HIGH VOLTAGE TECHNIQUES REVİEW: 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 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

𝝏

𝝏𝒕 partial derivative with respect to time

per second (factor contributed by applying the operator)

Fields

E electric field, also called the electric field intensity

volt per meter or, equivalently, [V/m] newton per coulomb [N/C]

B

magnetic field, also called: the magnetic induction

the magnetic field density the magnetic flux density

tesla, or equivalently, [T] weber per square meter, [Wb/m2] volt-second per square meter [V s/m2]

D electric displacement field, also called: the electric induction the electric flux density

coulombs per square meter or [C/m2] equivalently,

newton per volt-meter [N/V m]

H

magnetizing field, also called: auxiliary magnetic field

magnetic field intensity magnetic field

ampere per meter [A/m]

ε0 permittivity of free space, also called the dielectric constant, a universal constant

farads per meter [F/m]

μ0 permeability of free space, also called the magnetic constant, a universal constant

henries per meter, [H/m] or newtons per 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:

𝑟𝑜𝑡 𝐸 = 𝛻 × 𝐸 =𝜕𝐵

𝜕𝑡

𝑟𝑜𝑡 𝐻 = 𝛻 × 𝐻 = 𝐽

𝑑𝑖𝑣 𝐷 = 𝜌

𝑑𝑖𝑣 𝐵 = 0 Auxiliary equations (constructive equations):

𝐽 = 𝜎 𝐸 + 𝜌 𝑉 +𝜕𝐷

𝜕𝑡

𝐷 = 𝜀𝐸

µ: permeability µr: relative permeability µ0 = 4 π. 10-7 H/m magnetic constant, relative permeability of space

: Dielectric constant = 0 r r : relative dielectric constant (no unit), relative permittivity 0 = 8.854. 10-12 F/m dielectric constant for space, permittivity

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

𝑑𝑖𝑣 𝐷 = 𝛻.𝐷 = 𝜌

𝑑𝑖𝑣 𝐷 = 𝛻. 𝜀𝐸 = 𝛻 𝜀 −𝛻𝐸 = −𝜀𝛻2V = 𝜌

grad V

𝛻2V = −𝜌

𝜀

∆𝐕 = −𝝆

𝜺 Poisson equation

The electrostatics equations: Laplace’s equation

If there is no free charges at the medium 𝜌 = 0.

∆𝐕 = 𝟎 Laplace’s equation

Operators: Laplacian

∆𝑽 =𝝏𝟐𝑽

𝝏𝒙𝟐+

𝝏𝟐𝑽

𝝏𝒚𝟐+

𝝏𝟐𝑽

𝝏𝒛𝟐

Operators: Gradient

𝛁𝑽 =𝝏𝑽

𝝏𝒙𝒊𝒙 +

𝝏𝑽

𝝏𝒚𝒊𝒚 +

𝝏𝑽

𝝏𝒛𝒊𝒛

Laplace’s equation in Cartesian coordinates:

∆𝑽 =𝝏𝟐𝑽

𝝏𝒙𝟐+

𝝏𝟐𝑽

𝝏𝒚𝟐+

𝝏𝟐𝑽

𝝏𝒛𝟐= 𝟎

𝛁𝑽 =𝝏𝑽

𝝏𝒙𝒊𝒙 +

𝝏𝑽

𝝏𝒚𝒊𝒚 +

𝝏𝑽

𝝏𝒛𝒊𝒛

Laplace’s equation in Polar coordinates:

∆𝑽 =𝝏𝟐𝑽

𝝏𝒓𝟐+

𝟐

𝒓

𝝏𝑽

𝝏𝒓+

𝟏

𝒓𝟐

𝝏𝟐𝑽

𝝏𝜽𝟐+

𝟏

𝒓𝟐𝒔𝒊𝒏𝟐𝜽

𝝏𝟐𝑽

𝝏𝜶𝟐+

𝒄𝒐𝒕𝒂𝒏𝜽

𝒓𝟐

𝝏𝟐𝑽

𝝏𝜽𝟐= 𝟎

𝛁𝑽 =𝝏𝑽

𝝏𝒓𝒊𝒓 +

𝟏

𝒓. 𝒔𝒊𝒏𝜽

𝝏𝑽

𝝏𝜶𝒊𝜶 +

𝟏

𝒓

𝝏𝑽

𝝏𝜽𝒊𝜽

Laplace’s equation in Cylindrical coordinates:

∆𝑽 =𝝏𝟐𝑽

𝝏𝒓𝟐+

𝟏

𝒓

𝝏𝑽

𝝏𝒓+

𝟏

𝒓𝟐

𝝏𝟐𝑽

𝝏𝜽𝟐+

𝝏𝟐𝑽

𝝏𝒛𝟐= 𝟎

𝛁𝑽 =𝝏𝑽

𝝏𝒓𝒊𝒓 +

𝟏

𝒓

𝝏𝑽

𝝏𝜽𝒊𝜽 +

𝝏𝑽

𝝏𝒛𝒊𝒛