5. Conductors and dielectrics

Contents Current and current density Continuity of current Metallic conductors Conductor properties and boundary conditions The method of images Semiconductors Dielectric materials Boundary conditions for dielectric materials

Current and voltage

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

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

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.

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 + - Electron energy level External E-field - Tightly bound electron To accommodate lots of electrons, the discrete energy levels are broadened.

Insulator and conductor Insulator atoms Conductor atoms + + + + + + - - - - - - + + + + + + - - - - - - External E-field External E-field + + + + + + - - - - - - + + + + + + - - - - - - Empty energy level - Occupied energy level - Energy level of insulator atoms Energy level of conductor atom

Movement of electrons in a conductor

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

Example : calculation of resistance

Conductivities of materials

Electric field on a conductor due to external field tangential component normal component +q1 -q1 -q1 Conductor Conductor 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. 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. The uncompensated field component is a normal electric field whose value is proportional to the surface charge density. With zero tangential electric field, the conductor surface can be assumed to be equi-potential.

Charges on a conductor In equilibrium, there is no charge in the interior of a conductor due to repulsive forces between like charges. The charges are bound on the surface of a conductor. The electric field in the interior of a conductor is zero. The electric field emerges on the positive charges and sinks on negative charges. 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.

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. 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. - Perfect electric conductor +q1 -q1 Image charge

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. +q1 도체 +q1 -q1 Image charge

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

Dielectric material (1) No material (2) With 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.

Electric dipole +q -q

Electric field in dielectric material - + - + - + - + Induced dipole에 의해 물질 내부 전기장 세기 줄어듦. 도체 양단의 전압을 측정하면 전압이 줄어듦.

Gauss’ law in Dielectric material + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - Length : d Dipole + - + - + - + - + - + - - + - + - + - + - + - + +q1 - + + - - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + Induced dipole

Relative permittivity

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 Medium #2 Medium #1 Unit vector normal 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. Medium #2 Medium #1

Example – conductor surface tangential component normal component +q1 -q1 -q1 Conductor Conductor 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. 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. The uncompensated field component is a normal electric field whose value is proportional to the surface charge density. With zero tangential electric field, the conductor surface can be assumed to be equi-potential.

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

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

Static electric field : Conservative property 정전기장에 의한 potential difference VAB는 시작점과 끝점이 고정된 경우, 적분 경로와 상관없이 동일한 값을 갖는다.