Why Study Electromagnetics?

What is Electromagnetics?

What is a charge q?

Fundamental Laws of Electromagnetics

Steps in Studying Electromagnetics

SI (International System) of Units

Units Derived From the Fundamental Units

Fundamental Electromagnetic Field Quantities

Three Universal Constants

VECTOR REPRESENTATION

3 PRIMARY COORDINATE SYSTEMS:

• RECTANGULAR

• CYLINDRICAL

• SPHERICAL

Choice is based on symmetry of problem

Examples:

Sheets - RECTANGULAR

Wires/Cables - CYLINDRICAL

Spheres - SPHERICAL

Orthogonal Coordinate Systems: (coordinates mutually perpendicular)

Spherical Coordinates

Cylindrical Coordinates

Cartesian Coordinates

P (x,y,z)

P (r, Θ, Φ)

P (r, Θ, z)

x

y

zP(x,y,z)

θ

z

rx y

z

P(r, θ, z)

θ

Φ

r

z

yx

P(r, θ, Φ)

Page 108

Rectangular Coordinates

-Parabolic Cylindrical Coordinates (u,v,z)-Paraboloidal Coordinates (u, v, Φ)-Elliptic Cylindrical Coordinates (u, v, z)-Prolate Spheroidal Coordinates (ξ, η, φ)-Oblate Spheroidal Coordinates (ξ, η, φ)-Bipolar Coordinates (u,v,z)-Toroidal Coordinates (u, v, Φ)-Conical Coordinates (λ, μ, ν)-Confocal Ellipsoidal Coordinate (λ, μ, ν)-Confocal Paraboloidal Coordinate (λ, μ, ν)

Parabolic Cylindrical Coordinates

Paraboloidal Coordinates

Elliptic Cylindrical Coordinates

Prolate Spheroidal Coordinates

Oblate Spheroidal Coordinates

Bipolar Coordinates

Toroidal Coordinates

Conical Coordinates

Confocal Ellipsoidal Coordinate

Confocal Paraboloidal Coordinate

Cartesian CoordinatesP(x,y,z)

Spherical CoordinatesP(r, θ, Φ)

Cylindrical CoordinatesP(r, θ, z)

x

y

zP(x,y,z)

θ

z

rx y

z

P(r, θ, z)

θ

Φ

r

z

yx

P(r, θ, Φ)

VECTOR NOTATION

VECTOR NOTATION:

zzyyxx aAaAaAA ˆˆˆ Rectangular or

Cartesian Coordinate

System

x

z

y

zzyyxx BABABABA

Dot Product

zyx

zyx

zyx

BBB

AAA

aaa

BA

ˆˆˆ

Cross Product

21

222zyx AAAA

Magnitude of vector

(SCALAR)

(VECTOR)

Coulomb’s Law

1 2212

q qF k12 r

Coulomb’s law gives the force (in Newtons) between charges q1 and q2, where r12 is

the distance in meters between the charges, and k=9x109 N·m2/C2.

Coulomb’s law quantifies the magnitude of the electrostatic force.

To make this into a “really good” starting equation I should specify “repulsive for like,” but that makes it too wordy. You’ll just have to remember how to find the direction.

1 2212

q qF k ,12 r

attractive for unlike

Force is a vector quantity. The equation on the previous slide gives the magnitude of the force. If the charges are opposite in sign, the force is attractive; if the charges are the same in sign, the force is repulsive. Also, the constant k is equal to 1/40, where

0=8.85x10-12 C2/N·m2.

Remember, a vector has a magnitude and a direction.

One could write Coulomb’s Law like this…

The equation is valid for point charges. If the charged objects are spherical and the charge is uniformly distributed, r12 is the distance between the centers of the spheres.

If more than one charge is involved, the net force is the vector sum of all forces (superposition). For objects with complex shapes, you must add up all the forces acting on each separate charge (turns into calculus!).

+ -

r12

+

++

-

--

We could have agreed that in the formula for F, the symbols q1 and q2 stand for the

magnitudes of the charges. In that case, the absolute value signs would be unnecessary.

1 2212

q qF k ,12 r

However, in later equations the sign of the charge will be important, so we really need to keep the magnitude part.

On your homework diagrams, show both the magnitudes and signs of q1 and q2.

Your starting equation sheet has this version of the equation:

which gives you the magnitude F12 and tells you that you need to figure out the direction separately.

Gauss’ Law

Electric Flux

We have used electric field lines to visualize electric fields and indicate their strength.

We are now going to count the number of electric field lines passing through a surface, and use this count to determine the electric field. E

The electric flux passing through a surface is the number of electric field lines that pass through it.

Because electric field lines are drawn arbitrarily, we quantify electric flux like this: E=EA, except that…

If the surface is tilted, fewer lines cut the surface.

EA

Later we’ll learn about magnetic flux, which is why I will use the subscript E on electric flux.

E

E

A

The “amount of surface” perpendicular to the electric field is A cos .

A = A cos so E = EA = EA cos .

We define A to be a vector having a magnitude equal to the area of the surface, in a direction normal to the surface.

Because A is perpendicular to the surface, the amount of A parallel to the electric field is A

cos .

Remember the dot product?E E A

If the electric field is not uniform, or the surface is not flat…

divide the surface into infinitesimal surface elements and add the flux through each…

dAE

iE i i

A 0i

lim E A

E E dA

A

If the surface is closed (completely encloses a volume)…

E

…we count* lines going out as positive and lines going in as negative…

E E dA

dA

a surface integral, therefore a double integral

Question: you gave me five different equations for electric flux. Which one do I need to use?

E E dA

E E dA

E E A

E EAcos

E EA

Answer: use the simplest (easiest!) one that works.

Flat surface, E A, E constant over surface. Easy!

Flat surface, E not A, E constant over surface.

Flat surface, E not A, E constant over surface.

Surface not flat, E not uniform. Avoid, if possible.

Closed surface. Most general. Most complex.

If the surface is closed, you may be able to “break it up” into simple segments and still use E=E·A for each segment.