HIGH VOLTAGE TECHNIQUES REVΔ°EW: Electrostatics & Magnetostatics Assistant Professor Suna BOLAT KRÖGER Eastern Mediterranean University Department of Electric & Electronic Engineering Zap You walk across the rug, reach for the doorknob and..........ZAP!!! In the winter, when you change your pullover you hear and/or see sparks... you come inside from the cold, pull off your hat and......BOING!!! Static hair - that static electricity makes your hair stand straight out from your head. What is going on there? Static electricity... ...is the imbalance of positive and negative charges. • As you walk across a carpet, electrons move from the rug to you. Now you have extra electrons and a negative static charge. Touch a door knob and ZAP! The door knob is a conductor. The electrons jump from you to the knob, and you feel the static shock. • We usually only notice static electricity in the winter when the air is very dry. During the summer, the air is more humid. The water in the air helps electrons move off you more quickly, so you can not build up as big a static charge. Static Question: Static? What does that mean? Answer: constant with time Question: How constant is it? Does it really not move at all? Answer: there is movement of charges. In fact, when you get zapped, charges are actually moving between your fingers and the doorknob. However, the movement is only brief compared to the current in a closed circuit. Static regime In the static regime, electromagnetic quantities do not vary as a function of time. We have two main cases: • ELECTROSTATICS • MAGNETOSTATICS Definitions and units Symbol π. π× π ππ E B D H ε0 μ0 Meaning (first term is the most common) SI Unit of measure Differential operators the divergence operator per meter (factor contributed by applying either operator) the curl operator per second (factor contributed by applying the partial derivative with respect to time operator) Fields ο· electric field, also called volt per meter or, equivalently, [V/m] ο· the electric field intensity newton per coulomb [N/C] ο· magnetic field, also called: the magnetic tesla, or equivalently, [T] induction ο· weber per square meter, [Wb/m2] ο· the magnetic field density ο· volt-second per square meter [V s/m2] ο· the magnetic flux density ο· electric displacement field, also called: ο· coulombs per square meter or [C/m2] ο· the electric induction equivalently, ο· the electric flux density ο· newton per volt-meter [N/V m] ο· magnetizing field, also called: auxiliary magnetic field ampere per meter [A/m] ο· magnetic field intensity ο· magnetic field permittivity of free space, also called the farads per meter [F/m] dielectric constant, a universal constant permeability of free space, also called the henries per meter, [H/m] or newtons per magnetic constant, a universal constant ampere squared [N/A2] Electric field Electric field is said to exist in a region of space if a charge experiences a force when placed anywhere in that region. If a unit positive charge is placed at some point in the field, the force experienced by it is said to be the electric stress at that point. Electrostatics The electric charges do not change position in time. Therefore, ρ, E and D are constant and there is no magnetic field H, since there is no current density J. Magnetostatics The charge crossing a given cross-section (current) does not vary in time. Therefore, J, H and B are constant. Although charges are moving, the steady current maintains a constant charge density ρ in space and the electric field E is static. Electric Field & Magnetic Field Voltage is associated with electric field Current is associated with magnetic field Maxwell’s equations Basic equations: ππ΅ ππ‘ πππ‘ π» = π» × π» = π½ πππ£ π· = π πππ£ π΅ = 0 Auxiliary equations (constructive equations): πππ‘ πΈ = π» × πΈ = π½ =ππΈ+ππ+ π· = ππΈ ο₯: Dielectric constant = ο₯0 ο₯r ο₯r : relative dielectric constant (no unit), relative permittivity ο₯0 = 8.854. 10-12 F/m dielectric constant for space, permittivity ππ· ππ‘ µ: permeability µr: relative permeability µ0 = 4 π. 10-7 H/m magnetic constant, relative permeability of space The electrostatic equations The equations of electrostatics are obtained Maxwell’s equations, By assuming ∂/∂t , J, H and B are all zero: The electrostatics equations: Poisson equation πππ£ π· = π». π· = π πππ£ π· = π». ππΈ = π» π −π»πΈ = −ππ» 2 V = π grad V π»2V = π − π π βπ = − Poisson equation πΊ The electrostatics equations: Laplace’s equation If there is no free charges at the medium π = 0. βπ = π Laplace’s equation Operators: Laplacian ππ π½ ππ π½ ππ π½ βπ½ = π + π + π ππ ππ ππ Operators: Gradient ππ½ ππ½ ππ½ ππ½ = ππ + ππ + ππ ππ ππ ππ Laplace’s equation in Cartesian coordinates: π π π½ ππ π½ ππ π½ βπ½ = π + π + π = π ππ ππ ππ ππ½ ππ½ ππ½ ππ½ = ππ + ππ + ππ ππ ππ ππ Laplace’s equation in Polar coordinates: ππ π½ π ππ½ π ππ π½ π ππ π½ ππππππ½ ππ π½ βπ½ = π + + π π+ π + =π π π π π ππ π ππ π ππ½ π πππ π½ ππΆ π ππ½ ππ½ π ππ½ π ππ½ ππ½ = ππ + ππΆ + ππ½ ππ π. ππππ½ ππΆ π ππ½ Laplace’s equation in Cylindrical coordinates: ππ π½ π ππ½ π ππ π½ ππ π½ βπ½ = π + + π π+ π =π ππ π ππ π ππ½ ππ ππ½ π ππ½ ππ½ ππ½ = ππ + ππ½ + ππ ππ π ππ½ ππ
