: Electron density (number of electrons per unit volume. - • μ : mobility (ne)( e )E neeE J E ; Ohm’s law : Electric conductivity EMLAB 14 Example : calculation of resistance B A S J B A VBA E dr E dr A VBA J LBA B V R I B I L S BA VBA IR , R A B J dr J L BA A S l S E dr J da E da S E dr S EMLAB Conductivities of materials 15 EMLAB 16 Electric field on a conductor due to external field tangential component Et 0 E E n nˆ normal component +q1 -q1 S 0 -q1 Ein E Conductor Conductor 1. Tangential component of an external E-field causes a positive charge (+q) to move in the direction of the field. A negative charge (-q) moves in the opposite direction. 2. The movement of the surface charge compensates the tangential electric field of the external field on the surface, thus there is no tangential electric field on the surface of a conductor. 3. The uncompensated field component is a normal electric field whose value is proportional to the surface charge density. 4. With zero tangential electric field, the conductor surface can be assumed to be equi-potential. EMLAB 17 Charges on a conductor 1. In equilibrium, there is no charge in the interior of a conductor due to repulsive forces between like charges. 2. The charges are bound on the surface of a conductor. Ein 0 3. The electric field in the interior of a conductor is zero. 4. The electric field emerges on the positive charges and sinks on negative charges. 5. On the surface, tangential component of electric field becomes zero. If non-zero component exist, it induces electric current flow which generates heats on it. EMLAB 18 Image method +q1 • If a conductor is placed near the charge q1, the shape of electric field lines changes due to the induced charges on the conductor. • The charges on the conductor redistribute themselves until the tangential electric field on the surface becomes zero. Etan 0 E nˆ 0 n̂ - - - - Perfect electric conductor •If we use simple Coulomb’s law to solve the problem, charges on the conductors as well as the charge q1 should be taken into account. As the surface charges are unknown, this approach is difficult. • Instead, if we place an imaginary charge whose value is the negative of the original charge at the opposite position of the q1, the tangential electric field simply becomes zero, which solves the problem. +q1 Etan 0 E nˆ 0 -q1 Image charge EMLAB 19 Example : a point charge above a PEC plane • The electric field due to a point charge is influenced by a nearby PEC whose charge distribution is changed. In this case, an image charge method is useful in that the charges on the PEC need not be taken into account. •As shown in the figure on the right side, the presence of an image charge satisfies the boundary condition imposed on the PEC surface, on which tangential electric field becomes zero. • This method is validated by the uniqueness theorem which states that the solution that satisfy a given boundary condition and differential equation is unique. E +q1 n̂ Etan 0 E nˆ 0 도체 +q1 E( x 0) E tan 0 x̂ (a ,0,0) ẑ x̂ ẑ q1 xˆ ( x a ) yˆ y zˆ z 40 ( x a ) 2 y 2 z 2 3 / 2 q xˆ ( x a ) yˆ y zˆ z 1 40 ( x a ) 2 y 2 z 2 3 / 2 (a ,0,0) Etan 0 E nˆ 0 q1 2a xˆ 2 40 a y 2 z 2 3/ 2 -q1 (a ,0,0) Image charge EMLAB Dielectric material 20 molecule The charges in the molecules force the molecules aligned so that externally applied electric field be decreased. EMLAB 21 Dielectric material ẑ (1) No material D zˆ S S E0 (2) With dielectric material x̂ S d D zˆ S Ein E0 + + Ep S E0 zˆ S S 0 Ein E0 E p , E p e Ein Ein E0 e Ein (1 e )Ein E0 Ein D E 0 (1 e )E E0 1 e D 0 E0 0 (1 e )Ein Ein • D (electric flux density) is related with free charges, so D is the same despite of the dielectric material. • But the strength of electric field is changed by the induced dipoles inside. EMLAB 22 Electric dipole ẑ P (0, r sin , r cos ) θ Q (0, 0, d / 2) +q d -q Q (0, 0, d / 2) d (0, 0, d) N p qiri qd ; dipole moment . i 1 1 1 q PQ PQ PQ PQ 40 PQ PQ q PQ PQ qd cos p rˆ 2 2 40 r 40 r 40 r 2 qd cos E V rˆ 2 cos θˆ sin 40 r 3 V q q q 40 PQ 40 PQ 40 EMLAB 23 Electric field in dielectric material S E0 x̂ S d Ein E0 + + + Ep S S S 0 E ẑ E0 zˆ + S 0 r Induced dipole에 의해 물질 내부 전 기장 세기 줄어듦. 도체 양단의 전 압을 측정하면 전압이 줄어듦. EMLAB Gauss’ law in Dielectric material 24 E 0 in da Q total q free q bound q free P da S S + 0Ein P da q free D da S S + D 0Ein P + + + + + p Length : d P 0 e Ein + + + + + + + +q1 + + + + Dipole + + + + + Induced dipole d V V Sd d nˆ da S qbound (V ) N (q) Nqd nˆ da N p da P da S S S EMLAB Relative permittivity 25 EMLAB 26 Boundary conditions (1) Boundary condition on tangential electric field component Tangential boundary condition can be derived from the result of line integrals on a closed path. w C1 E1t E2 t τ̂ unit vector tangential to the surface Medium #2 V 0 E ds (E1 τˆ w E2 τˆ w ) C1 Medium #1 Unit vector normal to the surface E1 τˆ E2 τˆ (2) Boundary condition on normal component of electric field n̂ S Medium #2 h S Boundary condition on normal component can be obtained from the result of surface integrals on a closed surface. D2n D1n S 2E2n 1E1n S D da D d d S Medium #1 V V (D 2 nˆ D1 nˆ )S Dcurved side τˆ h hS If h 0, Dcurved side τˆ h 0 D 2 nˆ D1 nˆ h S (surface charge density) EMLAB Example – conductor surface tangential component 27 Et 0 E E n nˆ normal component +q1 -q1 S 0 -q1 Ein E Conductor Conductor 1. Tangential component of an external E-field causes a positive charge (+q) to move in the direction of the field. A negative charge (-q) moves in the opposite direction. 2. The movement of the surface charge compensates the tangential electric field of the external field on the surface, thus there is no tangential electric field on the surface of a conductor. 3. The uncompensated field component is a normal electric field whose value is proportional to the surface charge density. 4. With zero tangential electric field, the conductor surface can be assumed to be equi-potential. EMLAB Example – dielectric interface 28 Surface charge density of dielectric interface can not be infinite. 2 1 2 1 E1 E2 tangential component : Et1 Et 2 normal component : Dn1 Dn 2 E t 2 E t1 E 2 sin 2 E1 sin 1 Dn 2 Dn1 2E n 2 1E n1 2E 2 cos 2 1E1 cos 1 E2 E2 sin 2 E2 cos 2 2 2 E1 sin 1 2 1 E1 cos 1 2 2 E1 sin 1 1 cos 2 1 2 2 EMLAB 2 Example – dielectric interface D Dx Dy Dz Dz 0 x y z z 29 The normal component of D is equal to the surface charge density. Dz C Dz S ẑ ŷ D1z D2 z S x̂ 1E1z 2 E2 z S Capacitance : 0 V E dr E2 z d 2 E1z d1 d S d 2 S d1 2 1 Q D da ( zˆ )( s zˆ ) S s S S C Q sS S d 2 d1 V S d S d 2 1 2 1 2 1 EMLAB Static electric field : Conservative property 30 VB C3 VA B C2 A VBA E dr E dr C1 C1 C2 정전기장에 의한 potential difference VAB는 시작점과 끝점이 고정된 경우, 적분 경로와 상관없이 동일한 값을 갖는다. E dr E dr E dr 0 C C1 E dr 0 C C2 E 0 EMLAB Stokes’ theorem 31 E dr E da C • S 벡터함수를 임의의 닫혀진 경로에 대한 선적분을 하는 경우 그 경 로로 둘러싸인 면에 대한 면적분으로 바꿀 수 있다. • 이 때 피적분 함수는 원래 함수의 ‘curl’로 바꿔야 한다. E dr C E dr n Cn Cn 임의의 닫혀진 경로에 대한 선적분은 매우 작은 폐곡선의 선적분의 합으로 분해할 수 있다. EMLAB Line integral over an infinitesimally small closed path E E x xˆ E y yˆ E z zˆ 32 ẑ ( x, y , z ) ŷ x̂ 2 3 4 1 2 3 4 E dr E y dy Ex dx E y dy Ex dx Cn 1 2 3 4 1 x y x y E y x dy E x y dx E y x dy E x y dx 1 2 3 4 2 2 2 2 x x y y E y x E y x y E x y E x y x 2 2 2 2 E E y x xy E da S y x ( E) zˆxy E dr E y E x Cn ( E) z Lim y S 0 xy x E E E E E E E xˆ z y yˆ x z zˆ y x z z x x y y EMLAB