Electromagnetics I - 5EPA0

5EPA0 Electromagnetics I

• Area: Electromagnetics and Waves
• Topics: Vector Calculus · Maxwell Equations · Magnetic Circuits and Transformers
• Related courses: Electromagnetics II - 5EPB0 · Electromechanics - 5EWA0

Gauss Divergence Theorem: https://www.youtube.com/watch?v=zZqxbwl3Dno | https://www.youtube.com/watch?v=pY4t-ikhzhU
Stokes Theorem: https://www.youtube.com/watch?v=0UvNF_cfBJ4

Boundary Conditions (normal and tangential components):
https://www.youtube.com/watch?v=_FS6csrxBPU
https://www.youtube.com/watch?v=vlNUGyDzJ90
Tangential Magnetic Field Boundary Conditions
Normal Magnetic Field Boundary Conditions

https://www.youtube.com/watch?app=desktop&v=MPzdh9ZshLI

Lorentz Force: https://www.youtube.com/watch?v=grgNdIYP6zI
Hall Effect: https://www.youtube.com/watch?v=1OZCWetFCps
Lenz Law: https://www.youtube.com/watch?v=u7Rg0TcHQ4Y
Maxwell Equations: https://www.youtube.com/watch?v=hJD8ywGrXks
Motional & Transformer EMF: https://www.youtube.com/watch?v=Xi_wZ1IYLo8
flux: https://www.youtube.com/watch?v=m1PPujngqAw
Electromagnetic Wave Equation in Free Space
What Is a Plane Wave?
Here’s What Maxwell’s Equations ACTUALLY Mean.

Mathematical Tools

Topic: Vector Calculus

Vector calculations for EM shapes

• Dot product: cos
• Cross product: sin
• So think if things are parallel or perpendicular to each other in the plane, if its 0 or 1
• take into account the direction for negative/positive
• For angles clockwise → positive
• means a is thumb and b is point finger, middle finger is the output vector you get the + or -

GEOMETRY

Cylinder: Area: Volume:
Sphere: Area: Volume:
Circle: Area: Circumference:

• unit vector, vector divide by distance

Two Points

Distance:
Line through two points:
Projection: A onto B

Dot Product - Scalar

Dot Product of Unit Vectors

as they are unit vectors

Cross Product - Vector

Cartesian Coordinates

Unit vectors: , Position vector: , Vector field: , Line elements: , Incremental distance: , Volume element:

Circular Cylindrical Coordinates

Unit vector: , Position vector: , Vector field: , Line elements: , Incremental distance: , Volume element:

Spherical coordinates

Unit vectors: , Position vector: , Vector field: , Line elements: , Incremental distance: , Volume element:

Electrostatics

Topic: Maxwell Equations

• Total flux going through a closed surface = total charge inside that surface
• Gauss’s law for electricity states that the electric flux Φ across any closed surface is proportional to the net electric charge Q enclosed by the surface

Electric flux density

• in a closed surface it must always point outwards

Vector Operators

Divergence
Is a way to differentiate a vector field -> to a scalar field
For flux in= flux out of volume → if not enclosing any charge

Gradient
Curl

Work Done

force on particle:
force to hold particle:
work done by field:
work done by applied force:

→ work performed on the charge by the electric field
→ same but negative sign → work required to move the charge through the field

Line Integral

use of preregistration: for

Gradient

relationship between E(electric) and V(potential) field
Potential difference:
Line integral:
In a limit
integrate E to find V, differentiate V to find E

The gradient operator can only be applied to a scalar field, and gives a vector
field in return. Beware that the definition of the gradient operator is dependent
on the coordinate system. A common mistake is to forget the additional constants in cylindrical or and in spherical coordinate systems!

How are Kirchhoff’s current law (KCL) and Ampère-Maxwell’s law (AML) related?
AML reduces to KCL when quasi-static conditions are assumed.

Under quasi-static conditions, the assumption is that the fields and currents change sufficiently slowly so that the dynamic effects (like displacement currents) are negligible, thereby simplifying AML to essentially KCL.

cross product: tangential
dot product: normal
Electric flux density for dielectrics:

Faraday-Henry Law
Ampere-Maxwell Law



Uniform electric field accelerates a charge, while a uniform magnetic field makes keeps charge constant

Capacitance

capacitance for point charge: self capacitance
E field: independent of distance plates

Poisson/Laplace

In empty space ⇒ Laplace is special case of Poisson ⇒
Laplace/Poisson must satisfy boundary conditions to have a unique solution

Mesh Calculation:
Poisson:
Laplace: ~=0

Laplace
Poisson

Divergence / Gauss's Theorem

Gauss Law:

Magnetostatics

Topics: Maxwell Equations · Magnetic Circuits and Transformers

The magnetic field H might be thought of as the magnetic field produced by the flow of current in wires and the magnetic field B as the total magnetic field including also the contribution M made b OP for Live Preview. You can also use Minimal Theme which includes the snippet bundled in, and is kept up to date for new versions of Obsidian.y the magnetic properties of the materials in the field.

Ampere law easier: use when symmetry | like Gauss Law
Biot-Savart law: general | like Coulomb law

observation point source of field

Stokes Theorem

• The magnetic flux passing through a closed surface is zero.

• FBI: (F thumb) Left Hand Rule for (-), same but right hand for (+) charge & (conventional current)
• A circle with a dot in it, represents a vector out of the page. A circle with an x in it
• https://physics.stackexchange.com/questions/395851/left-or-right-hand-rule

The magnetic field H might be thought of as the magnetic field produced by the flow of current in wires and the magnetic field B as the total magnetic field including also the contribution M made by the magnetic properties of the materials in the field.

Magnetic permeability is the realitive strength of a magnetic field that a material can achieve. Electric permittivity is the realitive ease with which an electric field propagates through a material.
permittivity: opposition against electric field
permeability: easiness magnetic flux to pass when applied by external magnetic field

• If an exercise mentions a (perfect) conductor, you should immediately remember two things: (1) there cannot exist an electric field inside the conductor, and (2) all free charges accumulate at the boundary of the conductor.

PEC MATERIAL: ⇒ Gaussian Surface outside object

• If an exercise mentions a (perfect) dielectric, you should remember that (1) there cannot exist free charges inside the dielectric and that (2) the surface charge density at the border of the dielectric
  

• When an electrical conductor carrying a current is located in a magnetic field perpendicular to that current, a voltage difference will be generated across the conductor in a direction perpendicular to both the current and the magnetic field. This is called the Hall effect.

Boundary conditions for electromagnetic fields
The dot product boundary condition: Normal, while the cross product is the tangential fox exmaple:

Magnetic Flux Density

if normal component of flux density is constant across surface:

Magnetic field strength or field intensity (H) is the amount of magnetising force. Magnetic flux density (B) is the amount of magnetic force induced on the given body due to the magnetising force H.

• Inductance measures the ability of a circuit to generate a magnetic field when an electric current flows through it.

Time-Dependent Fields & Waves

Topic: Maxwell Equations

Lenz’s Law: The direction of the induced current is such as to oppose the very cause producing it

and electric and magnetic current densities

Steady Current

steady current
Current density:

→ accumulation of charges
→ accumulation of continuous point charge density

→ divergence of current density = of change over time of volume density → negative sink, positive source

Charge Conservation

potential:

Electric flux density for dielectrics:

Fraday-Hnery Law
Apere-Maxwell Law

coloumb potential referenced at infinity

N number of turns in coil

surface integral = area of the object

Example, if you wanted to find the volume of a cylindrical shell with inner radius R1​ and outer radius R2​, you might integrate dr from R1​ to R2​, and then integrate dz for the height of the shell.

Extra notes due to EM2

• hollow metal sphere; neutral; small negative charge on surface; → will distribute veenly over surface. if out of insulating material charge stays where it is.
  

• negative positive attraction

• there is equal magnitude where 2 things act on each other

• positive charge; uniform electric field; → constant acceleration

• electric potential energy: energy of opposition

• in uniform field force on particles is same but not zero, same for work done

• force by magnetic field on a charge
  

• Electromagnetic wave equation
  

Kirchoff and Maxwell law relation:

then and
as long as

An electric field ‘points’ from a high to a lower voltage. So, if the potential voltage difference is positive, and the distance is as well, the field is negative, and thus points in the direction of the lower voltage (to the left in our case).