EMLAB 1 5. Conductors and dielectrics

EMLAB 2 Contents 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

5 5.1 Current and 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. 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. Current Current density

EMLAB 6 Current density from velocity and charge density Charges with density ρ With known charge density and velocity, current density can be calculated.

EMLAB 7 Continuity equation : Kirchhoff ’s current law Charges going out through dS. For steady state, charges do not accumulate at any nodes, thus ρ become constant. differential form integral form Kirchhoff ’s current law

EMLAB Electron energy level atom Electrons in an isolated atom Tightly bound 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. - - More freely moving electron

EMLAB 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. Electrons in a solid Freely moving electron Tightly bound electron - Electron energy level To accommodate lots of electrons, the discrete energy levels are broadened. External E-field

EMLAB 10 - Energy level of insulator atoms Energy level of conductor atom Insulator and conductor Insulator atomsConductor atoms Occupied energy level Empty energy level External E-field

EMLAB 11 Movement of electrons in a conductor

EMLAB 12

EMLAB  : Electric conductivity ; Ohm’s law Electron flow in metal : Ohm’s law n: Electron density (number of electrons per unit volume. μ : mobility

EMLAB 14 Example : calculation of resistance

EMLAB 15 Conductivities of materials

EMLAB 16 -q 1 +q 1 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. -q 1 Conductor Electric field on a conductor due to external field normal component tangential component

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. 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 If a conductor is placed near the charge q 1, 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. If we use simple Coulomb’s law to solve the problem, charges on the conductors as well as the charge q 1 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 q 1, the tangential electric field simply becomes zero, which solves the problem. +q 1 Perfect electric conductor -q 1 Image charge +q

EMLAB 19 +q 1 도체 -q 1 Image charge +q 1 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. Example : a point charge above a PEC plane

EMLAB 20 molecule Dielectric material The charges in the molecules force the molecules aligned so that externally applied electric field be decreased.

EMLAB 21 Dielectric material 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. (1) No material (2) With dielectric material

EMLAB 22 +q -q Electric dipole

EMLAB Electric field in dielectric material Induced dipole 에 의해 물질 내부 전 기장 세기 줄어듦. 도체 양단의 전 압을 측정하면 전압이 줄어듦.

EMLAB q Gauss’ law in Dielectric material Dipole Length : d Induced dipole

EMLAB 25 Relative permittivity

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. unit vector tangential to the surface (2) Boundary condition on normal component of electric field Boundary condition on normal component can be obtained from the result of surface integrals on a closed surface. Unit vector normal to the surface Medium #1 Medium #2 Medium #1 Medium #2

EMLAB 27 -q 1 +q 1 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. -q 1 Conductor Example – conductor surface normal component tangential component

EMLAB 28 Surface charge density of dielectric interface can not be infinite. Example – dielectric interface

EMLAB 29 Example – dielectric interface The normal component of D is equal to the surface charge density. Capacitance :

EMLAB 30 Static electric field : Conservative property 정전기장에 의한 potential difference V AB 는 시작점과 끝점이 고정된 경우, 적분 경로와 상관없이 동일한 값을 갖는다.

EMLAB 31 임의의 닫혀진 경로에 대한 선적분은 매우 작은 폐곡선의 선적분의 합으로 분해할 수 있다. Stokes’ theorem 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.

EMLAB 32 Line integral over an infinitesimally small closed path