Wedn We dne esda day, y, No Nov vem embe berr 12, 12, 20 2014 14
9:16 9: 16 PM
Engr. Rogelio F. Delgado Jr. Engineering Electromagnetics by WH Hayt / Quantities having both magnitude and direction Quantities having magnitude only. Vector with magnitude of 1
Where ϴ is the smaller angle between vectors A and B
Cross Product: Where ϴ is the smaller angle between vectors A and B.
Rectangular: (x, y, z) Cylindrical: (ρ, φ, z) Spherical: (R, φ, ϴ) Where ρ is the distance from origin to point in xy plane Where φ is the bottom angle between x and y Where ϴ is the top angle between R and Z Tips and Tricks: 1. Convert everything first to rectangular and check if t here are missing components like z! 2. Projection of A on B:
Aa A X B = 0, if A and B are parallel. A o B = 0, if A and B are perpendicular
Wedn We dne esda day, y, No Nov vem embe berr 26, 26, 20 2014 14
9:43 9: 43 PM
cylindrical coordinates are a generalization of twodimensional polar coordinates to three dimensions by superposing a height (z) axis.
Where: = radius of the base (radial coordinate) = angle with respect to the x-axis (azimuthal coordinate) similar to rectangular rectangular coordinate's z - axis (height coordinate) Sv.
Conversion from Cylindrical to Rectangular: x = ρcos φ Y = ρsin φ Z=Z Dot Product Table: aρ
aφ
ax
cos φ
- sin sin φ
ay
sin φ
cos φ
Wedn We dne esda day, y, No Nov vem embe berr 26, 26, 20 2014 14
10:2 10 :25 5 PM PM
Spherical coordinates, also called spherical polar coordinates, are a system of curvilinear coordinates that are natural for describing positions on a sphere or spheroid. (r,φ,ϴ) where: r = radius from origin to the point (sphere's radius) φ = angle should be dependent on the signs of the coordinates ϴ = polar angle, less than 180°
Conversion from Rectangular to Spherical
Conversion from Spherical to Rectangular X = R sinϴ cosφ Y = R sin ϴ cosφ Z = R cosϴ
Conversion from Rectangular to Spherical unit vectors: aR
aφ
aϴ
ax
sinϴ cosφ
-sin φ
cosϴ cosφ
ay
sinϴ sinφ Cos φ
az
cosϴ
0
cosϴ sinφ -sinϴ
Schaum's Outline Sun unda day, y, De Dece cem mbe berr 07, 07, 20 2014 14
7:48 7: 48 PM
1. Use calculator to solve for polar to rectangular. 2. Distance should have units. 3. P1P2 = P2 - P1
Mon onda day, y, De Dece cemb mber er 08 08,, 20 2014 14
1:53 1: 53 AM
1.2. Given three points, A(4,3,2),B(−2,0,5), andC(7,−2,1):
Q
Midterm Monday, January 19 19, 201 2015 5
2:00 PM
There is a force between two charges cha rges which is directly proportional to the charge magnitudes and inversely proportional to the square of the separation distance.
Where = force in newtons = distance in meters permittivity of the medium C2/Nm or F/m
*unit of charge is in Coulombs
F = qE
*if charge is , the electric field is . *if charge is , the electric field is . *the intensity of the field gets weaker the farther the small charge *units is N/C or V/m
is.
Charge Density:
field of a single point charge
field with a charge that is distributed with uniform density (pe) along an infinite, straight line, which will be chosen as the z-axis.
field with a charge that is distributed with uniform density (p) over an infinite plane.
*get the absolute value, +E is for above the z-axis and -E is for below the z-axis.
Thu hurrsda day, y, Ja Jan nua uary ry 22 22,, 201 2015 5
pv is c/ m3
2:46 2: 46 AM
Wedn We dne esda day, y, Fe Febr brua uary ry 04 04,, 201 2015 5
3:18 3: 18 PM
2.1 Coulomb's Law
where F = force (Newtons) D = distance (meters) Q = charge (C) εo = permittivity of free space (F/m or C2 / Nm) ε0 = 8.854 x 10-12 F/m = 10-9/36π
2.3 Charge Distributions Volume Charge 2.4 Standard Charge Configurations Electric Fields of Point Charge:
Electric Fields of Infinite Straight Line Line Charge: where = uniform density (C/m)
ρl a = perpendicular distance from the point
Sample Problem 1: A uniform line charge, infinite in extend, with pl = 20nC/m, lies along the z-axis. Find E at (6,8,3)m Answer: 36ar V/m Electric Fields in Infinite Plane Charge:
Sample Problem 2: Charge is distributed uniformly over the plane z = 10cm with a density (1/3 π) nC/m2. Find E. Answer: Above the sheet (z > 10cm), 6 V/m. Below the sheet (z < 10cm), -6 V/m
charge and is coincident with the streamlines Electric Flux - streams away symetrically from a point charge whenever electric field is present.
I Dd
@A
Assignment Wedn We dne esda day, y, Ma Marc rch h 11 11,, 20 2015 15
10::42 PM 10
greek word of amber (1600)
: where
= coulombs
= meters
0
Both
and
follows the
2.4 Field of a Line Charge: Infinite Line Charge: where
is the linear charge density
where ρ is the shortest distance from the line to the point Finite Line Charge:
2.5 Field of a Sheet of Chargep11AN4pmpp Infinite Surface Charge:
2.6 Streamlines and Sketches of Fields Streamlines / Flux lines = pictorial representation of the concept of Electric Field Kapag everywhere, may three dimensions.
Introduction:
